\documentclass[10pt,twoside,fleqn]{article} \usepackage[margin=2cm]{geometry} \usepackage{graphicx} \usepackage{multirow} \usepackage [displaymath, mathlines]{lineno} \usepackage{titlesec} \usepackage{color} \usepackage{ifpdf} \ifpdf \usepackage[hyperindex]{hyperref}%,pagebackref \else \expandafter\ifx\csname dvipdfm\endcsname\relax \usepackage[hypertex,hyperindex]{hyperref} \else \usepackage[dvipdfm,hyperindex]{hyperref} \fi \fi \hypersetup{colorlinks, linkcolor=blue, urlcolor=blue, citecolor=blue} \titlelabel{\thetitle.\quad} %\linenumbers \usepackage[font=small,format=plain,labelfont=bf,up]{caption} \usepackage{amsmath,amssymb,amsthm} \setlength{\mathindent}{0cm} \renewcommand\linenumberfont{\normalfont\bfseries\small} \renewcommand{\baselinestretch}{1} \allowdisplaybreaks[4] \numberwithin{equation}{section} \theoremstyle{plain} \newtheorem{thm}{Theorem}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \theoremstyle{remark} \newtheorem{rem}{Remark}[section] \DeclareMathOperator{\td}{d} \newcommand{\cmdeg}[1]{\sideset{}{_\mathrm{cm}^{#1}}\deg} \setcounter{page}{1} \pagestyle{myheadings} \thispagestyle{empty} \markboth{\footnotesize \emph{\emph{Global Journal of Mathematical Analysis}}}{\footnotesize \emph{\emph{Global Journal of Mathematical Analysis}}} \date{} \begin{document} {\renewcommand{\arraystretch}{0.65} \begin{table}[ht] \begin{tabular}{ll} \multirow{8}{*}{\includegraphics[width=1.4cm]{logo}}&\\&{\scriptsize\emph{\textbf{Global Journal of Mathematical Analysis}, 2 (x) (2014) xxx-xxx}}\\ &{\scriptsize\emph{\copyright Science Publishing Corporation}}\\ &{\scriptsize\emph{www.sciencepubco.com/index.php/GJMA}}\\ &{\scriptsize\emph{doi: }}\\ &{\scriptsize\emph {Research paper}} \end{tabular} \end{table}} \renewcommand{\arraystretch}{1} \centerline{} \centerline{} \centerline{} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} \huge{\bf Absolute monotonicity of a function involving the exponential function} \end{center} \centerline{} \centerline{\bf {Feng Qi}} \centerline{} \begin{center}\small\em Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China\\ E-mail: \href{mailto: F. Qi }{qifeng618@gmail.com}, \href{mailto: F. Qi }{qifeng618@hotmail.com}, \href{mailto: F. Qi }{qifeng618@qq.com}\\ URL: \url{http://qifeng618.wordpress.com} \end{center} \centerline{} \centerline{} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent \hspace{-3 pt}{\scriptsize \textbf{ Copyright \copyright 2014 Author. This is an open access article distributed under the \href{http://creativecommons.org/licenses/by/3.0/}{% Creative Commons Attribution License} Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \smallskip \noindent \hrulefill %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent \textbf{Abstract}\\ \centerline{} In the paper, the author verifies the absolute monotonicity of a function involving the exponential function.\\ \centerline{} \noindent {\footnotesize \emph{\textbf{Keywords}}: \emph{absolute monotonicity; absolutely monotonic function; completely monotonic function; completely monotonic degree; exponential function}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent\hrulefill \section{Preliminaries and main results} Recall from~\cite{mpf-1993, Schilling-Song-Vondracek-2nd, widder} that a function $f$ is said to be completely monotonic on an interval $I$ if it has derivatives of all orders on $I$ such that \begin{equation} (-1)^{k}f^{(k)}(x)\ge0 \end{equation} for $x\in I$ and $k\ge0$. Recall also from~\cite{mpf-1993, Schilling-Song-Vondracek-2nd, widder} that a function $f$ is said to be absolutely monotonic on an interval $I$ if it has derivatives of all orders and \begin{equation} f^{(k-1)}(t)\ge0 \end{equation} for $t\in I$ and $k\in\mathbb{N}$, where $\mathbb{N}$ denotes the set of all positive integers. \par It is easy to see that a function $f(x)$ is completely monotonic in $(a,b)$ if and only if $f(-x)$ is absolutely monotonic in $(-b,-a)$. See~\cite[p.~145, Definition~2c]{widder}. \par Theorem~12a in~\cite[p.~160]{widder} reads that a necessary and sufficient condition that $f(x)$ should be completely monotonic in $0\le x<\infty$ is that \begin{equation}\label{berstein-1} f(x)=\int_0^{\infty}{e}^{-xt}\td \alpha(t), \end{equation} where $\alpha(t)$ is bounded and non-decreasing and the integral converges for $0\le x <\infty$. Theorem~12c in~\cite[p.~162]{widder} states that a necessary and sufficient condition that $f(x)$ should be absolutely monotonic in $-\infty0$ by \begin{equation*}%\label{gamma-dfn} \Gamma(z)=\int^\infty_0t^{z-1} e^{-t}\td t. \end{equation*} The logarithmic derivative of $\Gamma(z)$, denoted by $\psi(z)=\frac{\Gamma'(z)}{\Gamma(z)}$, is called the psi or di-gamma function. As a whole, the derivatives $\psi^{(i)}(z)$ for $i\ge0$ are called polygamma functions. \par Recently, when computing the completely monotonic degree of the function \begin{equation}\label{psi-der-square-psi-2der} \Psi(x)=[\psi'(x)]^2+\psi''(x) \end{equation} with respect to $x\in(0,\infty)$, we encounter the function \begin{equation}\label{R(t)-dfn-eq} \begin{split} R(t)&=e^{9 t} \bigl(t^5-12 t^4+70 t^3-160 t^2+192 t-128\bigr)-e^{8 t} \bigl(16 t^7-220 t^6+1219 t^5-3220 t^4+4490 t^3\\ &\quad-3248 t^2+1152 t-768\bigr)-4 e^{7 t} \bigl(37 t^7-423 t^6+1397 t^5-1409 t^4-1020 t^3+2632 t^2-732 t\\ &\quad+456\bigr)-4 e^{6 t} \bigl(225 t^7-1281 t^6+1213 t^5+3127 t^4-4372 t^3-2648 t^2+1020 t-504\bigr)\\ &\quad-2 e^{5 t} \bigl(908 t^7-1514 t^6-6493 t^5+8710 t^4+12754 t^3-1216 t^2-1656 t+336\bigr)-2 e^{4 t} \bigl(908 t^7\\ &\quad+1710 t^6-5489 t^5-12370 t^4+594 t^3+4880 t^2+696 t+336\bigr)-4 e^{3 t} \bigl(225 t^7+1263 t^6\\ &\quad+1771 t^5-887 t^4-3208 t^3-728 t^2+12 t-168\bigr)-4 e^{2 t} \bigl(37 t^7+353 t^6+1099 t^5+1337 t^4\\ &\quad+272 t^3-632 t^2-108 t+24\bigr)-e^t \bigl(16 t^7+180 t^6+827 t^5+1864 t^4+2226 t^3+1312 t^2+240 t\\ &\quad+96\bigr)+t^5+8 t^4+30 t^3+48 t^2+48 t+32 \end{split} \end{equation} on $(0,\infty)$. For more information on the notion ``completely monotonic degree'', please refer to~\cite{psi-proper-fraction-degree-two.tex, Geom-Mean-Deg-One-Guo.tex, degree-4-tri-tetra-gamma.tex, MIA-3997.tex, simp-exp-degree-ext-gjma.tex} and closely related references therein. For more information on results and properties of the function~\eqref{psi-der-square-psi-2der}, please refer to~\cite{UPB-1635.tex, BAustMS-5984-RV.tex, SCM-2012-0142.tex}. \par The main aim of this paper is to verify that the absolute monotonicity of the function $R(t)$ on $(0,\infty)$. \begin{thm}\label{prop-exp-am} The function $R(t)$ defined by~\eqref{R(t)-dfn-eq} is absolutely monotonic on $(0,\infty)$. In other words, the function $R(-t)$ defined by~\eqref{R(t)-dfn-eq} is completely monotonic on $(-\infty,0)$. \end{thm} \section{Proof of Theorem~\ref{prop-exp-am}}\label{appendix} Now we are in a position to spend a large amount of texts to detail the proof of the absolute monotonicity of the function $R(t)$ on $(0,\infty)$. \par Consecutive differentiation and straightforward simplification give \begin{align*} R'(t)&=5 t^4+32 t^3+90 t^2+96 t+48-e^t \bigl(16 t^7+292 t^6+1907 t^5+5999 t^4+9682 t^3+7990 t^2+2864 t+336\bigr)\\ &\quad-4 e^{2 t} \bigl(74 t^7+965 t^6+4316 t^5+8169 t^4+5892 t^3-448 t^2-1480 t-60\bigr)-4 e^{3 t} \bigl(675 t^7+5364 t^6\\ &\quad+12891 t^5+6194 t^4-13172 t^3-11808 t^2-1420 t-492\bigr)-2 e^{4 t} \bigl(3632 t^7+13196 t^6-11696 t^5\\ &\quad-76925 t^4-47104 t^3+21302 t^2+12544 t+2040\bigr)-2 e^{5 t} \bigl(4540 t^7-1214 t^6-41549 t^5+11085 t^4\\ &\quad+98610 t^3+32182 t^2-10712 t+24\bigr)-4 e^{6 t} \bigl(1350 t^7-6111 t^6-408 t^5+24827 t^4-13724 t^3-29004 t^2\\ &\quad+824 t-2004\bigr)-4 e^{7 t} \bigl(259 t^7-2702 t^6+7241 t^5-2878 t^4-12776 t^3+15364 t^2+140 t+2460\bigr)\\ &\quad-e^{8 t} \bigl(128 t^7-1648 t^6+8432 t^5-19665 t^4+23040 t^3-12514 t^2+2720 t-4992\bigr)+e^{9 t} \bigl(9 t^5-103 t^4\\ &\quad+582 t^3-1230 t^2+1408 t-960\bigr),\\ R''(t)&=4 \bigl(5 t^3+24 t^2+45 t+24\bigr)-e^t \bigl(16 t^7+404 t^6+3659 t^5+15534 t^4 +33678 t^3+37036 t^2+18844 t\\ &\quad+3200\bigr)-8 e^{2 t} \bigl(74 t^7+1224 t^6 +7211 t^5+18959 t^4+22230 t^3+8390 t^2-1928 t-800\bigr) -4 e^{3 t} \bigl(2025 t^7\\ &\quad+20817 t^6+70857 t^5+83037 t^4-14740 t^3-74940 t^2 -27876 t-2896\bigr)-8 e^{4 t} \bigl(3632 t^7+19552 t^6\\ &\quad+8098 t^5-91545 t^4 -124029 t^3-14026 t^2+23195 t+5176\bigr)-2 e^{5 t} \bigl(22700 t^7+25710 t^6 -215029 t^5\\ &\quad-152320 t^4+537390 t^3+456740 t^2+10804 t-10592\bigr) -8 e^{6 t} \bigl(4050 t^7-13608 t^6-19557 t^5+73461 t^4\\ &\quad+8482 t^3-107598 t^2 -26532 t-5600\bigr)-4 e^{7 t} \bigl(1813 t^7-17101 t^6+34475 t^5+16059 t^4 -100944 t^3\\ &\quad+69220 t^2+31708 t+17360\bigr)-4 e^{8 t} \bigl(256 t^7-3072 t^6 +14392 t^5-28790 t^4+26415 t^3-7748 t^2-817 t\\ &\quad-9304\bigr) +e^{9 t} \bigl(81 t^5-882 t^4+4826 t^3-9324 t^2+10212 t-7232\bigr),\\ R^{(3)}(t)&=12 \bigl(5 t^2+16 t+15\bigr)-e^t \bigl(16 t^7+516 t^6+6083 t^5+33829 t^4+95814 t^3 +138070 t^2+92916 t+22044\bigr)\\ &\quad-8 e^{2 t} \bigl(148 t^7+2966 t^6+21766 t^5 +73973 t^4+120296 t^3+83470 t^2+12924 t-3528\bigr)-12 e^{3 t} \bigl(2025 t^7\\ &\quad+25542 t^6+112491 t^5+201132 t^4+95976 t^3-89680 t^2-77836 t -12188\bigr)-8 e^{4 t} \bigl(14528 t^7\\ &\quad+103632 t^6+149704 t^5-325690 t^4-862296 t^3 -428191 t^2+64728 t+43899\bigr)-2 e^{5 t} \bigl(113500 t^7\\ &\quad+287450 t^6-920885 t^5 -1836745 t^4+2077670 t^3+3895870 t^2+967500 t-42156\bigr) -24 e^{6 t} \bigl(8100 t^7\\ &\quad-17766 t^6-66330 t^5+114327 t^4+114912 t^3-206714 t^2 -124796 t-20044\bigr)-4 e^{7 t} \bigl(12691 t^7\\ &\quad-107016 t^6+138719 t^5+284788 t^4 -642372 t^3+181708 t^2+360396 t+153228\bigr)-4 e^{8 t} \bigl(2048 t^7\\ &\quad-22784 t^6 +96704 t^5-158360 t^4+96160 t^3+17261 t^2-22032 t-75249\bigr) +e^{9 t} \bigl(729 t^5-7533 t^4\\ &\quad+39906 t^3-69438 t^2+73260 t-54876\bigr),\\ R^{(4)}(t)&=24 (5 t+8)-e^t \bigl(16 t^7+628 t^6+9179 t^5+64244 t^4+231130 t^3+425512 t^2 +369056 t+114960\bigr)\\ &\quad-32 e^{2 t} \bigl(74 t^7+1742 t^6+15332 t^5+64194 t^4 +134121 t^3+131957 t^2+48197 t+1467\bigr)-12 e^{3 t} \bigl(6075 t^7\\ &\quad+90801 t^6 +490725 t^5+1165851 t^4+1092456 t^3+18888 t^2-412868 t-114400\bigr) -16 e^{4 t} \bigl(29056 t^7\\ &\quad+258112 t^6+610304 t^5-277120 t^4-2375972 t^3 -2149826 t^2-298735 t+120162\bigr)-10 e^{5 t} \bigl(113500 t^7\\ &\quad+446350 t^6 -575945 t^5-2757630 t^4+608274 t^3+5142472 t^2+2525848 t+151344\bigr) -96 e^{6 t} \bigl(12150 t^7\\ &\quad-12474 t^6-126144 t^5+88578 t^4+286695 t^3-223887 t^2 -290551 t-61265\bigr)-4 e^{7 t} \bigl(88837 t^7\\ &\quad-660275 t^6+328937 t^5+2687111 t^4 -3357452 t^3-655160 t^2+2886188 t+1432992\bigr)-8 e^{8 t} \bigl(8192 t^7\\ &\quad-83968 t^6 +318464 t^5-391680 t^4+67920 t^3+213284 t^2-70867 t-312012\bigr) +e^{9 t} \bigl(6561 t^5-64152 t^4\\ &\quad+329022 t^3-505224 t^2+520464 t-420624\bigr),\\ R^{(5)}(t)&=120-e^t \bigl(16 t^7+740 t^6+12947 t^5+110139 t^4+488106 t^3+1118902 t^2 +1220080 t+484016\bigr)\\ &\quad-32 e^{2 t} \bigl(148 t^7+4002 t^6+41116 t^5+205048 t^4 +525018 t^3+666277 t^2+360308 t+51131\bigr)\\ &\quad-12 e^{3 t} \bigl(18225 t^7+314928 t^6 +2016981 t^5+5951178 t^4+7940772 t^3+3334032 t^2-1200828 t\\ &\quad -756068\bigr)-16 e^{4 t} \bigl(116224 t^7+1235840 t^6+3989888 t^5+1943040 t^4 -10612368 t^3-15727220 t^2\\ &\quad-5494592 t+181913\bigr)-10 e^{5 t} \bigl(567500 t^7 +3026250 t^6-201625 t^5-16667875 t^4-7989150 t^3\\ &\quad+27537182 t^2 +22914184 t+3282568\bigr)-96 e^{6 t} \bigl(72900 t^7+10206 t^6-831708 t^5 -99252 t^4\\ &\quad+2074482 t^3-483237 t^2-2191080 t-658141\bigr) -4 e^{7 t} \bigl(621859 t^7-4000066 t^6-1659091 t^5\\ &\quad+20454462 t^4-12753720 t^3 -14658476 t^2+18892996 t+12917132\bigr)-8 e^{8 t} \bigl(65536 t^7-614400 t^6\\ &\quad +2043904 t^5-1541120 t^4-1023360 t^3+1910032 t^2-140368 t -2566963\bigr)+3 e^{9 t} \bigl(19683 t^5\\ &\quad-181521 t^4+901530 t^3-1186650 t^2 +1224576 t-1088384\bigr), \\ R^{(6)}(t)&=e^t \bigl[81 e^{8 t} \bigl(6561 t^5-56862 t^4+273618 t^3-295380 t^2+320292 t -317440\bigr)-64 e^{7 t} \bigl(65536 t^7\\ &\quad-557056 t^6+1583104 t^5-263680 t^4 -1793920 t^3+1526272 t^2+337140 t-2584509\bigr)\\ &\quad-4 e^{6 t} \bigl(4353013 t^7 -23647449 t^6-35614033 t^5+134885779 t^4-7458192 t^3 -140870492 t^2\\ &\quad+102934020 t+109312920\bigr)-1728 e^{5 t} \bigl(24300 t^7 +31752 t^6-273834 t^5-264114 t^4+669438 t^3\\ &\quad+184668 t^2-784053 t -341107\bigr)-10 e^{4 t} \bigl(2837500 t^7+19103750 t^6+17149375 t^5 -84347500 t^4\\ &\quad-106617250 t^3+113718460 t^2+169645284 t +39327024\bigr)-64 e^{3 t} \bigl(116224 t^7+1439232 t^6\\ &\quad+5843648 t^5+6930400 t^4 -8669328 t^3-23686496 t^2-13358202 t-1191735\bigr)-108 e^{2 t} \bigl(6075 t^7\\ &\quad +119151 t^6+882279 t^5+3104271 t^4+5291892 t^3+3758268 t^2 +340620 t-385448\bigr)-64 e^t \bigl(148 t^7\\ &\quad+4520 t^6+53122 t^5+307838 t^4 +935114 t^3+1453804 t^2+1026585 t+231285\bigr)-16 t^7-852 t^6 \\ &\quad -17387 t^5-174874 t^4-928662 t^3-2583220 t^2-3457884 t-1704096\bigr]\\ &\triangleq e^tR_1(t),\\ R_1'(t)&=81 e^{8 t} \bigl(52488 t^5-422091 t^4+1961496 t^3-1542186 t^2+1971576 t-2219228\bigr)-64 e^{7 t} \bigl(458752 t^7\\ &\quad-3440640 t^6+7739392 t^5+6069760 t^4-13612160 t^3+5302144 t^2+5412524 t-17754423\bigr)\\ &\quad-4 e^{6 t} \bigl(26118078 t^7-111413603 t^6-355568892 t^5+631244509 t^4+494793964 t^3-867597528 t^2\\ &\quad+335863136 t+758811540\bigr)-1728 e^{5 t} \bigl(121500 t^7+328860 t^6-1178658 t^5-2689740 t^4+2290734 t^3\\ &\quad+2931654 t^2-3550929 t-2489588\bigr)-10 e^{4 t} \bigl(11350000 t^7+96277500 t^6+183220000 t^5\\ &\quad-251643125 t^4-763859000 t^3+135022090 t^2+906018056 t+326953380\bigr)-64 e^{3 t} \bigl(348672 t^7\\ &\quad+5131264 t^6+26166336 t^5+50009440 t^4+1713616 t^3-97067472 t^2-87447598 t-16933407\bigr)\\ &\quad-108 e^{2 t} \bigl(12150 t^7+280827 t^6+2479464 t^5+10619937 t^4+23000868 t^3+23392212 t^2+8197776 t\\ &\quad-430276\bigr)-64 e^t \bigl(148 t^7+5556 t^6+80242 t^5+573448 t^4+2166466 t^3+4259146 t^2+3934193 t\\ &\quad+1257870\bigr)-112 t^6-5112 t^5-86935 t^4-699496 t^3-2785986 t^2-5166440 t-3457884,\\ R_1''(t)&=-4 \bigl[168 t^5+6390 t^4+86935 t^3+524622 t^2+1392993 t+1291610+16 e^t \bigl(148 t^7+6592 t^6\\ &\quad+113578 t^5+974658 t^4+4460258 t^3+10758544 t^2+12452485 t+5192063\bigr)+54 e^{2 t} \bigl(12150 t^7\\ &\quad+323352 t^6+3321945 t^5+16818597 t^4+44240742 t^3+57893514 t^2+31589988 t+3668612\bigr)\\ &\quad+16 e^{3 t} \bigl(1046016 t^7+17834496 t^6+109286592 t^5+280860000 t^4+205178608 t^3-286061568 t^2\\ &\quad-456477738 t-138247819\bigr)+10 e^{4 t} \bigl(11350000 t^7+116140000 t^6+327636250 t^5-22618125 t^4\\ &\quad-1015502125 t^3-437872160 t^2+973529101 t+553457894\bigr)+432 e^{5 t} \bigl(607500 t^7+2494800 t^6\\ &\quad-3920130 t^5-19341990 t^4+694710 t^3+21530472 t^2-11891337 t-15998869\bigr)+2 e^{6 t} \bigl(78354234 t^7\\ &\quad-242827536 t^6-1400947485 t^5+1004811297 t^4+2746870910 t^3-1860601638 t^2+139991880 t\\ &\quad+2444366188\bigr)+16 e^{7 t} \bigl(3211264 t^7-20873216 t^6+33531904 t^5+81185280 t^4-71006080 t^3\\ &\quad-3721472 t^2+48491956 t-118868437\bigr)-81 e^{8 t} \bigl(104976 t^5-778572 t^4+3500901 t^3-1613250 t^2\\ &\quad+3172059 t-3945562\bigr)\bigr],\\ R_1^{(3)}(t)&=-4 \bigl[3 \bigl(280 t^4+8520 t^3+86935 t^2+349748 t+464331\bigr)+16 e^t \bigl(148 t^7+7628 t^6+153130 t^5\\ &\quad+1542548 t^4+8358890 t^3+24139318 t^2+33969573 t+17644548\bigr)+54 e^{2 t} \bigl(24300 t^7+731754 t^6\\ &\quad+8584002 t^5+50246919 t^4+155755872 t^3+248509254 t^2+178967004 t+38927212\bigr)\\ &\quad+48 e^{3 t} \bigl(1046016 t^7+20275200 t^6+144955584 t^5+463004320 t^4+579658608 t^3-80882960 t^2\\ &\quad-647185450 t-290407065\bigr)+10 e^{4 t} \bigl(45400000 t^7+544010000 t^6+2007385000 t^5+1547708750 t^4\\ &\quad-4152481000 t^3-4797995015 t^2+3018372084 t+3187360677\bigr)+432 e^{5 t} \bigl(3037500 t^7+16726500 t^6\\ &\quad-4631850 t^5-116310600 t^4-73894410 t^3+109736490 t^2-16395741 t-91885682\bigr)\\ &\quad+6 e^{6 t} \bigl(156708468 t^7-302828526 t^6-3287550042 t^5-325289881 t^4+6833490216 t^3-974332366 t^2\\ &\quad-960417332 t+4935396336\bigr)+16 e^{7 t} \bigl(22478848 t^7-123633664 t^6+109484032 t^5+735956480 t^4\\ &\quad-172301440 t^3-239068544 t^2+332000748 t-783587103\bigr)-81 e^{8 t} \bigl(839808 t^5-5703696 t^4\\ &\quad+24892920 t^3-2403297 t^2+22149972 t-28392437\bigr)\bigr],\\ R_1^{(4)}(t)&=-8 \bigl[3 \bigl(560 t^3+12780 t^2+86935 t+174874\bigr)+8 e^t \bigl(148 t^7+8664 t^6+198898 t^5\\ &\quad+2308198 t^4+14529082 t^3+49215988 t^2+82248209 t+51614121\bigr)+108 e^{2 t} \bigl(12150 t^7+408402 t^6\\ &\quad+5389632 t^5+35853462 t^4+128124855 t^3+241071531 t^2+213738129 t+64205357\bigr)\\ &\quad+24 e^{3 t} \bigl(3138048 t^7+68147712 t^6+556517952 t^5+2113790880 t^4+3590993104 t^3+1496326944 t^2\\ &\quad-2103322270 t-1518406645\bigr)+10 e^{4 t} \bigl(90800000 t^7+1246920000 t^6+5646800000 t^5+8113880000 t^4\\ &\quad-5209544500 t^3-15824711530 t^2+1238749153 t+7883907396\bigr)+216 e^{5 t} \bigl(15187500 t^7\\ &\quad+104895000 t^6+77199750 t^5-604712250 t^4-834714450 t^3+326999220 t^2+137494275 t\\ &\quad-475824151\bigr)+12 e^{6 t} \bigl(235062702 t^7-180002970 t^6-5385567852 t^5-4597372374 t^4+9924945443 t^3\\ &\quad+3663619113 t^2-1927792181 t+7162990171\bigr)+8 e^{7 t} \bigl(157351936 t^7-708083712 t^6\\ &\quad+24586240 t^5+5699115520 t^4+1737715840 t^3-2190384128 t^2+1845868148 t-5153108973\bigr)\\ &\quad-81 e^{8 t} \bigl(3359232 t^5-20715264 t^4+88164288 t^3+27726192 t^2+86196591 t-102494762\bigr)\bigr],\\ R_1^{(5)}(t)&=8 \bigl[81 e^{8 t} \bigl(26873856 t^5-148925952 t^4+622453248 t^3+486302400 t^2+745025112 t\\ &\quad-733761505\bigr)-8 e^{7 t} \bigl(1101463552 t^7-3855122432 t^6-4076398592 t^5+40016739840 t^4\\ &\quad+34960472960 t^3-10119541376 t^2+8540308780 t-34225894663\bigr)-12 e^{6 t} \bigl(1410376212 t^7\\ &\quad+565421094 t^6-33393424932 t^5-54512073504 t^4+41160183162 t^3+51756551007 t^2-4239514860 t\\ &\quad+41050148845\bigr)-1080 e^{5 t} \bigl(15187500 t^7+126157500 t^6+203073750 t^5-527512500 t^4\\ &\quad-1318484250 t^3-173829450 t^2+268293963 t-448325296\bigr)-10 e^{4 t} \bigl(363200000 t^7+5623280000 t^6\\ &\quad+30068720000 t^5+60689520000 t^4+11617342000 t^3-78927479620 t^2-26694426448 t+32774378737\bigr)\\ &\quad-24 e^{3 t} \bigl(9414144 t^7+226409472 t^6+2078440128 t^5+9123962400 t^4+19228142832 t^3+15261960144 t^2\\ &\quad-3317312922 t-6658542205\bigr)-108 e^{2 t} \bigl(24300 t^7+901854 t^6+13229676 t^5+98655084 t^4\\ &\quad+399663558 t^3+866517627 t^2+909619320 t+342148843\bigr)-8 e^t \bigl(148 t^7+9700 t^6+250882 t^5\\ &\quad+3302688 t^4+23761874 t^3+92803234 t^2+180680185 t+133862330\bigr)-15 \bigl(336 t^2+5112 t+17387\bigr)\bigr],\\ R_1^{(6)}(t)&=64 \bigl[162 e^{8 t} \bigl(13436928 t^5-66064896 t^4+273995136 t^3+359861184 t^2+433300356 t\\ &\quad-320316683\bigr)-e^{7 t} \bigl(7710244864 t^7-19275612160 t^6-51665524736 t^5+259735185920 t^4\\ &\quad+404790270080 t^3+34044629248 t^2+39543078708 t-231040953861\bigr)-9 e^{6 t} \bigl(1410376212 t^7\\ &\quad+2210860008 t^6-32828003838 t^5-82339927614 t^4+4818800826 t^3+72336642588 t^2+13012668809 t\\ &\quad+40343563035\bigr)-135 e^{5 t} \bigl(75937500 t^7+737100000 t^6+1772313750 t^5-1622193750 t^4\\ &\quad-8702471250 t^3-4824600000 t^2+993810915 t-1973332517\bigr)-5 e^{4 t} \bigl(363200000 t^7+6258880000 t^6\\ &\quad+38503640000 t^5+98275420000 t^4+72306862000 t^3-70214473120 t^2-66158166258 t+26100772125\bigr)\\ &\quad-9 e^{3 t} \bigl(9414144 t^7+248375808 t^6+2531259072 t^5+12588029280 t^4+31393426032 t^3+34490102976 t^2\\ &\quad+6857327174 t-7764313179\bigr)-27 e^{2 t} \bigl(24300 t^7+986904 t^6+15935238 t^5+131729274 t^4\\ &\quad+596973726 t^3+1466012964 t^2+1776136947 t+796958503\bigr)-e^t \bigl(148 t^7+10736 t^6+309082 t^5\\ &\quad+4557098 t^4+36972626 t^3+164088856 t^2+366286653 t+314542515\bigr)-45 (28 t+213)\bigr],\\ R_1^{(7)}(t)&=64 \bigl[648 e^{8 t} \bigl(26873856 t^5-115333632 t^4+481925376 t^3+925218720 t^2+1046531304 t-532308277\bigr)\\ &\quad-7 e^{7 t} \bigl(7710244864 t^7-11565367296 t^6-68187478016 t^5+222831239680 t^4+553210376320 t^3\\ &\quad+207526173568 t^2+49270115636 t-225391942617\bigr)-9 e^{6 t} \bigl(8462257272 t^7+23137793532 t^6\\ &\quad-183702862980 t^5-658179584874 t^4-300446905500 t^3+448476258006 t^2+222749298030 t\\ &\quad+255074047019\bigr)-675 e^{5 t} \bigl(75937500 t^7+843412500 t^6+2656833750 t^5+150120000 t^4\\ &\quad-10000226250 t^3-10046082750 t^2-936029085 t-1774570334\bigr)-10 e^{4 t} \bigl(726400000 t^7\\ &\quad+13788960000 t^6+95783920000 t^5+292809940000 t^4+341164564000 t^3-31968653240 t^2\\ &\quad-202530805636 t+19122461121\bigr)-9 e^{3 t} \bigl(28242432 t^7+811026432 t^6+9084032064 t^5\\ &\quad+50420383200 t^4+144532395216 t^3+197650587024 t^2+89552187474 t-16435612363\bigr)\\ &\quad-27 e^{2 t} \bigl(48600 t^7+2143908 t^6+37791900 t^5+343134738 t^4+1720864548 t^3+4722947106 t^2\\ &\quad+6484299822 t+3370053953\bigr)-e^t \bigl(148 t^7+11772 t^6+373498 t^5+6102508 t^4+55201018 t^3\\ &\quad+275006734 t^2+694464365 t+680829168\bigr)-1260\bigr],\\ R_1^{(8)}(t)&=64e^t\bigl[41472 e^{7 t} \bigl(3359232 t^5-12317184 t^4+53032320 t^3+138242592 t^2+159729498 t-50186483\bigr)\\ &\quad-49 e^{6 t} \bigl(7710244864 t^7-3855122432 t^6-78100649984 t^5+174125898240 t^4+680542513280 t^3\\ &\quad+444616334848 t^2+108563308084 t-218353354669\bigr)-432 e^{5 t} \bigl(1057782159 t^7+4126303377 t^6\\ &\quad-20070633681 t^5-101408163003 t^4-92404161927 t^3+37281600657 t^2+46530173004 t\\ &\quad+36524866253\bigr)-3375 e^{4 t} \bigl(75937500 t^7+949725000 t^6+3668928750 t^5+2806953750 t^4\\ &\quad-9880130250 t^3-16046218500 t^2-4954462185 t-1961776151\bigr)-1280 e^{3 t} \bigl(22700000 t^7\\ &\quad+470630000 t^6+3639605000 t^5+12891870000 t^4+19811703250 t^3+6997024055 t^2-6828597883 t\\ &\quad-984695009\bigr)-27 e^{2 t} \bigl(28242432 t^7+876925440 t^6+10706084928 t^5+65560436640 t^4\\ &\quad+211759572816 t^3+342182982240 t^2+221319245490 t+13415116795\bigr)-432 e^t \bigl(6075 t^7+289251 t^6\\ &\quad+5527953 t^5+54701811 t^4+300891753 t^3+913030491 t^2+1400905866 t+826525483\bigr)-148 t^7\\ &\quad-12808 t^6-444130 t^5-7969998 t^4-79611050 t^3-440609788 t^2-1244477833 t-1375293533\bigr]\\ &\triangleq 64e^tR_2(t),\\ R_2'(t)&=41472 e^{7 t} \bigl(23514624 t^5-69424128 t^4+321957504 t^3+1126795104 t^2+1394591670 t-191575883\bigr)\\ &\quad-98 e^{6 t} \bigl(23130734592 t^7+15420489728 t^6-245867317248 t^5+327126069760 t^4+2389879336320 t^3\\ &\quad+2354662774464 t^2+770306259100 t-600778409965\bigr)-432 e^{5 t} \bigl(5288910795 t^7+28035991998 t^6\\ &\quad-75595348143 t^5-607393983420 t^4-867653461647 t^3-90804482496 t^2+307214066334 t\\ &\quad+229154504269\bigr)-3375 e^{4 t} \bigl(303750000 t^7+4330462500 t^6+20374065000 t^5+29572458750 t^4\\ &\quad-28292706000 t^3-93825264750 t^2-51910285740 t-12801566789\bigr)-1280 e^{3 t} \bigl(68100000 t^7\\ &\quad+1570790000 t^6+13742595000 t^5+56873635000 t^4+111002589750 t^3+80426181915 t^2\\ &\quad-6491745539 t-9782682910\bigr)-108 e^{2 t} \bigl(14121216 t^7+487886976 t^6+6668430624 t^5+46162824480 t^4\\ &\quad+171440223048 t^3+329911170732 t^2+281751113865 t+62037369770\bigr)-432 e^t \bigl(6075 t^7+331776 t^6\\ &\quad+7263459 t^5+82341576 t^4+519698997 t^3+1815705750 t^2+3226966848 t+2227431349\bigr)-1036 t^6\\ &\quad-76848 t^5-2220650 t^4-31879992 t^3-238833150 t^2-881219576 t-1244477833,\\ R_2''(t)&=-4 \bigl[1554 t^5+96060 t^4+2220650 t^3+23909994 t^2+119416575 t+220304894+108 e^t \bigl(6075 t^7\\ &\quad+374301 t^6+9254115 t^5+118658871 t^4+849065301 t^3+3374802741 t^2+6858378348 t+5454398197\bigr)\\ &\quad+27 e^{2 t} \bigl(28242432 t^7+1074622464 t^6+16264183104 t^5+125667802080 t^4+527531744016 t^3\\ &\quad+1174143010608 t^2+1223324569194 t+405825853405\bigr)+320 e^{3 t} \bigl(204300000 t^7+5189070000 t^6\\ &\quad+50652525000 t^5+239333880000 t^4+560502309250 t^3+574286314995 t^2+141377127213 t\\ &\quad-35839794269\bigr)+3375 e^{4 t} \bigl(303750000 t^7+4862025000 t^6+26869758750 t^5+55040040000 t^4\\ &\quad+1279752750 t^3-115044794250 t^2-98822918115 t-25779138224\bigr)+108 e^{5 t} \bigl(26444553975 t^7\\ &\quad+177202335555 t^6-209760788727 t^5-3414946657815 t^4-6767843241915 t^3-3056982797421 t^2\\ &\quad+1354461366678 t+1452986587679\bigr)+49 e^{6 t} \bigl(69392203776 t^7+127219040256 t^6-691340482560 t^5\\ &\quad+366709916160 t^4+7823890148480 t^3+10648807327872 t^2+4665581551764 t-1417182100345\bigr)\\ &\quad-10368 e^{7 t} \bigl(164602368 t^5-368395776 t^4+1976006016 t^3+8853438240 t^2+12015731898 t\\ &\quad+53560489\bigr)\bigr],\\ R_2^{(3)}(t)&=-12 \bigl[2590 t^4+128080 t^3+2220650 t^2+15939996 t+39805525+36 e^t \bigl(6075 t^7+416826 t^6\\ &\quad+11499921 t^5+164929446 t^4+1323700785 t^3+5921998644 t^2+13607983830 t+12312776545\bigr)\\ &\quad+36 e^{2 t} \bigl(14121216 t^7+586735488 t^6+9744025248 t^5+83164129920 t^4+389433674088 t^3\\ &\quad+982720313316 t^2+1198733789901 t+508744069001\bigr)+320 e^{3 t} \bigl(204300000 t^7+5665770000 t^6\\ &\quad+61030665000 t^5+323754755000 t^4+879614149250 t^3+1134788624245 t^2+524234670543 t\\ &\quad+11285914802\bigr)+1125 e^{4 t} \bigl(1215000000 t^7+21574350000 t^6+136651185000 t^5\\ &\quad+354508953750 t^4+225279171000 t^3-456339918750 t^2-625381260960 t-201939471011\bigr)\\ &\quad+36 e^{5 t} \bigl(132222769875 t^7+1071123555600 t^6+14410069695 t^5-18123537232710 t^4\\ &\quad-47499002840835 t^3-35588443712850 t^2+658341238548 t+8619394305073\bigr)\\ &\quad+98 e^{6 t} \bigl(69392203776 t^7+208176611328 t^6-564121442304 t^5-209407152640 t^4+8068363425920 t^3\\ &\quad+14560752402112 t^2+8215183994388 t-639585175051\bigr)-3456 e^{7 t} \bigl(1152216576 t^5\\ &\quad-1755758592 t^4+12358459008 t^3+67902085728 t^2+101816999766 t+12390655321\bigr)\bigr],\\ R_2^{(4)}(t)&=-48 \bigl[2590 t^3+96060 t^2+1110325 t+3984999+9 e^t \bigl(6075 t^7+459351 t^6+14000877 t^5\\ &\quad+222429051 t^4+1983418569 t^3+9893100999 t^2+25451981118 t+25920760375\bigr)+9 e^{2 t} \bigl(28242432 t^7\\ &\quad+1272319488 t^6+23008463424 t^5+215048386080 t^4+1111523867856 t^3+3133741648896 t^2\\ &\quad+4362908206434 t+2216221927903\bigr)+80 e^{3 t} \bigl(612900000 t^7+18427410000 t^6+217086615000 t^5\\ &\quad+1276417590000 t^4+3933861467750 t^3+6043208320485 t^2+3842281260119 t+558092414949\bigr)\\ &\quad+1125 e^{4 t} \bigl(1215000000 t^7+23700600000 t^6+169012710000 t^5+525322935000 t^4+579788124750 t^3\\ &\quad-287380540500 t^2-853551220335 t-358284786251\bigr)+9 e^{5 t} \bigl(661113849375 t^7+6281177167125 t^6\\ &\quad+6498791682075 t^5-90545635815075 t^4-309989163135015 t^3-320439227086755 t^2\\ &\quad-67885181232960 t+43755312763913\bigr)+49 e^{6 t} \bigl(208176611328 t^7+867402547200 t^6\\ &\quad-1067834492928 t^5-2038525063680 t^4+23786275972480 t^3+55784802345216 t^2\\ &\quad+39206304385276 t+2188836472041\bigr)-864 e^{7 t} \bigl(8065516032 t^5-6529227264 t^4\\ &\quad+79486178688 t^3+512389977120 t^2+848523169818 t+188551587013\bigr)\bigr],\\ R_2^{(5)}(t)&=48 \bigl[864 e^{7 t} \bigl(56458612224 t^5-5377010688 t^4+530286341760 t^3+3825188375904 t^2\\ &\quad+6964442142966 t+2168384278909\bigr)-9 e^t \bigl(6075 t^7+501876 t^6+16756983 t^5+292433436 t^4\\ &\quad+2873134773 t^3+15843356706 t^2+45238183116 t+51372741493\bigr)-36 e^{2 t} \bigl(14121216 t^7\\ &\quad+685584000 t^6+13412710944 t^5+136284772320 t^4+770810320008 t^3+2400513725340 t^2\\ &\quad+3748324927665 t+2198838015560\bigr)-80 e^{3 t} \bigl(1838700000 t^7+59572530000 t^6+761824305000 t^5\\ &\quad+4914685845000 t^4+16907254763250 t^3+29931209364705 t^2+23613260421327 t+5516558504966\bigr)\\ &\quad-1125 e^{4 t} \bigl(4860000000 t^7+103307400000 t^6+818254440000 t^5+2946355290000 t^4+4420444239000 t^3\\ &\quad+589842212250 t^2-3988965962340 t-2286690365339\bigr)-45 e^{5 t} \bigl(661113849375 t^7+7206736556250 t^6\\ &\quad+14036204282625 t^5-84046844133000 t^4-382425671787075 t^3-506432724967764 t^2\\ &\quad-196060872067662 t+30178276517321\bigr)-98 e^{6 t} \bigl(624529833984 t^7+3330825781248 t^6\\ &\quad-601295837184 t^5-8785161423360 t^4+67281777790080 t^3+203033820994368 t^2\\ &\quad+173403715501044 t+26169661608761\bigr)-5 \bigl(1554 t^2+38424 t+222065\bigr)\bigr],\\ R_2^{(6)}(t)&=144 \bigl[2016 e^{7 t} \bigl(56458612224 t^5+34950569472 t^4+527213764224 t^3+4052453950944 t^2\\ &\quad+8057353107510 t+3163304585047\bigr)-196 e^{6 t} \bigl(624529833984 t^7+4059443920896 t^6\\ &\quad+2729529944064 t^5-9286241287680 t^4+61425003507840 t^3+236674709889408 t^2\\ &\quad+241081655832500 t+55070280858935\bigr)-15 e^{5 t} \bigl(3305569246875 t^7+40661479726875 t^6\\ &\quad+113421440750625 t^5-350053199251875 t^4-2248315735467375 t^3-3679440640200045 t^2\\ &\quad-1993169810273838 t-45169489481057\bigr)-1500 e^{4 t} \bigl(4860000000 t^7+111812400000 t^6\\ &\quad+973215540000 t^5+3969173340000 t^4+7366799529000 t^3+3905175391500 t^2-3694044856215 t\\ &\quad-3283931855924\bigr)-80 e^{3 t} \bigl(1838700000 t^7+63862830000 t^6+880969365000 t^5\\ &\quad+6184393020000 t^4+23460169223250 t^3+46838464127955 t^2+43567399997797 t+13387645312075\bigr)\\ &\quad-12 e^{2 t} \bigl(28242432 t^7+1470016512 t^6+30938925888 t^5+339633099360 t^4+2086759729296 t^3\\ &\quad+7113458410704 t^2+12297677306010 t+8146000958785\bigr)-3 e^t \bigl(6075 t^7+544401 t^6+19768239 t^5\\ &\quad+376218351 t^4+4042868517 t^3+24462761025 t^2+76924896528 t+96610924609\bigr)-20 (259 t+3202)\bigr],\\ R_2^{(7)}(t)&=144 \bigl[98784 e^{7 t} \bigl(8065516032 t^5+10754021376 t^4+78169359744 t^3+611200386720 t^2+1316456727642 t\\ &\quad+616336432711\bigr)-392 e^{6 t} \bigl(1873589501952 t^7+14364186181632 t^6+20366921594880 t^5\\ &\quad-21034899002880 t^4+165702527948160 t^3+802161634929984 t^2+959919677386908 t\\ &\quad+285751670493055\bigr)-15 e^{5 t} \bigl(16527846234375 t^7+226446383362500 t^6+811076082114375 t^5\\ &\quad-1183158792506250 t^4-12641791474344375 t^3-25142150407402350 t^2-17324730331769280 t\\ &\quad-2219017257679123\bigr)-1500 e^{4 t} \bigl(19440000000 t^7+481269600000 t^6+4563736560000 t^5\\ &\quad+20742771060000 t^4+45343891476000 t^3+37721100153000 t^2-6965828641860 t\\ &\quad-16829772279911\bigr)-80 e^{3 t} \bigl(5516100000 t^7+204459390000 t^6+3026085075000 t^5\\ &\quad+22958025885000 t^4+95118079749750 t^3+210895900053615 t^2+224379128249301 t\\ &\quad+83730335934022\bigr)-48 e^{2 t}\bigl(14121216 t^7+784432512 t^6+17674487712 t^5+208490207040 t^4\\ &\quad+1383012964008 t^3+5121799002324 t^2+9705567858357 t+7147419805895\bigr)-3 e^t \bigl(6075 t^7\\ &\quad+586926 t^6+23034645 t^5+475059546 t^4+5547741921 t^3+36591366576 t^2+125850418578 t\\ &\quad+173535821137\bigr)-5180\bigr],\\ R_2^{(8)}(t)&=432 e^t \bigl[230496 e^{6 t} \bigl(8065516032 t^5+16515104256 t^4+84314514816 t^3+644701540896 t^2\\ &\quad+1491085409562 t+804401679517\bigr)-784 e^{5 t} \bigl(1873589501952 t^7+16550040600576 t^6\\ &\quad+34731107776512 t^5-4062464340480 t^4+151679261946240 t^3+885012898904064 t^2\\ &\quad+1227306889030236 t+445738283390873\bigr)-25 e^{4 t} \bigl(16527846234375 t^7+249585368090625 t^6\\ &\quad+1082811742149375 t^5-372082710391875 t^4-13588318508349375 t^3-32727225292008975 t^2\\ &\quad-27381590494730220 t-5683963324032979\bigr)-16000 e^{3 t} \bigl(2430000000 t^7+64411200000 t^6\\ &\quad+660705120000 t^5+3305930220000 t^4+8260832817000 t^3+8966127345000 t^2\\ &\quad+1486840179330 t-2321403680047\bigr)-80 e^{2 t} \bigl(5516100000 t^7+217330290000 t^6+3435003855000 t^5\\ &\quad+28001501010000 t^4+125728780929750 t^3+306013979803365 t^2+364976394951711 t\\ &\quad+158523378683789\bigr)-16 e^t \bigl(28242432 t^7+1667713536 t^6+40055570496 t^5+505352852640 t^4\\ &\quad+3599986756176 t^3+14392636896672 t^2+29654733721362 t+24000407470147\bigr)-6075 t^7\\ &\quad-629451 t^6-26556201 t^5-590232771 t^4-7447980105 t^3-53234592339 t^2-199033151730 t\\ &\quad-299386239715\bigr]\\ &\triangleq 432 e^t R_3(t),\\ R_3'(t)&=2765952 e^{6 t} \bigl(4032758016 t^5+11618183808 t^4+47662292160 t^3+343429399152 t^2+852992961597 t\\ &\quad+526457957222\bigr)-784 e^{5 t} \bigl(9367947509760 t^7+95865329516544 t^6+272955782486016 t^5\\ &\quad+153343217180160 t^4+742146452369280 t^3+4880102280359040 t^2+7906560242959308 t\\ &\quad+3455998305984601\bigr)-25 e^{4 t} \bigl(66111384937500 t^7+1114036396003125 t^6+5828759177141250 t^5\\ &\quad+3925727869179375 t^4-55841604874965000 t^3-171673856693084025 t^2-174980812562938830 t\\ &\quad-50117443790862136\bigr)-48000 e^{3 t} \bigl(2430000000 t^7+70081200000 t^6+789527520000 t^5\\ &\quad+4407105420000 t^4+12668739777000 t^3+17226960162000 t^2+7464258409330 t-1825790286937\bigr)\\ &\quad-80 e^{2 t} \bigl(11032200000 t^7+473273280000 t^6+8173989450000 t^5+73178021295000 t^4\\ &\quad+363463565899500 t^3+989214302395980 t^2+1341980749510152 t+682023152319289\bigr)\\ &\quad-16 e^t \bigl(28242432 t^7+1865410560 t^6+50061851712 t^5+705630705120 t^4+5621398166736 t^3\\ &\quad+25192597165200 t^2+58440007514706 t+53655141191509\bigr)-3 \bigl(14175 t^6+1258902 t^5+44260335 t^4\\ &\quad+786977028 t^3+7447980105 t^2+35489728226 t+66344383910\bigr),\\ R_3''(t)&=-2 \bigl[3 \bigl(42525 t^5+3147255 t^4+88520670 t^3+1180465542 t^2+7447980105 t+17744864113\bigr)\\ &\quad+8 e^t \bigl(28242432 t^7+2063107584 t^6+61254315072 t^5+955939963680 t^4+8443920987216 t^3\\ &\quad+42056791665408 t^2+108825201845106 t+112095148706215\bigr)+80 e^{2 t} \bigl(11032200000 t^7\\ &\quad+511885980000 t^6+9593809290000 t^5+93612994920000 t^4+509819608489500 t^3\\ &\quad+1534409651245230 t^2+2331195051906132 t+1353013527074365\bigr)+24000 e^{3 t} \bigl(7290000000 t^7\\ &\quad+227253600000 t^6+2789069760000 t^5+17168953860000 t^4+55634641011000 t^3+89687099817000 t^2\\ &\quad+56846695551990 t+1986887548519\bigr)+25 e^{4 t} \bigl(132222769875000 t^7+2459462639287500 t^6\\ &\quad+14999627542291875 t^5+22423353681211875 t^4-103831754011571250 t^3-427110120698615550 t^2\\ &\quad-521635481818961685 t-187725293863193687\bigr)+392 e^{5 t} \bigl(46839737548800 t^7+544902280151040 t^6\\ &\quad+1939970889529344 t^5+2131494998330880 t^4+4324105130567040 t^3+26626950758903040 t^2\\ &\quad+49293005775514620 t+25186551772882313\bigr)-4148928 e^{6 t} \bigl(8065516032 t^5+29957630976 t^4\\ &\quad+110815496064 t^3+734521090464 t^2+1934938855962 t+1337246901643\bigr)\bigr],\\ R_3^{(3)}(t)&=-2 \bigl[81 \bigl(7875 t^4+466260 t^3+9835630 t^2+87441892 t+275851115\bigr)+8 e^t \bigl(28242432 t^7+2260804608 t^6\\ &\quad+73632960576 t^5+1262211539040 t^4+12267680841936 t^3+67388554627056 t^2+192938785175922 t\\ &\quad+220920350551321\bigr)+160 e^{2 t} \bigl(11032200000 t^7+550498680000 t^6+11129467230000 t^5\\ &\quad+117597518145000 t^4+697045598329500 t^3+2299139063979480 t^2+3865604703151362 t\\ &\quad+2518611053027431\bigr)+648000 e^{3 t} \bigl(810000000 t^7+27140400000 t^6+360397440000 t^5\\ &\quad+2424155940000 t^4+8725175499000 t^3+16146860092000 t^2+12959788381110 t+2326198451761\bigr)\\ &\quad+25 e^{4 t} \bigl(528891079500000 t^7+10763409946275000 t^6+74755286004892500 t^5\\ &\quad+164691552436306875 t^4-325633601321437500 t^3-2019935744829175950 t^2\\ &\quad-2940762168673077840 t-1272536657271736433\bigr)+1960 e^{5 t} \bigl(46839737548800 t^7\\ &\quad+610477912719360 t^6+2593853625710592 t^5+4071465887860224 t^4+6029301129231744 t^3\\ &\quad+29221413837243264 t^2+59943786079075836 t+35045152927985237\bigr)-49787136 e^{6 t} \bigl(4032758016 t^5\\ &\quad+18339447168 t^4+65393625024 t^3+394964419248 t^2+1089889609725 t+829868355485\bigr)\bigr],\\ R_3^{(4)}(t)&=-8 \bigl[81 \bigl(7875 t^3+349695 t^2+4917815 t+21860473\bigr)+2 e^t \bigl(28242432 t^7 +2458501632 t^6\\ &\quad+87197788224 t^5+1630376341920 t^4+17316526998096 t^3 +104191597152864 t^2\\ &\quad+327715894430034 t+413859135727243\bigr) +160 e^{2 t} \bigl(5516100000 t^7+294555690000 t^6\\ &\quad+6390481635000 t^5 +72710593110000 t^4+466120317309750 t^3+1672353730736865 t^2\\ &\quad +3082371883565421 t+2225706702301556\bigr)+162000 e^{3 t} \bigl(2430000000 t^7 +87091200000 t^6\\ &\quad+1244034720000 t^5+9074455020000 t^4 +35872150257000 t^3+74616106773000 t^2\\ &\quad+71173085327330 t +19938383736393\bigr)+25 e^{4 t} \bigl(528891079500000 t^7+11688969335400000 t^6\\ &\quad +90900400924305000 t^5+258135659942422500 t^4 -160942048885130625 t^3\\ &\quad-2264160945820254075 t^2-3950730041087665815 t -2007727199440005893\bigr)\\ &\quad+490 e^{5 t} \bigl(234198687744000 t^7+3380267726438400 t^6 +16632135604869120 t^5\\ &\quad+33326597567854080 t^4+46432369197599616 t^3+164194972573911552 t^2\\ &\quad+358161758069865708 t+235169550719002021\bigr)-37340352 e^{6 t} \bigl(8065516032 t^5+43400157696 t^4\\ &\quad+155239846272 t^3+855322463520 t^2+2443088832282 t+2023033247545\bigr)\bigr], \\ R_3^{(5)}(t)&=8 \bigl[448084224 e^{6 t} \bigl(4032758016 t^5+25060710528 t^4+92086642368 t^3+466471193328 t^2\\ &\quad+1364098160061 t+1215107359796\bigr)-490 e^{5 t} \bigl(1170993438720000 t^7+18540729446400000 t^6\\ &\quad+103442284382976000 t^5+249793665863616000 t^4+365468236259414400 t^3\\ &\quad+960271970462356608 t^2+2119198735497151644 t+1534009511664875813\bigr)\\ &\quad-25 e^{4 t} \bigl(2115564318000000 t^7+50458114898100000 t^6+433735419709620000 t^5\\ &\quad+1487044644391215000 t^4+388774444229167500 t^3-9539469929936408175 t^2\\ &\quad-20331242055991171410 t-11981638838847689387\bigr)-162000 e^{3 t} \bigl(7290000000 t^7\\ &\quad+278283600000 t^6+4254651360000 t^5+33443538660000 t^4+143914270851000 t^3\\ &\quad+331464771090000 t^2+362751469527990 t+130988236536509\bigr)-160 e^{2 t} \bigl(11032200000 t^7\\ &\quad+627724080000 t^6+14548297410000 t^5+177373594395000 t^4+1223083007059500 t^3\\ &\quad+4743068413402980 t^2+9509451228604572 t+7533785288168533\bigr)-2 e^t \bigl(28242432 t^7\\ &\quad+2656198656 t^6+101948798016 t^5+2066365283040 t^4+23838032365776 t^3\\ &\quad+156141178147152 t^2+536099088735762 t+741575030157277\bigr)\\ &\quad-405 \bigl(4725 t^2+139878 t+983563\bigr)\bigr],\\ R_3^{(6)}(t)&=16 \bigl[672126336 e^{6 t} \bigl(8065516032 t^5+56842684416 t^4+217587565440 t^3+1025029029024 t^2\\ &\quad+3039177115674 t+2884914106279\bigr)-245 e^{5 t} \bigl(5854967193600000 t^7+100900601303040000 t^6\\ &\quad+628455798593280000 t^5+1766179751232960000 t^4+2826515844751536000 t^3\\ &\quad+5897764561090026240 t^2+12516537618410471436 t+9789246293821530709\bigr)\\ &\quad-25 e^{4 t} \bigl(4231128636000000 t^7+108320704909200000 t^6+1018845184113540000 t^5\\ &\quad+4058427838056480000 t^4+3751638177240765000 t^3-18495778193529065100 t^2\\ &\quad-50201954041918750995 t-34128898705690964479\bigr)-729000 e^{3 t} \bigl(2430000000 t^7\\ &\quad+98431200000 t^6+1603739520000 t^5+13511541420000 t^4+62835218577000 t^3\\ &\quad+158459680647000 t^2+194575994529330 t+83968464348613\bigr)-160 e^{2 t} \bigl(11032200000 t^7\\ &\quad+666336780000 t^6+16431469650000 t^5+213744337920000 t^4+1577830195849500 t^3\\ &\quad+6577692923992230 t^2+14252519642007552 t+12288510902470819\bigr)-e^t \bigl(28242432 t^7\\ &\quad+2853895680 t^6+117885989952 t^5+2576109273120 t^4+32103493497936 t^3\\ &\quad+227655275244480 t^2+848381445030066 t+1277674118893039\bigr)-3645 (525 t+7771)\bigr],\\ R_3^{(7)}(t)&=16 \bigl[8065516032 e^{6 t} \bigl(4032758016 t^5+31781973888 t^4+127741344192 t^3+566911405872 t^2\\ &\quad+1690426729341 t+1695721812779\bigr)-245 e^{5 t} \bigl(29274835968000000 t^7+545487776870400000 t^6\\ &\quad+3747682600784640000 t^5+11973177749131200000 t^4+21197298228689520000 t^3\\ &\quad+37968370339704739200 t^2+74378217214232409660 t+61462769087518124981\bigr)\\ &\quad-25 e^{4 t} \bigl(16924514544000000 t^7+462900720088800000 t^6+4725304965909360000 t^5\\ &\quad+21327937272793620000 t^4+31240264061188980000 t^3-62728198242393965400 t^2\\ &\quad-237799372554733134180 t-186717548864682608911\bigr)-2187000 e^{3 t} \bigl(2430000000 t^7\\ &\quad+104101200000 t^6+1800601920000 t^5+16184440620000 t^4+80850607137000 t^3\\ &\quad+221294899224000 t^2+300215781627330 t+148827129191723\bigr)-320 e^{2 t} \bigl(11032200000 t^7\\ &\quad+704949480000 t^6+18430479990000 t^5+254823012045000 t^4+2005318871689500 t^3\\ &\quad+8944438217766480 t^2+20830212565999782 t+19414770723474595\bigr)-e^t \bigl(28242432 t^7\\ &\quad+3051592704 t^6+135009364032 t^5+3165539222880 t^4+42407930590416 t^3\\ &\quad+323965755738288 t^2+1303691995519026 t+2126055563923105\bigr)-1913625\bigr],\\ R_3^{(8)}(t)&=16 e^t \bigl[24196548096 e^{5 t} \bigl(8065516032 t^5+70285211136 t^4+297858653568 t^3+1261564155936 t^2\\ &\quad+3758794395930 t+3954919202005\bigr)-1225 e^{4 t} \bigl(29274835968000000 t^7+586472547225600000 t^6\\ &\quad+4402267933029120000 t^5+15720860349915840000 t^4+30775840427994480000 t^3\\ &\quad+50686749276918451200 t^2+89565565350114305340 t+76338412530364606913\bigr)\\ &\quad-800 e^{3 t} \bigl(2115564318000000 t^7+61564827567600000 t^6+677457005755320000 t^5\\ &\quad+3404321060022540000 t^4+6571025166747825000 t^3-4912250024562778800 t^2\\ &\quad-33645433959491264610 t-30770924000420736557\bigr)-19683000 e^{2 t} \bigl(810000000 t^7\\ &\quad+36590400000 t^6+669601440000 t^5+6395147940000 t^4+34143287099000 t^3\\ &\quad+100715168787000 t^2+149248571481110 t+82966352133611\bigr)-1280 e^t \bigl(5516100000 t^7\\ &\quad+371781090000 t^6+10272664215000 t^5+150449606010000 t^4+1257482447889750 t^3\\ &\quad+5976208262650365 t^2+14887325391883131 t+14914938503237243\bigr)-28242432 t^7\\ &\quad-3249289728 t^6-153318920256 t^5-3840586043040 t^4-55070087481936 t^3\\ &\quad-451189547509536 t^2-1951623506995602 t-3429747559442131\bigr]\\ &\triangleq 16 e^tR_4(t),\\ R_4'(t)&=-2 \bigl[98848512 t^6+9747869184 t^5+383297300640 t^4+7681172086080 t^3+82605131222904 t^2\\ &\quad+451189547509536 t+975811753497801+640 e^t \bigl(5516100000 t^7+410393790000 t^6\\ &\quad+12503350755000 t^5+201812927085000 t^4+1859280871929750 t^3+9748655606319615 t^2\\ &\quad+26839741917183861 t+29802263895120374\bigr)+39366000 e^{2 t} \bigl(405000000 t^7+19712700000 t^6\\ &\quad+389686320000 t^5+4034575770000 t^4+23466791489500 t^3+75965049717750 t^2\\ &\quad+124981870134055 t+78795318937083\bigr)+1200 e^{3 t} \bigl(2115564318000000 t^7+66501144309600000 t^6\\ &\quad+800586660890520000 t^5+4533416069614740000 t^4+11110119913444545000 t^3\\ &\quad+1658775142185046200 t^2-36920267309199783810 t-41986068653584491427\bigr)\\ &\quad+9800 e^{4 t} \bigl(7318708992000000 t^7+159425877542400000 t^6+1320494188466880000 t^5\\ &\quad+5305923816550560000 t^4+11624175194477580000 t^3+18442157399478577800 t^2\\ &\quad+28727234997143382735 t+24682450966973295812\bigr)-12098274048 e^{5 t} \bigl(40327580160 t^5\\ &\quad+391753635840 t^4+1770434112384 t^3+7201396740384 t^2+21317100291522 t+23533390405955\bigr)\bigr],\\ R_4''(t)&=-16 \bigl[18 \bigl(4118688 t^5+338467680 t^4+10647147240 t^3+160024418460 t^2+1147293489207 t\\ &\quad+3133260746594\bigr)+80 e^t \bigl(5516100000 t^7+449006490000 t^6+14965713495000 t^5\\ &\quad+264329680860000 t^4+2666532580269750 t^3+15326498222108865 t^2+46337053129823091 t\\ &\quad+56642005812304235\bigr)+4920750 e^{2 t} \bigl(810000000 t^7+42260400000 t^6+897648840000 t^5\\ &\quad+10017583140000 t^4+63071886059000 t^3+222330473904000 t^2+401893839703610 t\\ &\quad+282572508008221\bigr)+450 e^{3 t} \bigl(2115564318000000 t^7+71437461051600000 t^6\\ &\quad+933588949509720000 t^5+5867727171098940000 t^4+17154674672930865000 t^3\\ &\quad+12768895055629591200 t^2-35814417214409753010 t-54292824423317752697\bigr)\\ &\quad+1225 e^{4 t} \bigl(29274835968000000 t^7+688934473113600000 t^6+6238532019121920000 t^5\\ &\quad+27826166208536640000 t^4+67720396044112560000 t^3+108641155181347051200 t^2\\ &\quad+151793254787530686540 t+127457038865036565983\bigr)-1512284256 e^{5 t} \bigl(201637900800 t^5\\ &\quad+2160406080000 t^4+10419185105280 t^3+41318286039072 t^2+120988294938378 t\\ &\quad+138984052321297\bigr)\bigr],\\ R_4^{(3)}(t)&=-32 \bigl[27 \bigl(6864480 t^4+451290240 t^3+10647147240 t^2+106682945640 t+382431163069\bigr)\\ &\quad+40 e^t \bigl(5516100000 t^7+487619190000 t^6+17659752435000 t^5+339158248335000 t^4\\ &\quad+3723851303709750 t^3+23326095962918115 t^2+76990049574040821 t+102979058942127326\bigr)\\ &\quad+9841500 e^{2 t} \bigl(405000000 t^7+22547700000 t^6+512215020000 t^5+6130852620000 t^4\\ &\quad+41553526169500 t^3+158469151496250 t^2+312112156803805 t+241759713930013\bigr)\\ &\quad+675 e^{3 t} \bigl(2115564318000000 t^7+76373777793600000 t^6+1076463871612920000 t^5\\ &\quad+7423708753615140000 t^4+24978310901062785000 t^3+29923569728560456200 t^2\\ &\quad-27301820510656692210 t-66230963494787670367\bigr)+4900 e^{4 t} \bigl(14637417984000000 t^7\\ &\quad+370082718028800000 t^6+3635966864396160000 t^5+17812165616219520000 t^4\\ &\quad+47773281126324600000 t^3+79715726107215735600 t^2+103056916189102106070 t\\ &\quad+82702676280959618809\bigr)-756142128 e^{5 t} \bigl(1008189504000 t^5+11810219904000 t^4\\ &\quad+60737549846400 t^3+237848985511200 t^2+687578046770034 t+815908556544863\bigr)\bigr],\\ R_4^{(4)}(t)&=-32 \bigl[3240 \bigl(228816 t^3+11282256 t^2+177452454 t+889024547\bigr)+40 e^t \bigl(5516100000 t^7\\ &\quad+526231890000 t^6+20585467575000 t^5+427457010510000 t^4+5080484297049750 t^3\\ &\quad+34497649874047365 t^2+123642241499877051 t+179969108516168147\bigr)\\ &\quad+9841500 e^{2 t} \bigl(810000000 t^7+47930400000 t^6+1159716240000 t^5+14822780340000 t^4\\ &\quad+107630462819000 t^3+441598881501000 t^2+941162616600110 t+795631584663831\bigr)\\ &\quad+2025 e^{3 t} \bigl(2115564318000000 t^7+81310094535600000 t^6+1229211427200120000 t^5\\ &\quad+9217815206303340000 t^4+34876589239216305000 t^3+54901880629623241200 t^2\\ &\quad-7352774024949721410 t-75331570331673234437\bigr)+9800 e^{4 t} \bigl(29274835968000000 t^7\\ &\quad+791396399001600000 t^6+8382181882878720000 t^5+44714248393429440000 t^4\\ &\quad+131170893485088240000 t^3+231091373903918371200 t^2+285829558485419947740 t\\ &\quad+216933810656470290653\bigr)-756142128 e^{5 t} \bigl(5040947520000 t^5+64092047040000 t^4\\ &\quad+350928628848000 t^3+1371457577095200 t^2+3913588204872570 t+4767120829494349\bigr)\bigr],\\ R_4^{(5)}(t)&=160 \bigl[756142128 e^{5 t} \bigl(5040947520000 t^5+69132994560000 t^4+402202266480000 t^3\\ &\quad+1582014754404000 t^2+4462171235710650 t+5549838470468863\bigr)\\ &\quad-31360 e^{4 t} \bigl(7318708992000000 t^7+210656840486400000 t^6+2392319120345280000 t^5\\ &\quad+13797993936756960000 t^4+43971285469629420000 t^3+82367386004433637800 t^2\\ &\quad+100343811359344783335 t+72097800069456319397\bigr)-1215 e^{3 t} \bigl(2115564318000000 t^7\\ &\quad+86246411277600000 t^6+1391831616271320000 t^5+11266500918303540000 t^4\\ &\quad+47167009514287425000 t^3+89778469868839546200 t^2+29248479728132439390 t\\ &\quad-77782495006656474907\bigr)-7873200 e^{2 t} \bigl(405000000 t^7+25382700000 t^6+651753720000 t^5\\ &\quad+8861035470000 t^4+68638011749500 t^3+301522287864750 t^2+691380749050555 t\\ &\quad+633106446481943\bigr)-8 e^t \bigl(5516100000 t^7+564844590000 t^6+23742858915000 t^5\\ &\quad+530384348385000 t^4+6790312339089750 t^3+49739102765196615 t^2+192637541247971781 t\\ &\quad+303611350016045198\bigr)-3888 \bigl(114408 t^2+3760752 t+29575409\bigr)\bigr],\\ R_4^{(6)}(t)&=160 \bigl[3780710640 e^{5 t} \bigl(5040947520000 t^5+74173942080000 t^4+457508662128000 t^3\\ &\quad+1823336114292000 t^2+5094977137472250 t+6442272717610993\bigr)\\ &\quad-31360 e^{4 t} \bigl(29274835968000000 t^7+893858324889600000 t^6+10833217524299520000 t^5\\ &\quad+67153571348754240000 t^4+231077117625545520000 t^3+461383400426622811200 t^2\\ &\quad+566110017446246408940 t+388735011637170060923\bigr)-3645 e^{3 t} \bigl(2115564318000000 t^7\\ &\quad+91182728019600000 t^6+1564324438826520000 t^5+13586220278755740000 t^4\\ &\quad+62189010738692145000 t^3+136945479383126971200 t^2+89100792974025470190 t\\ &\quad-68033001763945661777\bigr)-7873200 e^{2 t} \bigl(810000000 t^7+53600400000 t^6+1455803640000 t^5\\ &\quad+20980839540000 t^4+172720165379000 t^3+808958610978000 t^2+1985806073830610 t\\ &\quad+1957593642014441\bigr)-8 e^t \bigl(5516100000 t^7+603457290000 t^6+27131926455000 t^5\\ &\quad+649098642960000 t^4+8911849732629750 t^3+70110039782465865 t^2+292115746778365011 t\\ &\quad+496248891264016979\bigr)-186624 (4767 t+78349)\bigr],\\ R_4^{(7)}(t)&=160 \bigl[18903553200 e^{5 t} \bigl(5040947520000 t^5+79214889600000 t^4+516847815792000 t^3\\ &\quad+2097841311568800 t^2+5824311583189050 t+7461268145105443\bigr)\\ &\quad-250880 e^{4 t} \bigl(14637417984000000 t^7+472544643916800000 t^6+6087002505816960000 t^5\\ &\quad+40347546627064320000 t^4+149115344487149880000 t^3+317345619322890975600 t^2\\ &\quad+398400858829778907270 t+265131257999365831579\bigr)-10935 e^{3 t} \bigl(2115564318000000 t^7\\ &\quad+96119044761600000 t^6+1746689894865720000 t^5+16193427676799940000 t^4\\ &\quad+80303971110366465000 t^3+199134490121819116200 t^2+180397779229443450990 t\\ &\quad-38332737439270505047\bigr)-31492800 e^{2 t} \bigl(405000000 t^7+28217700000 t^6+808302420000 t^5\\ &\quad+12310174320000 t^4+107340922229500 t^3+534019429523250 t^2+1397382342404305 t\\ &\quad+1475248339464873\bigr)-8 e^t \bigl(5516100000 t^7+642069990000 t^6+30752670195000 t^5\\ &\quad+784758275235000 t^4+11508244304469750 t^3+96845588980355115 t^2+432335826343296741 t\\ &\quad+788364638042381990\bigr)-889636608\bigr],\\ R_4^{(8)}(t)&=160 e^t \bigl[94517766000 e^{4 t} \bigl(5040947520000 t^5+84255837120000 t^4+580219727472000 t^3\\ &\quad+2407950001044000 t^2+6663448107816570 t+8626130461743253\bigr)\\ &\quad-501760 e^{3 t} \bigl(29274835968000000 t^7+996320250777600000 t^6+13591638943384320000 t^5\\ &\quad+95912599518671040000 t^4+378925782228428400000 t^3+858364255376506771200 t^2\\ &\quad+1114147336982448790140 t+729462945413621116793\bigr)-98415 e^{2 t} \bigl(705188106000000 t^7\\ &\quad+33685120501200000 t^6+646309328129640000 t^5+6368192500525380000 t^4\\ &\quad+33965069337588795000 t^3+93146153744061860400 t^2+104384701992440953930 t\\ &\quad+7266618545736881761\bigr)-31492800 e^t \bigl(810000000 t^7+59270400000 t^6+1785911040000 t^5\\ &\quad+28661860740000 t^4+263922541739000 t^3+1390061625735000 t^2+3862803543855110 t\\ &\quad+4347879021334051\bigr)-8 \bigl(5516100000 t^7+680682690000 t^6+34605090135000 t^5\\ &\quad+938521626210000 t^4+14647277405409750 t^3+131370321893764365 t^2+626027004304006971 t\\ &\quad+1220700464385678731\bigr)\bigr]\\ &\triangleq 160 e^t R_5(t),\\ R_5'(t)&=12 \bigl[15752961000 e^{4 t} \bigl(10081895040000 t^5+181114043040000 t^4+1328951129184000 t^3\\ &\quad+5686229593296000 t^2+15734846216677140 t+20583984977394791\bigr)\\ &\quad-125440 e^{3 t} \bigl(29274835968000000 t^7+1064628201369600000 t^6+15584279444939520000 t^5\\ &\quad+118565331090978240000 t^4+506809248253323120000 t^3+1237290037604935171200 t^2\\ &\quad+1686390173900119970940 t+1100845391074437380173\bigr)-32805 e^{2 t} \bigl(352594053000000 t^7\\ &\quad+18076639436100000 t^6+373682344816620000 t^5+3991982910424740000 t^4\\ &\quad+23350727169319777500 t^3+72046878875222526450 t^2+98765427868251407165 t\\ &\quad+29729484770978679363\bigr)-2624400 e^t \bigl(810000000 t^7+64940400000 t^6+2141533440000 t^5\\ &\quad+37591415940000 t^4+378569984699000 t^3+2181829250952000 t^2+6642926795325110 t\\ &\quad+8210682565189161\bigr)-2 \bigl(12870900000 t^6+1361365380000 t^5+57675150225000 t^4\\ &\quad+1251362168280000 t^3+14647277405409750 t^2+87580214595842910 t+208675668101335657\bigr)\bigr],\\ R_5''(t)&=180 \bigl[16803158400 e^{4 t} \bigl(2520473760000 t^5+48429102960000 t^4+377516293056000 t^3\\ &\quad+1670735735046000 t^2+4644490253331285 t+6129424132891019\bigr)\\ &\quad-25088 e^{3 t} \bigl(29274835968000000 t^7+1132936151961600000 t^6+17713535847678720000 t^5\\ &\quad+144539130165877440000 t^4+664896356374627440000 t^3+1744099285858258291200 t^2\\ &\quad+2511250198970076751740 t+1662975449041144037153\bigr)-2187 e^{2 t} \bigl(705188106000000 t^7\\ &\quad+38621437243200000 t^6+855824526249840000 t^5+9852377544932580000 t^4\\ &\quad+62669385980338515000 t^3+214145939258404385400 t^2+341624613486947867230 t\\ &\quad+158224397410208765891\bigr)-174960 e^t \bigl(810000000 t^7+70610400000 t^6+2531175840000 t^5\\ &\quad+48299083140000 t^4+528935648459000 t^3+3317539205049000 t^2+11006585297229110 t\\ &\quad+14853609360514271\bigr)-4 \bigl(2574180000 t^5+226894230000 t^4+7690020030000 t^3\\ &\quad+125136216828000 t^2+976485160360650 t+2919340486528097\bigr)\bigr],\\ R_5^{(3)}(t)&=2160 \bigl[1400263200 e^{4 t} \bigl(10081895040000 t^5+206318780640000 t^4+1703781584064000 t^3\\ &\quad+7815491819352000 t^2+21919432483417140 t+29162186784895361\bigr)\\ &\quad-6272 e^{3 t} \bigl(29274835968000000 t^7+1201244102553600000 t^6+19979408151601920000 t^5\\ &\quad+174061689912008640000 t^4+857615196595797360000 t^3+2408995642232885731200 t^2\\ &\quad+3673983056208915612540 t+2500058848697836287733\bigr)-729 e^{2 t} \bigl(352594053000000 t^7\\ &\quad+20544797807100000 t^6+485844418989720000 t^5+5995969430278590000 t^4\\ &\quad+41187070535101837500 t^3+154075009114456078950 t^2+277885276372676126315 t\\ &\quad+164518352076841349753\bigr)-14580 e^t \bigl(810000000 t^7+76280400000 t^6+2954838240000 t^5\\ &\quad+60954962340000 t^4+722131981019000 t^3+4904346150426000 t^2+17641663707327110 t\\ &\quad+25860194657743381\bigr)-50 \bigl(85806000 t^4+6050512800 t^3+153800400600 t^2\\ &\quad+1668482891040 t+6509901069071\bigr)\bigr],\\ R_5^{(4)}(t)&=6480 \bigl[3734035200 e^{4 t} \bigl(5040947520000 t^5+109460574720000 t^4+955050182352000 t^3\\ &\quad+4546664003700000 t^2+12913589196546570 t+17321022452874823\bigr)\\ &\quad-6272 e^{3 t} \bigl(29274835968000000 t^7+1269552053145600000 t^6+22381896356709120000 t^5\\ &\quad+207360703498011840000 t^4+1089697449811808880000 t^3+3266610838828683091200 t^2\\ &\quad+5279980151030839433340 t+3724719867434141491913\bigr)-243 e^{2 t} \bigl(705188106000000 t^7\\ &\quad+43557753985200000 t^6+1094957624822040000 t^5+14421160955505780000 t^4\\ &\quad+106358018791318035000 t^3+431711229834217670400 t^2+863920570974264410530 t\\ &\quad+606921980526358825821\bigr)-4860 e^t \bigl(810000000 t^7+81950400000 t^6+3412520640000 t^5\\ &\quad+75729153540000 t^4+965951830379000 t^3+7070742093483000 t^2+27450356008179110 t\\ &\quad+43501858365070491\bigr)-4000 \bigl(1430100 t^3+75631410 t^2+1281670005 t+6952012046\bigr)\bigr],\\ R_5^{(5)}(t)&=77760 \bigl[622339200 e^{4 t} \bigl(10081895040000 t^5+231523518240000 t^4+2129021514144000 t^3\\ &\quad+10525903280928000 t^2+30373842396793140 t+41098839504022931\bigr)\\ &\quad-1568 e^{3 t} \bigl(29274835968000000 t^7+1337860003737600000 t^6+24921000463000320000 t^5\\ &\quad+244663864092527040000 t^4+1366178387809158000000 t^3+4356308288640491971200 t^2\\ &\quad+7457720710249961494140 t+5484713251111087969693\bigr)-81 e^{2 t} \bigl(352594053000000 t^7\\ &\quad+23012956178100000 t^6+612815443388820000 t^5+8579277508780440000 t^4\\ &\quad+67600170351164797500 t^3+295624129010597361450 t^2+647815900404241040465 t\\ &\quad+519441133006745515543\bigr)-405 e^t \bigl(810000000 t^7+87620400000 t^6+3904223040000 t^5\\ &\quad+92791756740000 t^4+1268868444539000 t^3+9968597584620000 t^2+41591840195145110 t\\ &\quad+70952214373249601\bigr)-5000 \bigl(286020 t^2+10084188 t+85444667\bigr)\bigr],\\ R_5^{(6)}(t)&=233280 \bigl[3319142400 e^{4 t} \bigl(2520473760000 t^5+61031471760000 t^4+590136258096000 t^3\\ &\quad+3030667354134000 t^2+8909198509314285 t+12173075025805304\bigr)\\ &\quad-1568 e^{3 t} \bigl(29274835968000000 t^7+1406167954329600000 t^6+27596720470475520000 t^5\\ &\quad+286198864864194240000 t^4+1692396873265860720000 t^3+5722486676449649971200 t^2\\ &\quad+10361926236010289474940 t+7970620154527741801073\bigr)-27 e^{2 t} \bigl(705188106000000 t^7\\ &\quad+48494070727200000 t^6+1363708623846240000 t^5+20222632234504980000 t^4\\ &\quad+169517450737451355000 t^3+794048769074689115400 t^2+1886880058829676803830 t\\ &\quad+1686698166417732071551\bigr)-135 e^t \bigl(810000000 t^7+93290400000 t^6+4429945440000 t^5\\ &\quad+112312871940000 t^4+1640035471499000 t^3+13775202918237000 t^2+61529035364385110 t\\ &\quad+112544054568394711\bigr)-20000 (47670 t+840349)\bigr],\\ R_5^{(7)}(t)&=699840 \bigl[1106380800 e^{4 t} \bigl(10081895040000 t^5+256728255840000 t^4+2604670919424000 t^3\\ &\quad+13893078190824000 t^2+41698128745525140 t+57601498612535501\bigr)\\ &\quad-1568 e^{3 t} \bigl(29274835968000000 t^7+1474475904921600000 t^6+30409056379134720000 t^5\\ &\quad+332193398981653440000 t^4+2073995359751453040000 t^3+7414883549715510691200 t^2\\ &\quad+14176917353643389455740 t+11424595566531171626053\bigr)-36 e^{2 t} \bigl(352594053000000 t^7\\ &\quad+25481114549100000 t^6+754595418013920000 t^5+11815951897060290000 t^4\\ &\quad+104981357603230657500 t^3+524162472590433073950 t^2+1340464413952182959615 t\\ &\quad+1315069097916285236733\bigr)-45 e^t \bigl(810000000 t^7+98960400000 t^6+4989687840000 t^5\\ &\quad+134462599140000 t^4+2089286959259000 t^3+18695309332734000 t^2+89079441200859110 t\\ &\quad+174073089932779821\bigr)-317800000\bigr],\\ R_5^{(8)}(t)&=6298560 e^t \bigl[983449600 e^{3 t} \bigl(5040947520000 t^5+134665312320000 t^4+1430699587632000 t^3\\ &\quad+7923290690196000 t^2+24322333920468570 t+34013015399458393\bigr)\\ &\quad-1568 e^{2 t} \bigl(9758278656000000 t^7+514261285171200000 t^6+11119336062992640000 t^5\\ &\quad+127625053204514880000 t^4+838973297242330320000 t^3+3162959636488987910400 t^2\\ &\quad+6373391017817909972180 t+5383411561470767148211\bigr)-4 e^t \bigl(705188106000000 t^7\\ &\quad+53430387469200000 t^6+1662077523322440000 t^5+27404880884190180000 t^4\\ &\quad+257226522794702475000 t^3+1363269017990558120400 t^2+3729253773085232067130 t\\ &\quad+3970602609784753433081\bigr)-5 \bigl(810000000 t^7+104630400000 t^6+5583450240000 t^5\\ &\quad+159411038340000 t^4+2627137355819000 t^3+24963170210511000 t^2+126470059866327110 t\\ &\quad+263152531133638931\bigr)\bigr]\\ &\triangleq6298560 e^t R_6(t),\\ R_6'(t)&=2 \bigl[1475174400 e^{3 t} \bigl(5040947520000 t^5+143066891520000 t^4+1610253337392000 t^3\\ &\quad+9353990277828000 t^2+29604527713932570 t+42120460039614583\bigr)\\ &\quad-1568 e^{2 t} \bigl(9758278656000000 t^7+548415260467200000 t^6+12662119918506240000 t^5\\ &\quad+155423393361996480000 t^4+1094223403651360080000 t^3+4421419582352483390400 t^2\\ &\quad+9536350654306897882580 t+8570107070379722134301\bigr)-2 e^t \bigl(705188106000000 t^7\\ &\quad+58366704211200000 t^6+1982659848137640000 t^5+35715268500802380000 t^4\\ &\quad+366846046331463195000 t^3+2134948586374665545400 t^2+6455791809066348307930 t\\ &\quad+7699856382869985500211\bigr)-25 \bigl(567000000 t^6+62778240000 t^5+2791725120000 t^4\\ &\quad+63764415336000 t^3+788141206745700 t^2+4992634042102200 t+12647005986632711\bigr)\bigr],\\ R_6''(t)&=4 \bigl[2212761600 e^{3 t} \bigl(5040947520000 t^5+151468470720000 t^4+1801009192752000 t^3\\ &\quad+10964243615220000 t^2+35840521232484570 t+51988635944258773\bigr)\\ &\quad-1568 e^{2 t} \bigl(9758278656000000 t^7+582569235763200000 t^6+14307365699907840000 t^5\\ &\quad+187078693158262080000 t^4+1405070190375353040000 t^3+6062754687829523510400 t^2\\ &\quad+13957770236659381272980 t+13338282397533171075591\bigr)-e^t \bigl(705188106000000 t^7\\ &\quad+63303020953200000 t^6+2332860073404840000 t^5+45628567741490580000 t^4\\ &\quad+509707120334672715000 t^3+3235486725369055130400 t^2+10725688981815679398730 t\\ &\quad+14155648191936333808141\bigr)-7500 \bigl(5670000 t^5+523152000 t^4+18611500800 t^3\\ &\quad+318822076680 t^2+2627137355819 t+8321056736837\bigr)\bigr],\\ R_6^{(3)}(t)&=4 \bigl[6638284800 e^{3 t} \bigl(5040947520000 t^5+159870049920000 t^4+2002967153712000 t^3\\ &\quad+12765252807972000 t^2+43150016975964570 t+63935476355086963\bigr)\\ &\quad-3136 e^{2 t} \bigl(9758278656000000 t^7+616723211059200000 t^6+16055073407197440000 t^5\\ &\quad+222847107408031680000 t^4+1779227576691877200000 t^3+8170359973392553070400 t^2\\ &\quad+20020524924488904783380 t+20317167515862861712081\bigr)-e^t \bigl(705188106000000 t^7\\ &\quad+68239337695200000 t^6+2712678199124040000 t^5+57292868108514780000 t^4\\ &\quad+692221391300635035000 t^3+4764608086373073275400 t^2+17196662432553789659530 t\\ &\quad+24881337173752013206871\bigr)-7500 \bigl(28350000 t^4+2092608000 t^3+55834502400 t^2\\ &\quad+637644153360 t+2627137355819\bigr)\bigr],\\ R_6^{(4)}(t)&=4 \bigl[19914854400 e^{3 t} \bigl(5040947520000 t^5+168271629120000 t^4+2216127220272000 t^3\\ &\quad+14768219961684000 t^2+51660185514612570 t+78318815347075153\bigr)\\ &\quad-6272 e^{2 t} \bigl(9758278656000000 t^7+650877186355200000 t^6+17905243040375040000 t^5\\ &\quad+262984790926025280000 t^4+2224921791507940560000 t^3+10839201338430368870400 t^2\\ &\quad+28190884897881457853780 t+30327429978107314103771\bigr)-e^t \bigl(705188106000000 t^7\\ &\quad+73175654437200000 t^6+3122114225295240000 t^5+70856259104134980000 t^4\\ &\quad+921392863734694155000 t^3+6841272260274978380400 t^2+26725878605299936210330 t\\ &\quad+42077999606305802866401\bigr)-1800000 \bigl(472500 t^3+26157600 t^2+465287520 t+2656850639\bigr)\bigr],\\ R_6^{(5)}(t)&=4 \bigl[59744563200 e^{3 t} \bigl(5040947520000 t^5+176673208320000 t^4+2440489392432000 t^3\\ &\quad+16984347181956000 t^2+61505665489068570 t+95538877185279343\bigr)\\ &\quad-12544 e^{2 t} \bigl(9758278656000000 t^7+685031161651200000 t^6+19857874599440640000 t^5\\ &\quad+307747898526962880000 t^4+2750891373359991120000 t^3+14176584025692279710400 t^2\\ &\quad+39030086236311826724180 t+44422872427048043030661\bigr)-e^t \bigl(705188106000000 t^7\\ &\quad+78111971179200000 t^6+3561168151918440000 t^5+86466830230611180000 t^4\\ &\quad+1204817900151234075000 t^3+9605450851479060845400 t^2+40408423125849892971130 t\\ &\quad+68803878211605739076731\bigr)-108000000 \bigl(23625 t^2+871920 t+7754792\bigr)\bigr],\\ R_6^{(6)}(t)&=4 \bigl[179233689600 e^{3 t} \bigl(5040947520000 t^5+185074787520000 t^4+2676053670192000 t^3\\ &\quad+19424836574388000 t^2+72828563610372570 t+116040765681635533\bigr)\\ &\quad-25088 e^{2 t} \bigl(9758278656000000 t^7+719185136947200000 t^6+21912968084394240000 t^5\\ &\quad+357392585025564480000 t^4+3366387170413916880000 t^3+18302921085732266390400 t^2\\ &\quad+53206670262004106434580 t+63937915545203956392751\bigr)-e^t \bigl(705188106000000 t^7\\ &\quad+83048287921200000 t^6+4029839978993640000 t^5+104272670990203380000 t^4\\ &\quad+1550685221073678795000 t^3+13219904551932763070400 t^2+59619324828808014661930 t\\ &\quad+109212301337455632047861\bigr)-68040000000 (75 t+1384)\bigr],\\ R_6^{(7)}(t)&=4 \bigl[537701068800 e^{3 t} \bigl(5040947520000 t^5+193476366720000 t^4+2922820053552000 t^3\\ &\quad+22100890244580000 t^2+85778454659964570 t+140316953551759723\bigr)\\ &\quad-50176 e^{2 t} \bigl(9758278656000000 t^7+753339112243200000 t^6+24070523495235840000 t^5\\ &\quad+412175005236550080000 t^4+4081172340465045840000 t^3+23352501841353141710400 t^2\\ &\quad+71509591347736372824980 t+90541250676206009610041\bigr)-e^t \bigl(705188106000000 t^7\\ &\quad+87984604663200000 t^6+4528129706520840000 t^5+124421870885171580000 t^4\\ &\quad+1967775905034492315000 t^3+17871960215153799455400 t^2+86059133932673540802730 t\\ &\quad+168831626166263646709791\bigr)-5103000000000\bigr],\\ R_6^{(8)}(t)&=4 e^t \bigl[1613103206400 e^{2 t} \bigl(5040947520000 t^5+201877945920000 t^4 +3180788542512000 t^3\\ &\quad+25023710298132000 t^2+100512381489684570 t +168909771771747913\bigr)\\ &\quad-100352 e^t \bigl(9758278656000000 t^7 +787493087539200000 t^6+26330540831965440000 t^5\\ &\quad+472351313974639680000 t^4+4905522350938146000000 t^3 +29474260352050710470400 t^2\\ &\quad+94862093189089514535380 t +126296046350074196022531\bigr)-705188106000000 t^7\\ &\quad -92920921405200000 t^6-5056037334500040000 t^5 -147062519417775780000 t^4\\ &\quad-2465463388575178635000 t^3 -23775287930257276400400 t^2-121803054362981139713530 t \\ &\quad-254890760098937187512521\bigr]\\ &\triangleq 4 e^tR_7(t),\\ R_7'(t)&=14 \bigl[460886630400 e^{2 t} \bigl(2520473760000 t^5+107240157360000 t^4 +1792272217176000 t^3\\ &\quad+14897446555950000 t^2+62768045893908285 t +109582981258295099\bigr)\\ &\quad -7168 e^t \bigl(9758278656000000 t^7+855801038131200000 t^6+31055499357200640000 t^5 \\ &\quad+604004018134466880000 t^4+6794927606836704720000 t^3 +44190827404865148470400 t^2\\ &\quad+153810613893190935476180 t +221158139539163710557911\bigr)-5 \bigl(70518810600000 t^6 \\ &\quad+7964650406160000 t^5+361145523892860000 t^4 +8403572538158616000 t^3\\ &\quad+105662716653221941500 t^2 +679293940864493611440 t+1740043633756873424479\bigr)\bigr],\\ R_7''(t)&=784 \bigl[8230118400 e^{2 t} \bigl(5040947520000 t^5+227082683520000 t^4+4013505063792000 t^3\\ &\quad+35171709763428000 t^2+155330984899716570 t+281934008410498483\bigr)\\ &\quad-128 e^t \bigl(9758278656000000 t^7+924108988723200000 t^6+36190305585987840000 t^5\\ &\quad+759281514920470080000 t^4+9210943679374572240000 t^3+64575610225375262630400 t^2\\ &\quad+242192268702921232416980 t+374968753432354646034091\bigr)-225 \bigl(167901930000 t^5\\ &\quad+15802877790000 t^4+573246863322000 t^3+10004253021617400 t^2+83859298931128525 t\\ &\quad+269561087644640322\bigr)\bigr],\\ R_7^{(3)}(t)&=784 \bigl[32920473600 e^{2 t} \bigl(2520473760000 t^5+119842526160000 t^4+2233835215416000 t^3\\ &\quad+20595983679558000 t^2+95251347331572285 t+179799750430178384\bigr)\\ &\quad-128 e^t \bigl(9758278656000000 t^7+992416939315200000 t^6+41734959518327040000 t^5\\ &\quad+940233042850409280000 t^4+12248069739056452560000 t^3+92208441263498979350400 t^2\\ &\quad+371343489153671757677780 t+617161022135275878451071\bigr)-39375 \bigl(4797198000 t^4\\ &\quad+361208635200 t^3+9827089085520 t^2+114334320247056 t+479195993892163\bigr)\bigr],\\ R_7^{(4)}(t)&=12544 \bigl[2057529600 e^{2 t} \bigl(5040947520000 t^5+252287421120000 t^4+4947040535472000 t^3\\ &\quad+47893473005364000 t^2+231694662022260570 t+454850848191929053\bigr)\\ &\quad-8 e^t \bigl(9758278656000000 t^7+1060724889907200000 t^6+47689461154218240000 t^5\\ &\quad+1148907840442044480000 t^4+16009001910458089680000 t^3+128952650480668337030400 t^2\\ &\quad+555760371680669716378580 t+988504511288947636128851\bigr)-2480625 \bigl(19036500 t^3\\ &\quad+1075025700 t^2+19498192630 t+113426905007\bigr)\bigr],\\ R_7^{(5)}(t)&=25088 \bigl[4115059200 e^{2 t} \bigl(2520473760000 t^5+132444894960000 t^4+2725807688856000 t^3\\ &\quad+27657016904286000 t^2+139794067513812285 t+285349089601529669\bigr)\\ &\quad-4 e^t \bigl(9758278656000000 t^7+1129032840499200000 t^6+54053810493661440000 t^5\\ &\quad+1387355146213135680000 t^4+20604633272226267600000 t^3+176979656212042606070400 t^2\\ &\quad+813665672642006390439380 t+1544264882969617352507431\bigr)-86821875 \bigl(815850 t^2\\ &\quad+30715020 t+278545609\bigr)\bigr],\\ R_7^{(6)}(t)&=100352 \bigl[1028764800 e^{2 t} \bigl(5040947520000 t^5+277492158720000 t^4+5981394957552000 t^3\\ &\quad+63491456875140000 t^2+334902168836196570 t+710492246716871623\bigr)\\ &\quad-e^t \bigl(9758278656000000 t^7+1197340791091200000 t^6+60828007536656640000 t^5\\ &\quad+1657624198681442880000 t^4+26154053857078810320000 t^3+238793556028721408870400 t^2\\ &\quad+1167624985066091602580180 t+2357930555611623742946811\bigr)-27348890625 (1295 t+24377)\bigr],\\ R_7^{(7)}(t)&=100352 \bigl[4115059200 e^{2 t} \bigl(2520473760000 t^5+145047263760000 t^4+3268189637496000 t^3\\ &\quad+36231774655734000 t^2+199196812855668285 t+438971665567484954\bigr)\\ &\quad-e^t \bigl(9758278656000000 t^7+1265648741683200000 t^6+68012052283203840000 t^5\\ &\quad+1961764236364726080000 t^4+32784550651804581840000 t^3+317255717599957839830400 t^2\\ &\quad+1645212097123534420320980 t+3525555540677715345526991\bigr)-35416813359375\bigr],\\ R_7^{(8)}(t)&=100352 e^t \bigl[4115059200 e^t \bigl(5040947520000 t^5+302696896320000 t^4+7116568330032000 t^3\\ &\quad+82268118223956000 t^2+470857175022804570 t+1077140143990638193\bigr)-9758278656000000 t^7\\ &\quad-1333956692275200000 t^6-75605944733303040000 t^5-2301824497780745280000 t^4\\ &\quad-40631607597263486160000 t^3-415609369555371585350400 t^2-2279723532323450099981780 t\\ &\quad-5170767637801249765847971\bigr]\\ &\triangleq 100352 e^tR_8(t),\\ R_8'(t)&=980 \bigl[4199040 e^t \bigl(5040947520000 t^5+327901633920000 t^4+8327355915312000 t^3\\ &\quad+103617823214052000 t^2+635393411470716570 t+1547997319013442763\bigr)\\ &\quad-69701990400000 t^6-8167081789440000 t^5-385744615986240000 t^4-9395202031758144000 t^3\\ &\quad-124382472236520876000 t^2-848182386847697112960 t-2326248502370867448961\bigr],\\ R_8''(t)&=2822400 \bigl[1458 e^t \bigl(5040947520000 t^5+353106371520000 t^4+9638962450992000 t^3\\ &\quad+128599890959988000 t^2+842629057898820570 t+2183390730484159333\bigr)-49 \bigl(2963520000 t^5\\ &\quad+289366560000 t^4+10933804308000 t^3+199727934348600 t^2+1762790139406475 t\\ &\quad+6010362718591958\bigr)\bigr],\\ R_8^{(3)}(t)&=2822400 \bigl[1458 e^t \bigl(5040947520000 t^5+378311109120000 t^4+11051387937072000 t^3\\ &\quad+157516778312964000 t^2+1099828839818796570 t+3026019788382979903\bigr)-1225 \bigl(592704000 t^4\\ &\quad+46298649600 t^3+1312056516960 t^2+15978234747888 t+70511605576259\bigr)\bigr],\\ R_8^{(4)}(t)&=50803200 \bigl[81 e^t \bigl(5040947520000 t^5+403515846720000 t^4+12564632373552000 t^3\\ &\quad+190670942124180000 t^2+1414862396444724570 t+4125848628201776473\bigr)\\ &\quad-68600 \bigl(2352000 t^3+137793600 t^2+2603286740 t+15851423361\bigr)\bigr],\\ R_8^{(5)}(t)&=50803200 \bigl[81 e^t \bigl(5040947520000 t^5+428720584320000 t^4+14178695760432000 t^3\\ &\quad+228364839244836000 t^2+1796204280693084570 t+5540711024646501043\bigr)\\ &\quad-1372000 \bigl(352800 t^2+13779360 t+130164337\bigr)\bigr],\\ R_8^{(6)}(t)&=457228800 \bigl[9 e^t \bigl(5040947520000 t^5+453925321920000 t^4+15893578097712000 t^3\\ &\quad+270900926526132000 t^2+2252933959182756570 t+7336915305339585613\bigr)\\ &\quad-1536640000 (70 t+1367)\bigr],\\ R_8^{(7)}(t)&=457228800 \bigl[9 e^t \bigl(5040947520000 t^5+479130059520000 t^4+17709279385392000 t^3\\ &\quad+318581660819268000 t^2+2794735812235020570 t+9589849264522342183\bigr)-107564800000\bigr],\\ R_8^{(8)}(t)&=4115059200 e^t \bigl(5040947520000 t^5+504334797120000 t^4+19625799623472000 t^3\\ &\quad+371709498975444000 t^2+3431899133873556570 t+12384585076757362753\bigr)\\ &>0, \end{align*} and \begin{gather*} R^{(k)}(0)=0,\quad 0\le k\le6; \quad R_1^{(k)}(0)=0,\quad 0\le k\le8;\quad R_2(0)=0, \quad R_2'(0)=94594500, \\ R_2''(0)=5675670000,\quad R_2^{(3)}(0)=186994407600, \quad R_2^{(4)}(0)=4870671886080, \\ R_2^{(5)}(0)=115977230617248, \quad R_2^{(6)}(0)=2568045873032640, \quad R_2^{(7)}(0)=51468046246929312,\\ R_2^{(8)}(0)=919261925834668416; \quad R_3(0)=2127921124617288,\quad R_3'(0)=31808262868948566,\\ R_3''(0)=422643933990433848,\quad R_3^{(3)}(0)=5059127238393038844,\quad R_3^{(4)}(0)=55311317976132200928,\\ R_3^{(5)}(0)=559358124246419710080,\quad R_3^{(6)}(0)=5291254778485384410240,\\ R_3^{(7)}(0)=47275711643630010675648,\quad R_3^{(8)}(0)=402327284442724791735744;\\ R_4(0)=25145455277670299483484, \quad R_4'(0)=180175476735597623990958, \\ R_4''(0)=1233364679292092331540240, \quad R_4^{(3)}(0)=8128711084294241029973952, \\ R_4^{(4)}(0)=51948114141680687652017184, \quad R_4^{(5)}(0)=323999742368987506576507680, \\ R_4^{(6)}(0)=1983712454318091617900255520, \quad R_4^{(7)}(0)=11984168286345820788017615520, \\ R_4^{(8)}(0)=71752827530169076522475337120;\quad R_5(0)=448455172063556728265470857, \\ R_5'(0)=2222062109238254700527363412, \quad R_5''(0)=10966377777101822969342094780, \\ R_5^{(3)}(0)=54073563048190450871500371120, \quad R_5^{(4)}(0)=266769748534576065247741326480, \\ R_5^{(5)}(0)=1316887765697877890640258458880, \quad R_5^{(6)}(0)=6499339666080758476651443438720, \\ R_5^{(7)}(0)=32030330270749184706488159641920, \quad R_5^{(8)}(0)=157419994070058290561168403698880;\\ R_6(0)=24993013334803239242170973,\quad R_6'(0)=97363392902732249529746070,\\ R_6''(0)=376439498975614717003617884,\quad R_6^{(3)}(0)=1442729528728439890600508052,\\ R_6^{(4)}(0)=5477808342361637707363220348,\quad R_6^{(5)}(0)=20602476681796080169865157140,\\ R_6^{(6)}(0)=76776923758516348244316687404,\quad R_6^{(7)}(0)=283621637079942700292207261572,\\ R_6^{(8)}(0)=1039178314812864090945470799068;\quad R_7(0)=259794578703216022736367699767,\\ R_7'(0)=684880850316419203514174536198,\quad R_7''(0)=1781525699945699167445177663968,\\ R_7^{(3)}(0)=4578635505175348952464533634608,\quad R_7^{(4)}(0)=11640343378188197428722427451648,\\ R_7^{(5)}(0)=29304071938148606553453533106688,\quad R_7^{(6)}(0)=73113605516101190505198264803328,\\ R_7^{(7)}(0)=180921493369266920211167500832768,\quad R_7^{(8)}(0)=444290887449456146125411745425408;\\ R_8(0)=4427324691580199160210177629,\quad R_8'(0)=6367820885649279115456587820,\\ R_8''(0)=8983950893934449574356812800,\quad R_8^{(3)}(0)=12452006779921850992338297600,\\ R_8^{(4)}(0)=16978056111501152993675481600,\quad R_8^{(5)}(0)=22800344803799642868441945600,\\ R_8^{(6)}(0)=30191839866409652462819289600,\quad R_8^{(7)}(0)=39462797393404173379462233600,\\ R_8^{(8)}(0)=50963300758293091764469977600. \end{gather*} The proof of Theorem~\ref{prop-exp-am} is thus completed. \section{Remarks} \begin{rem} The method in the proof of Theorem~\ref{prop-exp-am} has been effectively used in several articles such as~\cite{x-4-di-tri-gamma-p(x)-Slovaca.tex, exp-beograd, Mortici-monoburn.tex, MIA-3997.tex}. Generally speaking, this method may be applied to deal with the functions like \begin{equation} Q_{n;m_0,m_1,\dotsc,m_n}(t)=\sum_{k=0}^ne^{kt}P_{m_k}(t), \end{equation} where $n,m_k\in\mathbb{N}\cup\{0\}$ and $P_{m_k}(t)$ are polynomials of degree $m_k$. \end{rem} \begin{rem} With the help of the famous software \textsc{Mathematica}, we may obtain that \begin{equation} R(t)=\frac1{216}t^{15}+\frac1{48}t^{16}+\frac{71}{1440} t^{17}+\frac{38599}{453600} t^{18}+\frac{1114147}{9072000} t^{19}+\frac{1414531}{9072000} t^{20}+\frac{200428987}{1143072000} t^{21}+\dotsm. \end{equation} This implies that Theorem~\ref{prop-exp-am} may also be proved by carefully expanding the function $R(t)$ into the power series at the point $0$ and minutely verifying the positivity of the coefficients of $t^k$ for $k\ge15$. \end{rem} \begin{rem} This paper is abstracted/adapted from the preprint~\cite[Appendix]{degree-4-tri-tetra-gamma.tex}. \end{rem} \begin{thebibliography}{99} \bibitem{psi-proper-fraction-degree-two.tex} B.-N. 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