\documentclass[12pt,reqno]{article} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \usepackage{epsf,psfig} \usepackage{amsmath,amssymb} \usepackage{color} \def\Ei{\mbox{\rm Ei\,}} \def\NN{{\mathbb N}} \def\RR{{\mathbb R}} \def\Re{\mbox{\rm Re}\,} \def\Im{\mbox{\rm Im}\,} \def\dj{d\kern-0.4em\char"16\kern-0.1em} \def\Dj{\mbox{\raise0.3ex\hbox{-}\kern-0.4em D}} \def\ra{\; \; \;} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\topmargin}{-.5in} \setlength{\textheight}{8.9in} \newcommand{\seqnum}[1]{\href{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=#1}{\underline{#1}}} \newtheorem{theorem}{Theorem} \newtheorem{tabel}{Table} \newtheorem{definition}{Definition} \newtheorem{lemma}{Lemma} \newtheorem{corollary}{Corollary} \newtheorem{remark}{Remark} \renewcommand{\thefootnote}{} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \vskip .5in \begin{center} {\LARGE \bf The function $_{\lowercase{v}} M_{\lowercase{m}}(\lowercase{s};\lowercase{a},\lowercase{z})$ and some well-known sequences}\vskip 1.5cm {\large Aleksandar Petojevi\'{c}} \smallskip \\ Aleja Mar\v{s}ala Tita 4/2\\ 24000 Subotica\\ Yugoslavia\medskip \\ Email address:\; \href{mailto:apetoje@ptt.yu}{apetoje@ptt.yu}\\ \vskip2cm {\bf Abstract} \end{center} \footnotetext{This work was supported in part by the Serbian Ministry of Science, Technology and Development under Grant \# 2002: {\it Applied Orthogonal Systems, Constructive Approximation and Numerical Methods}.} {\em In this paper we define the function $_vM_m(s;a,z)$, and we study the special cases $_1M_m(s;a,z)$ and $_nM_{-1}(1;1,n+1)$. We prove some new equivalents of Kurepa's hypothesis for the left factorial. Also, we present a generalization of the alternating factorial numbers.} \section{Introduction} Studying the Kurepa function \[K(z)= \ !z =\int_{0}^{+\infty}\frac{t^{z}-1}{t-1}\,e^{-t}\,dt \quad (\Re z>0),\] G. V. Milovanovi\'{c} gave a generalization of the function \[M_m(z)= \int_{0}^{+\infty}\frac{t^{z+m}-Q_{m}(t;z)}{(t-1)^{m+1}}e^{-t}dt\quad (\Re z > -(m+1)),\] where the polynomials $Q_{m}(t;z)$, $m = -1,0,1,2,...$ are given by \[Q_{-1}(t;z)=0 \quad Q_{m}(t;z)=\sum_{k=0}^{m} {m+z \choose k}(t-1)^k.\] The function $\{ M_{m}(z)\}_{m=-1}^{+\infty}$ has the integral representation \[M_m(z)=\frac{z(z+1)\cdots(z+m)}{m!}\int_0^1\xi^{z-1}(1-\xi)^m e^{(1-\xi)/\xi}\Gamma\Bigl(z,\frac{1-\xi}{\xi}\Bigr)\,d\xi,\] where $\Gamma(z,x)$, the incomplete gamma function, is defined by \begin{equation}\label{i1} \Gamma(z,x)=\int_x^{+\infty}t^{z-1}e^{-t}\,dt. \end{equation} Special cases include \begin{equation}\label{po1} M_{-1}(z)=\Gamma(z) \qquad \mbox{and}\qquad M_0(z)=K(z), \end{equation} where $\Gamma(z)$ is the gamma function \[\Gamma(z)=\int_{0}^{+\infty}t^{z-1}e^{-t}\,dt. \] The numbers $M_m(n)$ were introduced by Milovanovi\'c \cite{Mil2} and Milovanovi\'c and Petojevi\'{c} \cite{MiPe}. For non-negative integers $n,m\in \NN$ the following identities hold: \[ M_m(0)=0,\quad M_{m}(n)=\sum_{i=0}^{n-1}\frac{(-1)^i}{i!}\sum_{k=i}^{n-1}k!{m+n\choose k+m+1}. \] For the numbers $M_m(n)$ the following relations hold: \begin{eqnarray*} M_m(n+1)&=&n!+ \sum_{\nu=0}^m M_\nu(n),\\[3mm] \lim_{n\to+\infty}\frac{M_\nu(n)}{M_{\nu-1}(n)}&=&1,\\[3mm] \lim_{n\to+\infty}\frac{M_m(n)}{(n-1)!}&=&1. \end{eqnarray*} The generating function of the numbers $\{M_m(n)\}_{n=0}^{+\infty}$ is given by \[\frac1{m!}\left(A_m(x)e^{x-1}\left(\Ei(1)-\Ei(1-x)+B_m(x)e^x-C_m(x)\right)\right) =\sum_{n=0}^{+\infty}M_m(n)\,\frac{x^n}{n!}\,\] where $A_m(x)$, $B_m(x)$, and $C_m(x)$ are polynomials defined as follows: \begin{eqnarray*} \frac{A_m(x)}{m!}&=& \sum_{k=0}^m{\binom{m}{k}}\frac{(x-1)^k}{k!}\,,\\ [3mm] \frac{B_m(x)}{m!}&=& \sum_{\nu=0}^{m-1}\left(\sum_{k=1}^{m-\nu} {\binom{m}{k+\nu}}\frac{(-1)^{k-1}}{k}\sum_{j=0}^{k-1}\frac{(-1)^j}{j!}\right) \frac{(x-1)^\nu}{\nu!}\,,\\ [3mm] \frac{C_m(x)}{m!}&=& \sum_{j=0}^{m-1}\left(\sum_{\nu=0}^{j}(-1)^\nu {\binom{j}{\nu}}\sum_{k=j+1}^{m}\frac{(-1)^{k-1}}{k-\nu} {\binom{m}{k}}\right)\frac{(x-1)^j}{j!}\,, \end{eqnarray*} Here $\Ei(x)$ is the exponential integral defined by \begin{equation}\label{ei} \Ei(x)=\mbox{\rm p.v.}\int_{-\infty}^{x}\frac{e^t}{t}\,dt \qquad(x>0). \end{equation} In this paper, we give a generalization of the function $M_m(z)$ which we denote as $_vM_m(s;a,z)$. These generalization are of interest because its special cases include: \begin{eqnarray*} _1M_1(1;1,n+1)&=&n!,\\[3mm] _1M_0(1;1,n)&=&!n,\\[3mm] _nM_{-1}(1;1,n+1)&=&A_n. \end{eqnarray*} where $n!$, $!n$, and $A_n$ are the {\it right factorial numbers} (sequence A000142 in \cite{Slo}), the {\it left factorial numbers} (sequence A003422 in \cite{Slo}) and the {\it alternating factorial numbers} (sequence A005165 in \cite{Slo}), respectively. They are defined as follows: \begin{equation}\label{eq1.1} 0!=1,\; n!=n\cdot (n-1)!; \qquad \ !0=0, \; \ !n=\sum_{k=0}^{n-1}k! \quad\mbox{and}\quad A_n=\sum_{k=1}^{n}(-1)^{n-k}k!. \end{equation} \section{Definitions} We now introduce a generalization of the function $M_m(z)$. \vspace{3mm} \begin{definition}\label{d1} For $m=-1,0,1,2,...,$ and $\Re z> v-m-2$ the function $_vM_m(s;a,z)$ is defined by \begin{eqnarray*} &&_vM_m(s;a,z)=\\[3mm] &&=\sum_{k=1}^v\frac{(-1)^{2v+1-k}\Gamma(m+z+2-k)}{\Gamma(z+1-k)\Gamma(m+2)}\, \mathcal{L}[s;\,{}_2F_1(a,k-z,m+2,1-t)], \end{eqnarray*} where $v$ is a positive integer, and $s,\; a,\; z$ are complex variables. \end{definition}\vspace{3mm} The hypergeometric function $_2F_1(a,b;c,x)$ is defined by the series \[ _2F_1(a,b,c;x)=\sum_{n=0}^\infty \frac{(a)_n(b)_n}{(c)_n}\frac{x^n}{n!}\qquad (|x|<1), \] and has the integral representation \[_2F_1(a,b,c;x)=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} \int_0^1t^{b-1}(1-t)^{c-b-1}(1-tx)^{-a}dt,\] in the $x$ plane cut along the real axis from $1$ to $\infty$, if $\Re c>\Re b>0$.\\ The symbols $(z)_n$ and $\mathcal{L}[s;F(t)]$ represent the Pochhammer symbol \[(z)_n=\frac{\Gamma(z+n)}{\Gamma(z)},\] and Laplace transform \[\mathcal{L}[s;F(t)]=\int_0^\infty e^{-st}F(t)dt.\] \begin{table}[ht]\label{table1} \caption{The numbers $_1M_m(1;a,n)$ for $m=1,2,3,4$ and $a=0,1,2$}\vspace{3mm} \begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}llllll} \hline $a=0$&$\mbox{in \cite{Slo}}$&$a=1$&$\mbox{in \cite{Slo}}$&$a=2$&$\mbox{in \cite{Slo}}$\cr \hline $_1M_{1}(1,0,n)$&$A000217$&$_1M_{1}(1,1,n)$&$A014144$&$_1M_{1}(1,2,n)$&$A007489$\cr \\ $_1M_{2}(1,0,n)$&$A000292$&$_1M_{2}(1,1,n)$&$\mbox{unlisted}$&$_1M_{2}(1,2,n)$&$A014145$\cr \\ $_1M_{3}(1,0,n)$&$A000332$&$_1M_{3}(1,1,n)$&$\mbox{unlisted}$&$_1M_{3}(1,2,n)$&$\mbox{unlisted}$\cr \\ $_1M_{4}(1,0,n)$&$A000389$&$_1M_{4}(1,1,n)$&$\mbox{unlisted}$&$_1M_{4}(1,2,n)$&$\mbox{unlisted}$ \\ \hline \end{tabular*} \end{table} The term ``unlisted" in Table 1 means that the sequence cannot currently be found in Sloane's on-line encyclopedia of integer sequences\cite{Slo}.\vspace{3mm} \begin{lemma}\label{l1} Let $m=-1,0,1,2,\ldots\; .$ Then \[_1M_m(1;1,z)=M_m(z).\] \end{lemma}\vspace{3mm} \noindent {\it Proof.} \, The proof presented here is due to Professor G. V. Milovanovi\'{c}. \\ Since \[(k+m+1)!=(m+1)!(m+2)_k\quad\mbox{and}\quad (1-z)_k=\frac{(-1)^k\Gamma(z)}{\Gamma(z-k)},\] we have \[{m+z\choose k+m+1}=\frac{\Gamma(m+z+1)}{\Gamma(z-k)(k+m+1)!}= \frac{\Gamma(m+z+1)}{\Gamma(z)(m+1)!}\cdot\frac{(1-z)_k(-1)^k}{(m+2)_k}\,,\] so that \begin{eqnarray*} \frac{t^{z+m}-Q_{m}(t;z)}{(t-1)^{m+1}}&=&\sum_{k=0}^{+\infty} {m+z\choose k+m+1}(t-1)^k\qquad (|t-1|<1)\\[3mm] &=&\frac{\Gamma(m+z+1)}{\Gamma(z)(m+1)!}\sum_{k=0}^{+\infty} \frac{(1-z)_k(1)_k}{(m+2)_k}\cdot\frac{(1-t)^k}{k!}\\ [3mm] &=&\frac{\Gamma(m+z+1)}{\Gamma(z)(m+1)!}\,{}_2F_1(1,1-z,m+2;1-t). \end{eqnarray*} \section{The function $_{\lowercase{1}}M_{\lowercase{m}}(\lowercase{s};\lowercase{a}, \lowercase{z})$} \subsection{The numbers $\{_{\lowercase{1}}M_{\lowercase{m}}(1;-\lowercase{n},\lowercase{r})\} _{r=0}^{+\infty}\, _{\lowercase{n=0}}^{+\infty}\, _{\lowercase{m=-1}}^{+\infty}$} We have \[_1M_{m}(1;-n,z)=\frac{\Gamma(m+z+1)}{\Gamma(z)\Gamma(m+2)}\, \mathcal{L}[1;\,{}_2F_1(-n,1-z,m+2;1-t)],\] starting with the polynomials \[ \sum_{k=0}^\infty {m+z\choose k+m+1}{n\choose k}(1-t)^k, \quad n\in \NN.\] Since \[(-n)_k=(-1)^k\frac{n!}{(n-k)!}\] we have \begin{equation}\label{s1} \sum_{k=0}^\infty {m+z\choose k+m+1}{n\choose k}(1-t)^k=\frac{\Gamma(m+z+1)}{\Gamma(z)\Gamma(m+2)}\,{}_2F_1(-n,1-z,m+2;1-t) \end{equation} or, by continuation, \[ =\pi\; {\rm cosec}\;\pi z \int_0^1\xi^{1-z}(1-\xi)^{m+z+1} \bigl(1-(1-t)\xi\bigr)^{n}d\xi.\] Hence, the following definition is reasonable.\vspace{3mm} \begin{definition}\label{d2} For $n\in \NN$ and $m=-1,0,1,2,...,$ the polynomials $z\mapsto {_mP_n(z)}$ are defined by \[_mP_n(z)=\mathcal{L}[1;\,{}_2F_1(-n,1-z,m+2;1-t)].\] \end{definition} \begin{table}[ht]\label{table2} \caption{The polynomials $_{m}P_2(z)$, $_{m}P_3(z)$ and $_{m}P_4(z)$ }\vspace{3mm} \begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}llll} \hline $m$&$_{m}P_2(z)$&$_{m}P_3(z)$&$_{m}P_4(z)$\cr \hline $-1$&$\frac{1}{2}z^2-\frac{3}{2}z+2$&$-\frac{1}{3}z^3+\frac{7}{2}z^2-\frac{49}{6}z+6$& $\frac{3}{8}z^4-\frac{61}{12}z^3+\frac{193}{8}z^2-$\cr \\ $\;\;\;0$&$\frac{1}{6}z^2-\frac{1}{2}z+\frac{4}{3}$& $-\frac{1}{12}z^3+z^2-\frac{29}{12}z+\frac{5}{2}$&$-\frac{509}{12}z+24 $\cr \\ $\;\;\;1$&$\frac{1}{12}z^2-\frac{1}{4}z+\frac{7}{6}$& $-\frac{1}{30}z^3+\frac{9}{20}z^2-\frac{67}{60}z+\frac{17}{10}$&$ $\cr \hline \end{tabular*} \end{table}\vspace{3mm} The polynomials $_mP_n(z)$ can be expressed in terms of the {\it derangement numbers\/} (sequence A000166 in \cite{Slo}) \[S_k=k!\sum_{\nu=0}^k\frac{(-1)^\nu}{\nu!}\qquad (k\ge0)\] in the form \begin{theorem}\label{t1}For $m=-1,0,1,2,...$ and $n\in \NN$ we have \[_mP_n(z)={n+m+1\choose n}^{-1}\sum_{k=0}^\infty{n+m+1\choose k+m+1}{z-1\choose k}(-1)^kS_k.\] \end{theorem}\vspace{3mm} \noindent {\it Proof.} \, Using the relation (\ref{s1}) we have \begin{eqnarray*} _mP_n(z)&=&\frac{\Gamma(z)\Gamma(m+2)}{\Gamma(m+z+1)} \int_0^\infty e^{-t}\sum_{k=0}^\infty {m+z\choose k+m+1}{n\choose k}(1-t)^kdt\\[3mm] &=&\frac{\Gamma(z)(m+1)!}{\Gamma(m+z+1)} \sum_{k=0}^\infty \frac{\Gamma(m+z+1)}{\Gamma(z-k)(k+m+1)!}{n\choose k}\int_0^\infty e^{-t}(1-t)^kdt\\[3mm] &=&{n+m+1\choose n}^{-1}\sum_{k=0}^\infty{n+m+1\choose k+m+1}{z-1\choose k}\int_0^\infty e^{-t}(1-t)^kdt. \end{eqnarray*} Now use \[\mathcal{L}[s;(t+\alpha)^{z-1}]=\frac{e^{\alpha s}\Gamma(z,\alpha s)}{s^z}\qquad (\Re s>0),\] to obtain \[_mP_n(z)={n+m+1\choose n}^{-1}\sum_{k=0}^\infty{n+m+1\choose k+m+1}{z-1\choose k}(-1)^k\frac{\Gamma(k+1,-1)}{e}.\] Here $\Gamma(z,x)$ is the incomplete gamma function. The result follows from \[ \Gamma(k+1,-1)=eS_k. \] \begin{lemma}\label{le1}For $n\in \NN$ we have \[_{-1}P_n(z)=1+n!\sum_{k=1}^n\frac{(-1)^kS_k}{(n-k)!\,(k!)^2}\prod_{i=1}^k(z-i).\] \end{lemma}\vspace{3mm} \noindent {\it Proof.} \, Applying Theorem \ref{t1} for $m=-1$, we have \begin{eqnarray*} _{-1}P_n(z)&=&\sum_{k=0}^n{n\choose k}{z-1\choose k}(-1)^kS_k,\\[3mm] &=&1+n!\sum_{k=1}^n\frac{(-1)^kS_k}{(n-k)!(k!)^2}\prod_{i=1}^k(z-i). \end{eqnarray*}\vspace{3mm} \begin{remark} Let the sequence $X_{n,k}$ be defined by \begin{displaymath} X_{n,k} = \begin{cases} Y_n, & \quad {\rm if}\ n=k, \\ (n-k)X_{n-1,k}, & \quad {\rm if}\ n>k, \end{cases} \end{displaymath} \\ where $Y_n=(n!)^2.$ Since $X_{n,k}=(n-k)!\,(k!)^2$ we have \[_{-1}P_n(z)=1+n!\sum_{k=1}^n\frac{(-1)^kS_k}{X_{n,k}}\prod_{i=1}^k(z-i).\] \end{remark}\vspace{3mm} \begin{theorem}\label{t2}For $m=-1,0,1,2,...$ and $n\in \NN$ we have \begin{eqnarray*} _mP_0(z)&=& _mP_1(z)=1\\[3mm] _mP_n(z)&=& \frac{1}{n+m+1}\,{}[(m+1)\,{}_{m-1}P_n(z)+n\,{} _mP_{n-1}(z)],\quad m>-1. \end{eqnarray*} \end{theorem}\vspace{3mm} \noindent {\it Proof.} \, According to Theorem \ref{t1} we have \[\frac{1}{n+m+1}\,{}[(m+1)\,{}_{m-1}P_n(z)+n\,{} _mP_{n-1}(z)]=\] \begin{eqnarray*} &&\frac{(m+1)!n!}{(n+m+1)!}\sum_{k=0}^\infty{n+m\choose k+m}{z-1\choose k}(-1)^kS_k+\\[3mm] &+&\frac{(m+1)!n!}{(n+m+1)!}\sum_{k=0}^\infty{n+m\choose k+m+1}{z-1\choose k}(-1)^kS_k . \end{eqnarray*} The recurrence for $_mP_n(z)$ now follows from \[{a\choose b}+{a\choose b+1}={a+1\choose b+1}.\]\vspace{3mm} \begin{corollary}\label{co1} For $m=-1,0,1,2,...$ and $n\in \NN$ we have \[_1M_m(1;-n,z)=\frac{n!\, z(z+1)\ldots (z+m)}{(n+m+1)!}\sum_{k=0}^\infty{n+m+1\choose k+m+1}{z-1\choose k}(-1)^kS_k.\] \end{corollary}\vspace{3mm} \begin{corollary}\label{co2} For $n\in \NN$ the result is as follows \begin{eqnarray*} _1M_{-1}(1;-n,z)&=&1+n!\sum_{k=1}^n\frac{(-1)^kS_k}{(n-k)!\,(k!)^2}\prod_{i=1}^k(z-i),\\[3mm] _1M_{m}(1;0,z)&=&\,_1M_{m}(1;-1,z)=\frac{\Gamma(m+z+1)}{\Gamma(z)\Gamma(m+2)},\\[3mm] _1M_{m}(1;-n,z)&=&\frac{1}{n+m+1}\cdot[\,(m+z)\cdot\, _1M_{m-1}(1;-n,z)+\\[2mm] &+&n\cdot \, _1M_m(1;-n+1,z)], \quad m>-1. \end{eqnarray*} \end{corollary}\vspace{3mm} The numbers $_1M_m(1;-n,r)_{r=0}^{\infty}$ can now be evaluated recursively. \vspace{3mm} \begin{table}[ht]\label{table3} \caption{The numbers $_1M_m(1;-n,r)$ for $m=0,1,2,3,4$ and $n=1,2$}\vspace{3mm} \begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}llll} \hline $_1M_{m}(1,-n,r)$&$\mbox{sequence in \cite{Slo}}$&$_1M_{m}(1,-n,r)$&$\mbox{sequence in \cite{Slo}}$\cr \hline $_1M_{0}(1,-1,r)$&$0,1,2,3,4...$&$_1M_{0}(1,-2,r)$&$A000125$\cr \\ $_1M_{1}(1,-1,r)$&$A000217$&$_1M_{1}(1,-2,r)$&$A055795$\cr \\ $_1M_{2}(1,-1,r)$&$A000292$&$_1M_{2}(1,-2,r)$&$A027660$\cr \\ $_1M_{3}(1,-1,r)$&$A000332$&$_1M_{3}(1,-2,r)$&$A055796$\cr \\ $_1M_{4}(1,-1,r)$&$A000389$&$_1M_{4}(1,-2,r)$&$A055797$ \\ \hline \end{tabular*} \end{table} \subsection{The numbers $\{_{\lowercase{1}}M_{\lowercase{m}} (1/\lowercase{n};\lowercase{m+2},\lowercase{r}) \}_{\lowercase{r=0}}^{+\infty}\, _{\lowercase{n=1}}^{+\infty}\, _{\lowercase{m=-1}}^{+\infty}$} In Table 4 twelve well-known sequences from \cite{Slo} are given. These sequences are special cases of the function $_vM_m(s;a,r)$ for $v=1,\, s=1/n,$ and $a=m+2$. The sequences have the following common characteristic.\vspace{3mm} \begin{lemma} For $m=-1,0,1,2,...,$ we have \[_1M_{m}(1/n;m+2,z)=\frac{n^z\Gamma(z+m+1)}{(m+1)!}.\] \end{lemma}\vspace{3mm} \noindent {\it Proof.} \, Since \[_2F_1(m+2,1-z,m+2,1-t)=t^{z-1}\] we have \[\mathcal{L}[1/n;\,{}_2F_1(m+2,1-z,m+2,1-t)]=n^z\Gamma(z).\] \begin{table}[htb]\label{table4} \caption{The numbers $_1M_{m}(1/n;m+2,r)$ for $m=-1,0,1,2,3,4$ and $n=1,2,3,4$}\vspace{3mm} \begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}llll} \hline $_1M_{m}(1/n,m+2,r)$&$\mbox{in \cite{Slo}}$&$_1M_{m}(1/n,m+2,r)$&$\mbox{in \cite{Slo}}$\cr \hline $_1M_{-1}(1,1,r)$&$A000142$&$_1M_{-1}(1/2,1,r)$&$A066318$\cr \\ $_1M_{-1}(1/3,1,r)$&$A032179$&$_1M_{-1}(1/4,1,r)$&$\mbox{unlisted}$\cr \\ $_1M_{0}(1,2,r)$&$A000142$&$_1M_{0}(1/2,2,r)$&$A000165$\cr \\ $_1M_{0}(1/3,2,r)$&$A032031$&$_1M_{0}(1/4,2,r)$&$A047053$\cr \\ $_1M_{1}(1,3,r)$&$A001710$&$_1M_{1}(1/2,3,r)$&$A014297$\cr \\ $_1M_{1}(1/3,3,r)$&$\mbox{unlisted}$&$_1M_{1}(1/4,3,r)$&$\mbox{unlisted}$\cr \\ $_1M_{2}(1,4,r)$&$A001715$&$_1M_{2}(1/2,4,r)$&$\mbox{unlisted}$\cr \\ $_1M_{2}(1/3,4,r)$&$\mbox{unlisted}$&$_1M_{2}(1/4,4,r)$&$\mbox{unlisted}$\cr \\ $_1M_{3}(1,5,r)$&$A001720$&$_1M_{3}(1/2,5,r)$&$\mbox{unlisted}$\cr \\ $_1M_{3}(1/3,5,r)$&$\mbox{unlisted}$&$_1M_{3}(1/4,5,r)$&$\mbox{unlisted}$\cr \\ $_1M_{4}(1,6,r)$&$A001725$&$_1M_{4}(1/2,6,r)$&$\mbox{unlisted}$\cr \\ $_1M_{4}(1/3,6,r)$&$\mbox{unlisted}$&$_1M_{4}(1/4,6,r)$&$\mbox{unlisted}$ \\ \hline \end{tabular*} \end{table} \subsection{Some equivalents of Kurepa's hypothesis} The special values $M_{-1}(z)=\Gamma(z)$ and $M_0(z)=K(z)$ given in (\ref{po1}) yield \begin{equation}\label{kh1} _1M_{-1}(1,1,n+1)=n! \quad\mbox{and}\quad _1M_0(1;1,n)= !n \end{equation} where $n!$ and $!n$ are the right factorial numbers and the left factorial numbers given in (\ref{eq1.1}). The function $n!$ and $!n$ are linked by Kurepa's hypothesis:\vspace{3mm} \\{\bf KH hypothesis.} {\it For $n\in \NN \backslash \{1 \}$ we have} \[ \gcd( \, !n,\, n!\,)=2\] {\it where $\gcd(a,b)$ denotes the greatest common divisor of integers $a$ and $b$}.\vspace{3mm} \\This is listed as Problem B44 of Guy's classic book \cite{Gu}. In \cite{Du2}, it was proved that the KH is equivalent to the following assertion \[!p \not\equiv 0 \pmod p , \ra {\rm for \; all \; primes} \; p>2.\] \\The sequences $a_{n},\, b_n,\, c_n,\, d_n$ and $e_n$ (sequences A052169, A051398, \\ A051403, A002467 and A002720 in \cite{Slo}) are defined as follows: \begin{eqnarray*} a_{2}=1\quad a_{3}=2\quad a_{n}&=&(n-2)a_{n-1}+(n-3)a_{n-2},\\[1mm] b_{3}=2\quad b_{n}&=&-(n-3)b_{n-1}+2(n-2)^2,\\[1mm] c_{1}=3\quad c_{2}=8\quad c_{n}&=&(n+2)(c_{n-1}-c_{n-2}),\\[1mm] d_{0}=0\quad d_{1}=1\quad d_{n}&=&(n-1)(d_{n-1}+d_{n-2}),\\[1mm] e_{0}=1 \quad e_{1}=2\quad e_{n}&=&2ne_{n-1}-(n-1)^2e_{n-2}. \end{eqnarray*} They are related to the left factorial function. For instance, let $p>3$ be a prime number. Then \[!p\equiv -3a_{p-2}\equiv -b_{p}\equiv -2c_{p-3}\equiv d_{p-2}\equiv e_{p-1} \pmod p.\] We give the details for the last congruence.\vspace{3mm} \noindent {\it Proof.} \, Let \[L_{n}^{\nu}(x)=\sum_{k=0}^{n}\frac{\Gamma(\nu + n+1)} {\Gamma(\nu +k+1)} \frac{(-x)^{k}}{k!(n-k)!}\] be the Laguerre polynomials, and set $L_{n}^{0}(x)=L_{n}(x).$ Using the relation ${p-1 \choose k}\equiv (-1)^k \pmod p$, we have \[L_{p-1}(x)\equiv -(p-1)!\sum_{k=0}^{p-1}\frac{x^k}{k!} \pmod p .\] Wilson's theorem yields \[!p\equiv -L_{p-1}(-1) \pmod p.\] The recurrence for Laguerre polynomials \[(n+1)L_{n+1}^{\nu} (x)=(\nu+2n+1-x)L_{n}^{\nu}(x)-(\nu +n)L_{n-1}^{\nu}(x),\] for $\nu =0,\; x=-1$ produces \[h_{p-1}(p-1)!\equiv \!p \pmod p,\] where \[h_{1}=2 \quad h_{2}=\frac{7}{2}\quad h_{n}=2h_{n-1}-\frac{n-1}{n}h_{n-2}.\] The identity $e_n=h_{n}n!$ finally yields \[!p\equiv e_{p-1} \pmod p .\] \section{The numbers $_{\lowercase{n}}M_{\lowercase{-1}}(\lowercase{1};\lowercase{1}, \lowercase{n+1})$} The special values \[_nM_{-1}(1;1,n+1)=A_n,\] are the alternating factorial numbers given in (\ref{eq1.1}). This sequence satisfies the recurrence relation \[A_{0}=0,\; A_{1}=(-1)^{n-1},\; A_{n}=-(n-1)A_{n-1}+nA_{n-2}.\] These numbers can be expressed in terms of the gamma function as follows \begin{eqnarray*} A_{n} &=& \sum_{k=1}^{n}(-1)^{n-k}\Gamma(k+1)=\int_0^\infty e^{-x}\left(\sum_{k=1}^{n}(-1)^{n-k}x^{k}\right)dx \\ [3mm] &=& \int_0^\infty e^{-x}\frac{x^{n+1}-(-1)^{n}x}{x+1}dx. \end{eqnarray*} The same relation is now used in order to define the function $A_z$:\vspace{3mm} \begin{definition}\label{de4.1}For every complex number $z, \; Re\;z>0,$ the function $A_z$ is defined by \[A_{z}\stackrel{\rm def}{=}\int_0^\infty e^{-x}\frac{x^{z+1}-(-1)^{z}x}{x+1}dx.\] \end{definition}\vspace{3mm} The identity $\frac{x^{z+1}-(-1)^{z}x}{x+1}=x^{z}-\frac{x^{z}-(-1)^{z-1}x}{x+1}$ gives \[\int_0^\infty e^{-x}\frac{x^{z+1}-(-1)^{z}x}{x+1}dx=\int_0^\infty e^{-x}x^zdx-\int_0^\infty e^{-x}\frac{x^z-(-1)^{z-1}x}{x+1}dx,\] i.e., \begin{equation}\label{eq4.0} A_{z}=\Gamma(z+1)-A_{z-1}. \end{equation} This gives $A_{0}=\Gamma(2)-A_{1}=0$ and $ A_{-1}=\Gamma(1)-A_{0}=1.$ An inductive argument shows that $A_{-n},$ the residue of $A_z$ at the pole $z=-n$, is given by \[res\; A_{-n}=(-1)^{n}\sum_{k=0}^{n-2}\frac{1}{k!},\qquad n=2,3,4,...\] The derivation employs the fact that $\Gamma(z)$ is meromorphic with simple poles at $z=-n$ and residue $(-1)^n/n!$ there. The function $A_z$ can be expressed in terms of the exponential integral $\Ei(x)$ and the incomplete gamma function $\Gamma(z,x)$ by \[ A_{z}= \mathcal{L}[1;\,{}\frac{t^{z+1}-(-1)^z}{t+1}]= e\Gamma(z+2)\Gamma(-z-1,1)-(-1)^ze\Ei(-1)-(-1)^z.\] \subsection{The generating function for $A_{n-1}$} The {\it total number of arrangements of a set with $n$ elements} (sequence A000522 in \cite{Slo}) is defined (see \cite{Cam}, \cite{Ga}, \cite{Ri} and \cite{Si}) by: \begin{equation}\label{eq4.1} a_{0}=1,\ra a_{n}=na_{n-1}+1, \quad \mbox{or} \quad a_{n}=n!\sum_{k=0}^n\frac{1}{k!}. \end{equation} The sequence $\{a_n\}$ satisfies: \begin{equation}\label{eq4.2} a_{0}=1,\ra a_{1}=2,\ra a_{n}=(n+1)a_{n-1}-(n-1)a_{n-2}, \end{equation} and \begin{equation}\label{eq4.3} a_0=1,\ra a_{n}=\sum_{k=0}^{n-1}{n\choose k}(-1)^{n+1-k}(n+1-k)a_k. \end{equation}\medskip \\Relation (\ref{eq4.2}) comes from the theory of continued fractions and (\ref{eq4.3}) follows directly from (\ref{eq4.1}). We now establish a connection between the sequence $\{a_n\}$ and the alternating factorial numbers $A_n.$\vspace{3mm} \begin{lemma}\label{le4.1}Let $\; a_{-1}=1$ and $n\in \NN\backslash \{1\}$. Then $$ A_{n-1}=\sum_{k=0}^{n}{n \choose k}(-1)^{n-k}a_{k-1}. $$ \end{lemma}\vspace{3mm} \noindent {\it Proof.} \, Using (\ref{eq4.3}) and induction on $n$ we have \[ n!=\sum_{k=0}^n(-1)^{n-k}{n \choose k}a_k.\] Inversion yields \[a_n=n!-\sum_{k=1}^n{n \choose k-1}(-1)^{n+1-k}a_{k-1}.\] The relation ${n+1 \choose k}-{n\choose k}={n\choose k-1},\; k\geq 1$ produces \[\sum_{k=0}^{n+1}{n+1\choose k}(-1)^{n+1-k}a_{k-1}+\sum_{k=0}^{n}{n\choose k}(-1)^{n-k}a_{k-1}=n!.\] The result now follows from (\ref{eq4.0}).\vspace{3mm} \begin{theorem}\label{th4.1}The exponential generating function for $\{ A_{n-1}\}$ is given by \[g(x)=e^{1-x}[E_i(-1)-E_i(x-1)+e^{-1}]-1=\sum_{n=2}^\infty A_{n-1}\frac{x^n}{n!},\] where $E_{i}(x)$ is the exponential integral (\ref{ei}). \end{theorem}\vspace{3mm} \noindent {\it Proof.} \,The expansion of the exponential integral \[E_{i}(x)=\gamma +\ln(x)+\sum_{k=1}^\infty \frac{x^k}{k\cdot k!},\] where $\gamma$ is Euler's constant, appears in \cite[p.~57,~5.1.10.]{AS}. The statement of the theorem can be written as \[e[E_i(-1)-E_i(x-1)+e^{-1}]=e\left[-\ln(x-1)+\sum_{k=1}^\infty \frac{(-1)^k-(x-1)^k)}{k\cdot k!}\right]=\] \begin{equation}\label{eq4.4} = \sum_{k=0}^\infty a_{k-1}\frac{x^k}{k!}. \end{equation} Expand $e^{-x}$ as a Taylor series to obtain \[e^{-x}=\sum_{k=0}^\infty (-1)^k\frac{x^k}{k!}.\] Then \begin{eqnarray*} e^{1-x}[E_i(-1)-E_i(x-1)+e^{-1}]-1&=&\sum_{k=0}^\infty a_{k-1}\frac{x^k}{k!}\sum_{k=0}^\infty (-1)^k\frac{x^k}{k!}-1\\[3mm] &=&\sum_{n=0}^\infty\sum_{k=0}^n a_{k-1}\frac{x^k}{k!}(-1)^{n-k}\frac{x^{n-k}}{(n-k)!}-1\\[3mm] &=&\sum_{n=0}^\infty\sum_{k=0}^n {n \choose k}a_{k-1}(-1)^{n-k}\frac{x^{n}}{n!}-1\\[3mm] &=&\sum_{n=0}^\infty A_{n-1}\frac{x^n}{n!}-1. \end{eqnarray*} By induction on $n$ we get\vspace{3mm} \begin{lemma}Let $n\in N$. The function $g(x)$ in Theorem 4.1 satisfies \[g^{(n)}(x)=\frac{d}{dx}g^{(n-1)}(x)=(-1)^n\left[ g(x)+1+\sum_{k=0}^{n-1}\frac{k!}{(x-1)^{k+1}}\right]. \] \end{lemma}\vspace{3mm} This identity gives the algorithm for computing the $n$-th derivation of the function $g(x).$ \vspace{3mm} \subsection{The AL hypothesis} In \cite[p.~100]{Gu} the following problem is given:\vspace{3mm} \\{\bf Problem B43.} Are there infinitely many numbers $n$ such that $A_{n-1}$ is a prime?\vspace{3mm} Here \[A_n=\sum_{k=1}^{n}(-1)^{n-k}k!.\] If there is a value of $n-1$ such that $n$ divides $A_{n-1}$, then $n$ will divide $A_{m-1}$ for all $m>n$, and there would be only a finite number of prime values. The required condition for the existence of infinitely many numbers $n$ such that $A_{n-1}$ is a prime may be expressed as follows:\vspace{3mm} \\{\bf AL hypothesis.} {\it For every prime number} $p$ \[A_{p-1}\not\equiv 0 \pmod p,\] {\it holds.}\vspace{3mm} Let $p$ be a prime number and $n,m\in \NN \backslash \{1\}$. It is not difficult to prove the following results: \[A_{n-1}\equiv -1-\sum_{k=2}^{n}[k-1-(-1)^{n-k}]\Gamma(k) \pmod n,\] \begin{eqnarray*} A_{n-1}&=& \frac{\Gamma(n+1)-1+\sum_{k=2}^{n-1}[(-1)^{n-k}n-k+1-(-1)^{n-k}] \Gamma(k)}{n-1},\\[3mm] &=&1-!n+2\sum_{k=1}^{n-1}A_{k}\\[3mm] &=&3-!n-!(n-1)\cdot 2n+4\sum_{i=2}^{n}\sum_{k=1}^{i-1}A_{k}. \end{eqnarray*} \[\sum_{k=1}^{n}\sum_{i=0}^{m-1}(-1)^i(\Gamma(k+1))^{m-i}A_{k-1}^{i}= \left\{\begin{array}{rl}\ra A_{n}^m,\ra\ra\ra\ra m \; {\rm even}\\[1mm] \\A_{n}^m+2\sum_{j=1}^{n-1}A_{j}^m,\; m \; {\rm odd} \end{array}\right. ,\] \[A_{p-1}=-\sum_{i=1}^{p}\frac{1}{\Gamma(i)}+p\sum_{i=1}^{p}\frac{n_i(-1)^{i-1}} {\Gamma(i)}, \quad n_i\in \NN \quad (i=1,2,...p)\] The first step in solving problem B43 is proving the AL hypothesis. Using the previous identities, equivalents for the AL hypothesis can be given. \section{Conclusions} The main contribution is to define the function $_vM_m(s;a,z)$ by which problems B43 and B44 are connected. The Kurepa hypothesis is an unsolved problem since 1971 and there seems to be no significant progress in solving it, apart from numerous equivalents, such as these in Section 3.3. Further details can be found in \cite{IvMi}. However, apart from $n!, \ !n$, and $A_n$, twenty-five more well-known sequences in \cite{Slo} are special cases of the function $_vM_m(s;a,z).$ The first study of the function gave the author the idea to find an algorithm for computing some special cases (Corollary 2 and Lemma 3) before solving the above mentioned problems. The definition of the function $_vM_m(s;a,z)$ suggest another method of studying the function by using the characteristics of the inverse Laplace transform. \section*{Acknowledgements} I would like to thank the referee for the numerous comments and suggestions. \begin{thebibliography}{20} \bibitem{AS} M. Abramowitz, I. A. Stegun, {\it Handbook of Mathematical Functions}, Dover Publications, Inc. New York, 1965. \bibitem{Br} C. Brezinski, {\it History of Continued Fractions and Pad\'e Approximants}, Springer-Verlag, Berlin, 1991. \bibitem{Cam} P. J. Cameron, Sequences realized by oligomorphic permutation groups, {\it J. Integer Sequences} {\bf 3} (1) (2000), Article 00.1.5. \bibitem{Com} L. Comtet, {\it Advanced Combinatorics,} Reidel, Dordrecht, 1974. \bibitem{Ga} J. M. Gandhi, On logarithmic numbers, {\it Math.\ Student} {\bf 31} (1963), 73--83. \bibitem{Gu} R. Guy, {\it Unsolved Problems in Number Theory}, Springer-Verlag, 1994. \bibitem{IvMi} A. Ivi\'c and \v Z. Mijajlovi\'c, On Kurepa problems in number theory, {\it Publ.\ Inst.\ Math.\/} (N.S.) {\bf 57 (71)} (1995), 19--28. \bibitem{Du2} \Dj. Kurepa, On the left factorial function $!n$, {\it Math.\ Balkanica} {\bf 1} (1971), 147--153. \bibitem{Du3} \Dj. Kurepa, Left factorial function in complex domain, {\it Math.\ Balkanica} {\bf 3} (1973), 297--307. \bibitem{Mil2} G. V. Milovanovi\'c, A sequence of Kurepa's functions, {\it Sci.\ Rev.\ Ser.\ Sci.\ Eng.} No.~{\bf 19--20} (1996), 137--146. \bibitem{MiPe} G. V. Milovanovi\'c and A. Petojevi\'{c}, Generalized factorial function, numbers and polynomials and related problems, {\it Math.\ Balkanica}, to appear. \bibitem{Pe} O. Perron, {\it Die Lehren von den Kettenb\"{u}chen}, Chelsea Publishing Company, New York, 1954. \bibitem{Pet1} A. Petojevi\'c, On Kurepa's hypothesis for left factorial, {\it Filomat (Nis)}, {\bf 12} (1) (1998), 29--37. \bibitem{Pr} A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, {\it Integrals and Series. Elementary Functions,} Nauka, Moscow, 1981. (Russian) \bibitem{Ri} J. Riordan, {\it An Introduction to Combinatorial Analysis,} Wiley, 1958. \bibitem{Si} D. Singh, The numbers {\it L(m,n)} and their relations with prepared Bernoulli and Eulerian numbers, {\it Math.\ Student} {\bf 20} (1952), 66--70. \bibitem{Slo} N. J. A. Sloane, {\it The On-Line Encyclopedia of Integer Sequences}, published electronically at\hfil\break {\tt http://www.research.att.com/\char'176 njas/sequences/} \bibitem{Wilf} H. S. Wilf, {\it Generatingfunctionology,} Academic Press, New York, 1990. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2000 {\it Mathematics Subject Classification}: \ \ Primary 11B83; Secondary 33C05, 44A10. \noindent \emph{Keywords: left factorial, alternating factorial, hypergeometric function, Laplace transform} \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum{A000142}, \seqnum{A003422}, \seqnum{A005165}, \seqnum{A000217}, \seqnum{A014144}, \seqnum{A007489}, \seqnum{A000292}, \seqnum{A000332}, \seqnum{A000389}, \seqnum{A014145}, \seqnum{A000166}, \seqnum{A000125}, \seqnum{A000217}, \seqnum{A000389}, \seqnum{A055795}, \seqnum{A027660}, \seqnum{A055796}, \seqnum{A055797}, \seqnum{A032179}, \seqnum{A032031}, \seqnum{A001710}, \seqnum{A001715}, \seqnum{A001720}, \seqnum{A001725}, \seqnum{A066318}, \seqnum{A000165}, \seqnum{A047053}, \seqnum{A014297}, \seqnum{A052169}, \seqnum{A051398}, \seqnum{A051403}, \seqnum{A002467}, \seqnum{A002720}, and \seqnum{A000522}.) \vspace*{+.1in} \noindent Received March 21, 2002; revised version received August 14, 2002. Published in Journal of Integer Sequences August 31, 2002. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.math.uwaterloo.ca/JIS/}. \end{document}