\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{amscd} \usepackage{graphicx} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{psfig} \usepackage{graphics} \usepackage{latexsym} \usepackage{epsf} \usepackage{breakurl} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.1in} \setlength{\textheight}{8.4in} \newcommand{\seqnum}[1]{\href{https://oeis.org/#1}{\underline{#1}}} \def\modd#1 #2{#1\ \mbox{\rm (mod}\ #2\mbox{\rm )}} \DeclareMathOperator{\Res}{Res} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \begin{center} \vskip .5cm{\LARGE\bf A Short Proof of the Binomial Identities \\ \vskip .1in of Frisch and Klamkin } \vskip 1cm \large Ulrich Abel\\ Department MND\\ Technische Hochschule Mittelhessen\\ Wilhelm-Leuschner-Stra{\ss}e 13 \\ 61169 Friedberg\\ Germany \\ \href{mailto:ulrich.abel@mnd.thm.de}{\tt ulrich.abel@mnd.thm.de} \\ \end{center} \vskip .2 in \begin{abstract} We present short proofs for Frisch's identity and Klamkin's identity. Furthermore, we deduce variants of Frisch's and Klamkin's identities involving infinite series. \end{abstract} \section{Introduction} In their recent paper \cite{Gould-Quaintance-2016}, Gould and Quaintance gave new proofs of Frisch's identity \begin{equation} \sum_{k=0}^{n}\left( -1\right) ^{k}{\binom{n}{k}\binom{b+k}{c}}^{-1}=\frac{c}{n+c}{\binom{n+b}{b-c}}^{-1}\text{ }\qquad \left( b\geq c>0\right) \label{Identity-Frisch} \end{equation} and Klamkin's identity \begin{equation} \sum_{k=0}^{n}{\binom{n}{k}\binom{x}{k+b}}^{-1}=\frac{x+1}{x-n+1}{\binom{x-n}{b}}^{-1}\text{ }\qquad \left(x-n \geq b\geq 0\right) . \label{Identity-Klamkin} \end{equation} As usual we put ${\binom{x}{0}=1}$. For suitable real numbers $x,y$, the binomial coefficient ${\binom{x}{y}}$ is to be read in terms of the gamma function, i.e., ${\binom{x+y}{y}=\Gamma }\left( x+y+1\right) /\left( {\Gamma }\left( x+1\right) {\Gamma }\left( y+1\right) \right) $. The proofs in \cite{Gould-Quaintance-2016} are based on the well-known formula of Gauss for the hypergeometric function $_{2}F_{1}\left( a,b;c;1\right) $. In 1969, Ragnar Frisch (1895--1973) was awarded the first Nobel Prize in Economic Sciences. Identity $\left( \ref{Identity-Frisch}\right) $ appeared in Frisch's 1926 dissertation \cite{Frisch-Dissertation-1926}. It was cited and proved in the 2nd edition 1927 of the book \cite[pp.~337--338]{netto-book} by Netto. A further proof of Frisch's identity $\left( \ref{Identity-Frisch}\right) $, a two-page calculation involving an application of Melzak's formula, can be found in the new book \cite[Section 7.2]{Quaintance-Gould-Book-2016}. In \cite{Gould-Quaintance-2016} the authors report that identity $\left( \ref{Identity-Klamkin}\right) $ in its original form \cite[Eq. (1)]{Gould-Quaintance-2016} with $x=n+a$ was stated by Murray S. Klamkin in a letter to Henry W. Gould on May 16, 1966. It is tabulated as Formula (4.2) in Gould's collection \cite{Gould-collection-1972}. Identity (4.6) in \cite{Gould-collection-1972} is a special case of this. The purpose of this note are elementary short proofs without application of hypergeometric functions. They are based on the use of the Euler beta function \begin{equation*} B\left( x,y\right) =\int_{0}^{1}t^{x-1}\left( 1-t\right) ^{y-1}dt=\frac{\Gamma \left( x\right) \Gamma \left( y\right) }{\Gamma \left( x+y\right) }\ \qquad \left( x,y>0\right) . \end{equation*} Furthermore, we present variants of Frisch's and of Klamkin's identities involving infinite series. \section{Proof of Frisch's and Klamkin's formulas} By direct calculation, we obtain, for $n=0,1,2,\ldots $ and $b\geq c>0$, \begin{eqnarray*} \sum_{k=0}^{n}\left( -1\right) ^{k}{\binom{n}{k}\binom{b+k}{c}}^{-1} &=&c\sum_{k=0}^{n}\left( -1\right) ^{k}{\binom{n}{k}}B\left( b-c+1+k,c\right) \\ &=&c\sum_{k=0}^{n}\left( -1\right) ^{k}{\binom{n}{k}}\int_{0}^{1}t^{b-c+k} \left( 1-t\right) ^{c-1}dt \\ &=&c\cdot B\left( b-c+1,c+n\right) =\frac{c}{n+c}{\binom{n+b}{b-c}}^{-1}, \end{eqnarray*} which is Frisch's identity $\left( \ref{Identity-Frisch}\right) $. For suitable $x\neq -1$, we have \begin{eqnarray*} \frac{1}{x+1}\sum_{k=0}^{n}{\binom{n}{k}\binom{x}{k+b}}^{-1} &=&\sum_{k=0}^{n}{\binom{n}{k}}B\left( b+k+1,x-b-k+1\right) \\ &=&\sum_{k=0}^{n}{\binom{n}{k}}\int_{0}^{1}t^{b}\left( 1-t\right) ^{x-b}\left( \frac{t}{1-t}\right) ^{k}dt \\ &=&B\left( b+1,x-b-n+1\right) =\frac{1}{x-n+1}{\binom{x-n}{b}}^{-1}, \end{eqnarray*} which proves Klamkin's identity $\left( \ref{Identity-Klamkin}\right) $. \section{Variants of Frisch's and Klamkin's identities} Since ${\binom{n}{k}=0}$, for integers $k>n\geq 0$, the left-hand sides in both identities $\left( \ref{Identity-Frisch}\right) $, $\left( \ref{Identity-Klamkin}\right) $ can be written as infinite sums. Formally replacing $n$ with $-n$ in those equations and using the obvious binomial identity \begin{equation*} \left( -1\right) ^{k}{\binom{-n}{k}=\binom{n+k-1}{k}} \end{equation*} yields \begin{eqnarray*} \sum_{k=0}^{\infty }{\binom{n+k-1}{k}\binom{b+k}{c}}^{-1} &=&\frac{c}{c-n}{\binom{-n+b}{b-c}}^{-1}, \\ \sum_{k=0}^{\infty }\left( -1\right) ^{k}{\binom{n+k-1}{k}\binom{x}{k+b}}^{-1} &=&\frac{x+1}{x+n+1}{\binom{x+n}{b}}^{-1}. \end{eqnarray*} We shall show that both equations are valid also for positive integers $n$, if the parameters $a,b,c$ are chosen in an appropriate manner. First we deduce the variant of Frisch's identity. \begin{theorem} \label{theorem-variant-Frisch} For $b\geq c>n\geq 1$, \begin{equation} \sum_{k=0}^{\infty }{\binom{n+k-1}{k}\binom{b+k}{c}}^{-1}=\frac{c}{c-n}{\binom{b-n}{b-c}}^{-1}. \label{Identity-variant-Frisch} \end{equation} \end{theorem} The infinite series in Eq. $\left( \ref{Identity-variant-Frisch}\right) $ is convergent since \begin{equation*} {\binom{n+k-1}{k}\binom{b+k}{c}}^{-1}=\Gamma \left( n\right) \frac{\Gamma \left( n+k\right) }{\Gamma \left( 1+k\right) }\Gamma \left( c+1\right) \frac{\Gamma \left( b-c+1+k\right) }{\Gamma \left( b+1+k\right) }\sim \frac{\Gamma \left( n\right) \Gamma \left( c+1\right) }{k^{1-n+c}} \end{equation*} as $k\rightarrow \infty $ (see, e.g., \cite[(6.1.46)]{Abramowitz}) and $c>n$. \begin{proof}[Proof of Theorem \protect \ref{theorem-variant-Frisch}] By using the well-known power series expansion \begin{equation*} \sum_{k=0}^{\infty }{\binom{n+k-1}{k}}z^{k}=\left( 1-z\right) ^{-n}\ \qquad \left( \left \vert z\right \vert <1\right) \end{equation*} instead of the binomial formula, it follows in the same manner as in the preceding section that \begin{equation*} \sum_{k=0}^{\infty }{\binom{n+k-1}{k}}B\left( x+k,y\right) =B\left( x,y-n\right) \ \qquad \left( x>0,\text{ }y>n\right) . \end{equation*} Rewritten in terms of binomial coefficients the latter identity takes the form\ \begin{equation*} x\sum_{k=0}^{\infty }{\binom{n+k-1}{k}\binom{x+y+k-1}{y}}^{-1}=y{\binom{x+y-n-1}{x}}^{-1}\ \qquad \left( x>0,\text{ }y>n\right) . \end{equation*} Replacing $x$ with $b-c+1$ and $y$ with $c$, we obtain \begin{equation*} \sum_{k=0}^{\infty }{\binom{n+k-1}{k}\binom{b+k}{c}}^{-1}=\frac{c}{b-c+1}{\binom{b-n}{b-c+1}}^{-1}\ \qquad \left( b>c-1,\text{ }c>n\right) . \end{equation*} Now Eq. $\left( \ref{Identity-variant-Frisch}\right) $ follows since \begin{equation*} \frac{c}{b-c+1}{\binom{b-n}{b-c+1}}^{-1}=\frac{c}{c-n}{\binom{b-n}{b-c}} ^{-1}. \end{equation*} This completes the proof. \end{proof} \begin{remark} An alternative approach is the observation \begin{equation*} \sum_{k=0}^{\infty }{\binom{n+k-1}{k}\binom{b+k}{c}}^{-1}={\binom{b}{c}}^{-1}\ _{2}F_{1}\left( 1+b-c,n;b+1;1\right) . \end{equation*} This formula immediately implies Eq. $\left( \ref{Identity-variant-Frisch} \right) $ since, by Gauss's formula \cite[(15.1.20)]{Abramowitz}, \begin{equation*} _{2}F_{1}\left( 1+b-c,n;b+1;1\right) =\frac{\Gamma \left( c-n\right) \Gamma \left( b+1\right) }{\Gamma \left( c\right) \Gamma \left( b-n+1\right) }\ \qquad \left( c>n,\text{ }b>-1\right) . \end{equation*} \end{remark} We close with the variant of Klamkin's identity. \begin{theorem} \label{theorem-variant-Klamkin}Let $-a\notin \mathbb{N}$ and $-b\notin \mathbb{N}$. For $-a-1>n\geq 1$ and $a-b+1\notin \mathbb{N}$, \begin{equation} \sum_{k=0}^{\infty }\left( -1\right) ^{k}{\binom{n+k-1}{k}\binom{a}{b+k}} ^{-1}=\frac{a+1}{a+n+1}{\binom{a+n}{b}}^{-1}. \label{Identity-variant-Klamkin} \end{equation} \end{theorem} The method of proof using the beta integral does not work in the case of Theorem~\ref{theorem-variant-Klamkin} because the arising sum becomes divergent. Therefore, we use the Gauss formula as in the preceding remark. \begin{proof}[Proof of Theorem \protect \ref{theorem-variant-Klamkin}] Noting that \begin{equation*} \left( -1\right) ^{k}{\binom{a}{b+k}^{-1}={\binom{a}{b}}^{-1}}\frac{\Gamma \left( b+1+k\right) }{\Gamma \left( b+1\right) }\frac{\Gamma \left( b-a\right) }{\Gamma \left( b-a+k\right) }. \end{equation*} we have \begin{equation*} \left( -1\right) ^{k}{\binom{n+k-1}{k}\binom{a}{b+k}}^{-1}=\gamma \frac{\Gamma \left( n+k\right) }{\Gamma \left( 1+k\right) }\frac{\Gamma \left( b+1+k\right) }{\Gamma \left( b-a+k\right) }\sim \gamma k^{n+a} \end{equation*} as $k\rightarrow \infty $, where $\gamma = \Gamma \left( n\right) \frac{\Gamma \left( b-a\right) }{\Gamma \left( b+1\right) }{{\binom{a}{b}}^{-1}}$. If $a+1+n<0$ we have \begin{equation*} \left( -1\right) ^{k}{\binom{n+k-1}{k}\binom{a}{b+k}}^{-1}=O\left( k^{-\eta }\right) \text{\ \qquad }\left( k\rightarrow \infty \right) \end{equation*} with a certain constant $\eta >1$. This shows the convergence of the infinite series in Eq.~$\left( \ref{Identity-variant-Klamkin}\right) $. Finally, we observe \begin{equation*} \sum_{k=0}^{\infty } \left( -1\right) ^{k} {\binom{n+k-1}{k}\binom{a}{b+k}}^{-1}={\binom{a}{b}}^{-1} \text{ }_{2}F_{1}\left( n,b+1;b-a;1\right) , \end{equation*} for $a+1+n<0$ and $a-b+1\notin \mathbb{N}$. Using \begin{equation*} _{2}F_{1}\left( n,b+1;b-a;1\right) =\frac{\Gamma \left( -a-n-1\right) \Gamma \left( b-a\right) }{\Gamma \left( b-a-n\right) \Gamma \left( -a-1\right) }= \frac{\Gamma \left( a-b+n+1\right) \Gamma \left( a+2\right) }{\Gamma \left( a-b+1\right) \Gamma \left( a+2+n\right) } \end{equation*} the desired formula Eq.~$\left( \ref{Identity-variant-Klamkin} \right) $ follows after a short calculation. \end{proof} \section{Acknowledgment} The author is grateful to the anonymous reviewer for a thorough reading of the manuscript and valuable comments. \begin{thebibliography}{30} \bibitem{Abramowitz} M.~Abramowitz and I.~A.~Stegun, \newblock \textit{Handbook of Mathematical Functions}, \newblock Appl. Math. Ser. 55, National Bureau of Standards, 1972. \bibitem{Frisch-Dissertation-1926} R. Frisch, Sur les semi-invariants et moments employ\'{e}s dans l'\'{e}tude des distributions statistiques, \textit{Skrifter utgitt av Det Norske Videnskaps-Akademi i Oslo, II, Historisk-Filosofisk Klasse} \textbf{3} (1926), 1--87. Quoted by Th. Skolem, p.~337, in Netto's \textit{Lehrbuch}. \bibitem{Gould-collection-1972} H. W. Gould, \textit{Combinatorial Identities}, Morgantown, West Virginia, 1972. \bibitem{Gould-Quaintance-2016} H. W. Gould and J. Quaintance, On the binomial identities of Frisch and Klamkin, {\it J. Integer Sequences} \textbf{19} (2016) \href{https://cs.uwaterloo.ca/journals/JIS/VOL19/Gould/gould8.html}{Article 16.7.7}. \bibitem{netto-book} E. Netto, \textit{Lehrbuch der Kombinatorik}, 2nd edition, Chelsea Publications, 1927. \bibitem{Quaintance-Gould-Book-2016} J. Quaintance and H. W. Gould, \textit{Combinatorial Identities for Stirling Numbers}, World Scientific Press, 2016. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2010 {\it Mathematics Subject Classification}: Primary 05A19. \noindent \emph{Keywords: } combinatorial identity, binomial coefficient, beta function. \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received October 29 2019; revised version received March 30 2020. Published in {\it Journal of Integer Sequences}, June 12 2020. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}