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\begin{center}
\vskip 1cm{\Large\bf On the Binomial Identities of Frisch and Klamkin}
\vskip 1cm
H. W. Gould\\
Department of Mathematics\\
West Virginia University\\
Morgantown, WV 26506\\
USA\\
\href{mailto:henrygou@gmail.com}{\tt henrygou@gmail.com}\\
\ \\
Jocelyn Quaintance\\
Department of Computer Science\\
University of Pennsylvania\\
Philadelphia, PA 19104\\
USA\\
\href{mailto:jocelynq@seas.upenn.edu}{\tt jocelynq@seas.upenn.edu} \\
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\begin{abstract}
In this paper we investigate two somewhat similar identities for sums of ratios of binomial coefficients.
We give several proofs,
and note that the identities all follow from a hypergeometric identity of Gauss. Inverse identities are also given.
\end{abstract}

\section{Introduction}

Klamkin \cite{msk} stated the following identity in a letter to Gould in 1966:
\begin{align}\label{klam}
\sum_{k=0}^n\frac{{n\choose k}}{{n+a\choose k+b}} &amp;= \frac{a+1+n}{(a+1){a\choose b}}
\end{align}
Identity (4.6) in Gould's book \cite{hwg} is a special case of this,
corresponding to $a=2x$ and $b=x$.


Frisch \cite{rf}, in his dissertation in 1926, gave the curious formula
\begin{align}\label{eq8}
\sum_{k=0}^n(-1)^k{n\choose k}\frac{1}{{b+k\choose c}} &amp;= \frac{c}{n+c}\frac{1}{{n+b\choose b-c}},\qquad b\geq c &gt; 0.
\end{align}
This was cited and proved by Netto \cite[pp.~337--338]{en} and is tabulated
as Formula (4.2)  in Gould's book \cite{hwg}.

(It is interesting to note that 
Frisch's research laid the foundations for modern econometrics theory
and micro- and macro-economics, work for which he later received the
Nobel Prize.)

Klamkin's identity is actually a special case of  the identity
\begin{align}\label{klam1a}
\sum_{k=0}^n\frac{{n\choose k}}{{x\choose k+b}} &amp;= \frac{x+1}{(x-n+1){x-n\choose b}}.
\end{align}

Here is how we obtain Eq.~(\ref{klam1a}).  Let $t_k = \frac{{n\choose k}}{{x\choose k+b}}$.  Then
\begin{align*}
\frac{t_{k+1}}{t_k} &amp;= \frac{{n\choose k+1}{x\choose k+b}}{{x\choose k+1+b}{n\choose k}} = \frac{(n-k)(k+1+b)}{(k+1)(x-k-b)}.
\end{align*}
Therefore,
\begin{align}\label{eq1}
\sum_{k=0}^n\frac{{n\choose k}}{{x\choose k+b}} &amp;= \frac{1}{{x\choose b}}\,_2F_1\left[\begin{array}{cc} 1+b, &amp; -n\\ -x + b &amp; \end{array}|1\right].
\end{align}
Applying Gauss's $_2F_1$ formula, namely,
\begin{align*}
_2F_1\left[\begin{array}{cc}a, &amp; b\\ c &amp; \end{array}|1\right] &amp;= \frac{\Gamma(c-a-b)\Gamma(c)}{\Gamma(c-a)\Gamma(c-b)},
\end{align*}
we have
\begin{align*}
\sum_{k=0}^n\frac{{n\choose k}}{{x\choose k+b}} &amp;= \frac{1}{{x\choose b}}\cdot\frac{\Gamma(-x+n-1)\Gamma(-x+b)}{\Gamma(-x-1)\Gamma(-x+b+n)}\\
&amp;= \frac{b!}{x(x-1)\dots(x-b+1)}\cdot\frac{(x+1)x(x-1)\dots(x-n+3)(x-n+2)}{(x-b)(x-b-1)\dots(x-n-b+2)(x-n-b+1)}\\
&amp;= \frac{b!(x+1)\prod_{j=0}^{n-2}(x-j)}{\prod_{j=0}^{n+b-1}(x-j)}\\
&amp;= \frac{b!(x+1)}{\prod_{j=n-1}^{n+b-1}(x-j)}\\
&amp;= \frac{x+1}{x-n+1}\cdot\frac{b!}{(x-n)(x-n-1)\dots(x-n-b+1)}\\
&amp;= \frac{x+1}{x-n+1}\cdot\frac{1}{{x-n\choose b}}.
\end{align*}

In this proof we assumed that $n\geq 2$.  If $n = 0$, Eq.~(\ref{klam}) is obviously true, while if $n = 1$, we obtain easily verified identity
\begin{align*}
\frac{1}{{x\choose b}} + \frac{1}{{x\choose b+1}} &amp;= \frac{x+1}{x{x-1\choose b}}.
\end{align*}

\section{Proof of Klamkin's and Frisch's formulas}

We shall now prove Klamkin's formula directly from Formula (7.1) in Gould \cite{hwg}, namely
\begin{align}\label{klamaa}
\sum_{k=0}^n(-1)^k{n\choose k}\frac{{z\choose k}}{{y\choose k}} &amp;= \frac{{y-z\choose n}}{{y\choose n}},
\end{align}
and the easy binomial identity
\begin{align}
{x\choose k+b}{k+b\choose k} = {x\choose b}{x-b\choose k} 
\end{align}

Indeed, using these we have
\begin{align*}
\sum_{k=0}^n\frac{{n\choose k}}{{x\choose k+b}} &amp;=  \frac{1}{{x\choose b}}\sum_{k=0}^n{n\choose k}\frac{{k+b\choose k}}{{x-b\choose k}}\\
&amp; = \frac{1}{{x\choose b}}\sum_{k=0}^n(-1)^k{n\choose k}\frac{{-1-b\choose k}}{{x-b\choose k}}\\
&amp; =  \frac{1}{{x\choose b}}\frac{{x+1\choose n}}{{x-b\choose n}}= \frac{x+1}{(x+1-n){x-n\choose b}}
\end{align*}
by Formula (7.1), and our proof is complete. We remark that Formula (7.1) is equivalent to using the Gauss $_2F_1$ formula.\\

Simple binomial inversion yields the inverse Klamkin identity
\begin{align}
\sum_{k=0}^n(-1)^{n-k}{n\choose k}\frac{x+1}{(x+1-k){x-k\choose b}}=\frac{1}{{x\choose n+b}}.
\end{align}

We now proceed in a similar way to prove Frisch's identity. It is easy to verify the binomial identity 
\begin{align*}
 \frac{1}{{k+b\choose c}} = \frac{1}{{b\choose c}}\frac{{c-b-1\choose k}}{{-b-1\choose k}},
\end{align*}
so that
\begin{align*}
\sum_{k=0}^n(-1)^k{n\choose k}\frac{1}{{k+b\choose c}} &amp;=\frac{1}{{b\choose c}}\sum_{k=0}^n(-1)^k{n\choose k}\frac{{c-b-1\choose k}}{{-b-1\choose k}}.
\end{align*}
Upon applying Eq.~(\ref{klamaa}), we find the sum equals
\begin{align*}
\frac{1}{{b\choose c}}\frac{{-c\choose n}}{{-b-1\choose n}}= \frac{c}{n+c}\frac{1}{{n+b\choose b-c}},
\end{align*}
as desired to show.

This new proof of Frisch's identity should be compared to Gould's
original proof \cite[Section 7.2]{jg}, a two-page
calculation involving an application of Melzak's formula.

An inverse Frisch identity then follows by simple binomial inversion,
and we have
\begin{align}
\sum_{k=0}^n(-1)^k{n\choose k} \frac{c}{k+c}\frac{1}{{k+b\choose b-c}}&amp;=\frac{1}{{n+b\choose c}}.
\end{align}

\begin{thebibliography}{9}

\bibitem{rf} Ragnar Frisch, Sur les semi-invariants et moments
employ\'{e}s dans l'\'{e}tude des distributions statistiques, {\it
Skrifter utgitt av Det Norske Videnskaps-Akademi i Oslo, II,
Historisk-Filosofisk Klasse} {\bf 3} (1926), 1--87.  Quoted by Th.
Skolem, p.~337, in Netto's {\it Lehrbuch}.

\bibitem{hwg} H. W. Gould, {\it Combinatorial Identities}, published by
the author, Morgantown, WV, 1972.

\bibitem{msk} Murray S. Klamkin, Letter to H. W. Gould, May 16, 1966.

\bibitem{en} Eugen Netto, {\it Lehrbuch der Kombinatorik}, 2nd edition,
Chelsea Publications, 1927.

\bibitem{jg} Jocelyn Quaintance and H. W. Gould, {\it Combinatorial
Identities for Stirling Numbers}, World Scientific Press, 2016.

\end{thebibliography}



\bigskip
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\noindent 2010 {\it Mathematics Subject Classification}:
Primary 05A19; Secondary 11B65, 33C10.

\noindent \emph{Keywords: } binomial identity,
hypergeometric series,
Gauss' hypergeometric summation formula, binomial inversion.

\bigskip
\hrule
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\vspace*{+.1in}
\noindent
Received July 4 2016;
revised versions received  August 31 2016; September 1 2016; September
7 2016.
Published in {\it Journal of Integer Sequences}, September 7 2016.

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