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\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} \\
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\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&gt;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&gt;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 $b\geq c&gt;0$, 
\begin{eqnarray*}
\sum_{k=0}^{n}\left( -1\right) ^{k}{\binom{n}{k}\binom{b+k}{c}}^{-1}
&amp;=&amp;c\sum_{k=0}^{n}\left( -1\right) ^{k}{\binom{n}{k}}B\left( b-c+1+k,c\right)
\\
&amp;=&amp;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 \\
&amp;=&amp;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}
&amp;=&amp;\sum_{k=0}^{n}{\binom{n}{k}}B\left( b+k+1,x-b-k+1\right) \\
&amp;=&amp;\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 \\
&amp;=&amp;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 $k&gt;n\geq 0$, 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} &amp;=&amp;\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} &amp;=&amp;\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 $b\geq c&gt;n\geq 1$, 
\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 $c&gt;n$.

\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 &lt;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&gt;0,\text{ }y&gt;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&gt;0,\text{ }y&gt;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&gt;c-1,\text{ }c&gt;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&gt;n,\text{ }b&gt;-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 $-a-1&gt;n\geq 1$ 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 $a+1+n&lt;0$ 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 $\eta &gt;1$. 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 $a+1+n&lt;0$ 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}

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\noindent 2010 {\it Mathematics Subject Classification}:
Primary 05A19.

\noindent \emph{Keywords: } 
combinatorial identity, binomial coefficient, beta function.


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\vspace*{+.1in}
\noindent
Received October 29 2019;
revised version received March 30 2020.
Published in {\it Journal of Integer Sequences}, June 12 2020.

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