A hypergeometric proof for a binomial identity related to $1/\pi$

TL;DR

This paper presents a hypergeometric proof for a binomial identity connected to series expansions of 1/π, linking it to Whipple's second theorem for hypergeometric series.

Contribution

It provides a novel hypergeometric proof of a binomial identity related to 1/π series, connecting it to classical hypergeometric theorems.

Findings

Identifies a binomial identity as an instance of Whipple's second theorem

Establishes a hypergeometric framework for understanding 1/π series expansions

Bridges combinatorial identities with hypergeometric function theory

Abstract

We show that a binomial identity arising in the context of the study of series expansions of $1/\pi$ can be seen as an incarnation of Whipples second theorem for hypergeometric series.

Equations

$$\sum _{n=0}^m\left(\genfrac{}{}{0ex}{}{m}{n}\right)\left(\genfrac{}{}{0ex}{}{k+n}{m}\right){\left(\genfrac{}{}{0ex}{}{k+m+n}{m+n}\right)}^{-1}\frac{2n+k+1}{m+n+k+1}=1$$

$$\sum _{n=0}^m$$

$$=\sum _{n=0}^m\frac{m!}{n!(m-n)!}\frac{(n+k)!}{m!(k-m+n)}\frac{(m+n)!k!}{(k+m+n)!}\frac{2n+k+1}{m+n+k+1}$$

$$=\frac{(k+1)!k!}{(k-m)!(k+m+1)!}\sum _{n=0}^m\frac{(k+n)!}{k!}\frac{2n+k+1}{k+1}\frac{(m+n)!}{(m-n)!}\frac{(k+m+n+1)!}{(k+m+n+1)!}\frac{(k-m)!}{(k-m+n)!}\frac{1}{n!}$$

$$=\frac{(k+1)!k!}{(k-m)!(k+m+1)!}\sum _{n\ge 0}(k+1{)}_n\frac{(1+\frac{k+1}{2}{)}_n}{(\frac{k+1}{2}{)}_n}\frac{(-m{)}_n(-1{)}^n(m+1{)}_n}{(k+m+2{)}_n(k-m+1{)}_n}\frac{1}{n!}$$

$$\displaystyle=\frac{(k+1)!\,k!}{(k-m)!\,(k+m+1)!}{}_{4}F_{3}\biggl{(}\genfrac{.}{.}{0.0pt}{}{k+1\mathchar 44\relax\mkern 10.0mu1+\frac{k+1}{2}\mathchar 44\relax\mkern 10.0mu-m\mathchar 44\relax\mkern 10.0mum+1}{\frac{k+1}{2}\mathchar 44\relax\mkern 10.0muk+m+2\mathchar 44\relax\mkern 10.0muk-m+1}\bigg{|}-1\biggr{)}.$$

$${}_{4}F_{3}\biggl{(}\genfrac{.}{.}{0.0pt}{}{a\mathchar 44\relax\mkern 10.0mu1+a/2\mathchar 44\relax\mkern 10.0mub\mathchar 44\relax\mkern 10.0muc}{a/2\mathchar 44\relax\mkern 10.0mua-b+1\mathchar 44\relax\mkern 10.0mua-c+1}\bigg{|}-1\biggr{)}=\frac{\Gamma(a-b+1)\Gamma(a-c+1)}{\Gamma(a+1)\Gamma(a-b-c+1)}.$$

$$\frac{(k+1)!k!}{(k-m)!(k+m+1)!}\frac{\Gamma (k+m+2)\Gamma (k-m+1)}{\Gamma (k+2)\Gamma (k+1)}$$

$$=1,$$

Full text

B. Hackl is supported by the Austrian Science Fund (FWF): P 28466-N35 and by a distinction grant from the Austrian Federal Ministry of Education, Science and Research.

In [2], Sesma studies a family of series expansions for $1/\pi$ based on Heaviside’s exponential series. As a side product of these investigations, the hypergeometric identity

[TABLE]

with $k\geq m\in\mathbb{Z}_{\geq 0}$ arises and was conjectured to be new.

We observe that rewriting the left-hand side of (1) with the help of factorials, regrouping the terms and simplifying quotients with the help of the Pochhammer symbol $(\ell)_{n}=\ell(\ell+1)\cdots(\ell+n-1)$ yields

[TABLE]

The second Whipple theorem (cf. [1, Section 4.4]) for ${}_{4}F_{3}$-hypergeometric series states that for suitable complex-valued variables $a$, $b$, $c$ we have

[TABLE]

By comparing this to the hypergeometric ${}_{4}F_{3}$-term in (2) we can immediately see that we can use the second Whipple theorem with $a=k+1$, $b=-m$, $c=m+1$ to rewrite (2) further to

[TABLE]

where we only had to use the relation $\Gamma(\ell+1)=\ell!$ for non-negative integers $\ell$. This proves that the binomial identity (1) is a consequence of the second Whipple theorem.