[Paper Review] A note on some summations due to Ramanujan, their generalization and some allied series

Arjun K. Rathie, R. B. Paris|arXiv (Cornell University)|Jan 18, 2013

Advanced Mathematical Identities4 references3 citations

TL;DR

This paper generalizes Ramanujan's hypergeometric series summations using the generalized Karlsson-Minton formula and Dixon's theorem, deriving closed-form evaluations for ${}_3F_2$ and ${}_4F_3$ series involving Pochhammer symbols and gamma functions. Key results include exact evaluations of Ramanujan-type series in terms of $π$, $Γ$-functions, and the digamma function, with applications to special cases like $S_p = \Gamma(p)/\Gamma^2(p+1/2)$.

ABSTRACT

In this short note, we aim to discuss some summations due to Ramanujan, their generalizations and some allied series

Motivation & Objective

• To generalize Ramanujan's summations involving hypergeometric series using advanced summation theorems.
• To evaluate generalized ${}_3F_2$ and ${}_4F_3$ hypergeometric series with parameters related to $\frac{1}{2}$, $\frac{1}{4}$, and $p+1$.
• To derive exact closed-form expressions for series involving Pochhammer symbols and reciprocal linear terms in the denominator.
• To unify and extend Ramanujan’s original results using the generalized Karlsson-Minton summation formula.
• To provide a systematic method for evaluating infinite series with quadratic and higher-order Pochhammer ratios.

Proposed method

• Utilizes the generalized Karlsson-Minton summation formula for ${}_{r+2}F_{r+1}$ series with integer parameter shifts.
• Applies Dixon’s summation theorem for ${}_3F_2$ series with parameters $a=b=\frac{1}{2}$, $c=\frac{1}{4}$, and related combinations.
• Employs Gauss’s summation theorem for ${}_2F_1$ series to evaluate cases where $c-a-b>0$.
• Transforms series with denominators like $(n+1)\cdots(n+p)$ into hypergeometric forms using Pochhammer symbol identities.
• Uses the representation $S = \sum_{n=0}^\infty \frac{(\frac{1}{2})_n}{n!} \frac{1}{b + n\mu}$ and reduces it to a ${}_2F_1$ function via hypergeometric transformation.
• Applies limiting processes and special values of the digamma function $\psi(z)$ to handle cases where parameters coincide.

Experimental results

Research questions

• RQ1 How can Ramanujan’s original summations involving $\left(\frac{1\cdot3}{2\cdot4}\right)^2$ be generalized to higher-order hypergeometric series?
• RQ2 What closed-form evaluation can be derived for the series $\sum_{n=0}^\infty \left(\frac{(\frac{1}{2})_n}{n!}\right)^2 \frac{1}{(n+1)\cdots(n+p)}$?
• RQ3 Can the generalized Karlsson-Minton formula be applied to evaluate ${}_3F_2$ and ${}_4F_3$ series with shifted parameters?
• RQ4 What is the exact value of $S = \sum_{n=0}^\infty \frac{(\frac{1}{2})_n}{n!} \frac{1}{(1+n/b)(1+n/c)}$ for $b \neq c$ and $b = c$?
• RQ5 How do the results reduce to known identities such as Ramanujan’s $\frac{\pi^{3/2}}{2\sqrt{2}\Gamma^2(\frac{3}{4})}$ when specific parameters are substituted?

Key findings

• The series $S_p = \sum_{n=0}^\infty \left(\frac{(\frac{1}{2})_n}{n!}\right)^2 \frac{1}{(n+1)\cdots(n+p)}$ evaluates exactly to $\frac{\Gamma(p)}{\Gamma^2(p+\frac{1}{2})}$ for integer $p \geq 1$.
• For $p=1$, $S_1 = \frac{4}{\pi}$; for $p=2$, $S_2 = \frac{16}{9\pi}$; and for $p=3$, $S_3 = \frac{128}{225\pi}$.
• When $b=c=\frac{1}{4}$, the generalized series $S = {}_3F_2\left[\frac{1}{2},\frac{1}{4},\frac{1}{4};\frac{5}{4},\frac{5}{4};1\right]$ reduces to Ramanujan’s identity $\frac{\pi^{5/2}}{8\sqrt{2}\Gamma^2(\frac{3}{4})}$.
• The case $b \neq c$ yields $S = \frac{\pi^{1/2}bc}{b-c}\left(\frac{\Gamma(c)}{\Gamma(c+\frac{1}{2})} - \frac{\Gamma(b)}{\Gamma(b+\frac{1}{2})}\right)$, with a limiting form for $b=c$ involving the digamma function.
• For $m=1$, the series $\sum_{n=0}^\infty \left(\frac{(\frac{1}{2})_n}{n!}\right)^2 \frac{n+f}{(n+1)\cdots(n+p)} = \frac{\Gamma(p)}{\Gamma^2(p+\frac{1}{2})}\left(f + \frac{1}{4(p-1)}\right)$ for $p \geq 2$.
• For $m=2$, the evaluation yields $\sum_{n=0}^\infty \left(\frac{(\frac{1}{2})_n}{n!}\right)^2 \frac{(n+f)(n+f+1)}{(n+1)\cdots(n+p)} = \frac{\Gamma(p)}{\Gamma^2(p+\frac{1}{2})}\left(f(f+1) + \frac{f+1}{2(p-1)} + \frac{9}{16(p-1)(p-2)}\right)$ for $p \geq 3$.