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On Askey's extension of Clausen's identity and its polynomial perturbation

Dmitrii Karp, Vinay Shukla

TL;DR

This work extends Clausen's identity by deriving a Contiguous/Askey-style formula for the square of a Gauss hypergeometric function with a shifted bottom parameter, ${}_2F_{1}(a,b;a+b+m+ frac{1}{2};x)$, to any natural $m$ via a polynomial perturbation of the ${}_3F_{2}$ RHS. It then generalizes further by considering the product of this square with a polynomial perturbation of degree $s\le 2m+1$, showing the product remains hypergeometric and giving an explicit form in terms of a new polynomial perturbation $\,hat{P}_{2m+s}(t)$ of degree $2m+s$. Key contributions include a detailed expression and interpolation representation for the perturbing polynomial $P_{2m}^{a,b}$ (and its $s$-perturbed analogue $\,hat{P}_{2m+s}$), along with explicit cases for small $m$ (recovering Askey’s $m=1$ result and illustrating higher $m$). The results advance understanding of when products of hypergeometric series remain hypergeometric and provide concrete tools for evaluating these perturbations in closed form.

Abstract

The celebrated Clausen's identity expresses the square of the Gauss hypergeometric series ${}_2F_{1}(a,b;a+b+1/2;x)$ as a single hypergeometric ${}_3F_2$ series. Goursat showed in 1883 that replacing $1/2$ by $m+1/2$ leads to a hypergeometric series for the square whenever $m$ is a positive integer. Askey found this series explicitly for $m=1$. The first goal of this paper is to extend this result by treating the case of any natural $m$. The ${}_3F_{2}$ series on the right-hand side is thereby replaced by its perturbation by an explicit characteristic polynomial of degree $2m$, i.e., its coefficients are multiplied by values of this polynomial at nonnegative integers. The second goal of this paper is to make one further step and replace the square of the Gauss function by its product with its perturbation by an arbitrary polynomial of degree $s\le{2m+1}$. We show that such product remains hypergeometric and find its explicit form in terms of a polynomial perturbation of the ${}_3F_2$ series. We present an explicit formula for the characteristic polynomial whose degree is shown to be $2m+s$.

On Askey's extension of Clausen's identity and its polynomial perturbation

TL;DR

This work extends Clausen's identity by deriving a Contiguous/Askey-style formula for the square of a Gauss hypergeometric function with a shifted bottom parameter, , to any natural via a polynomial perturbation of the RHS. It then generalizes further by considering the product of this square with a polynomial perturbation of degree , showing the product remains hypergeometric and giving an explicit form in terms of a new polynomial perturbation of degree . Key contributions include a detailed expression and interpolation representation for the perturbing polynomial (and its -perturbed analogue ), along with explicit cases for small (recovering Askey’s result and illustrating higher ). The results advance understanding of when products of hypergeometric series remain hypergeometric and provide concrete tools for evaluating these perturbations in closed form.

Abstract

The celebrated Clausen's identity expresses the square of the Gauss hypergeometric series as a single hypergeometric series. Goursat showed in 1883 that replacing by leads to a hypergeometric series for the square whenever is a positive integer. Askey found this series explicitly for . The first goal of this paper is to extend this result by treating the case of any natural . The series on the right-hand side is thereby replaced by its perturbation by an explicit characteristic polynomial of degree , i.e., its coefficients are multiplied by values of this polynomial at nonnegative integers. The second goal of this paper is to make one further step and replace the square of the Gauss function by its product with its perturbation by an arbitrary polynomial of degree . We show that such product remains hypergeometric and find its explicit form in terms of a polynomial perturbation of the series. We present an explicit formula for the characteristic polynomial whose degree is shown to be .
Paper Structure (3 sections, 5 theorems, 97 equations)

This paper contains 3 sections, 5 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Contiguous extension of Clausen's identity
  3. Polynomial perturbation of the extended Askey's formula

Key Result

Theorem 2.1

Suppose $m\ge0$ is an integer, and no bottom parameter equals a non-positive integer. Then, the following product formula holds: where the polynomial $P_{2m}^{a,b}(t)$ of degree $2m$ is given by

Theorems & Definitions (15)

  • Theorem 2.1
  • Remark 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 3.1
  • Remark 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 5 more