Extensions of the Classical Transformations of 3F2

TL;DR

This paper extends classical quadratic and cubic transformations of the hypergeometric function ${}_3F_2$ to include additional parameters constrained by orthogonal polynomials, leading to new identities and applications.

Contribution

It introduces extended transformation identities for ${}_3F_2$ involving parameters linked to orthogonal polynomial roots, broadening the scope of classical hypergeometric transformations.

Findings

Extended quadratic and cubic identities for ${}_3F_2$ functions.

New relations involving orthogonal polynomial roots.

Applications to Whipple's identity and other summation formulas.

Abstract

It is shown that the classical quadratic and cubic transformation identities satisfied by the hypergeometric function ${}_3F_2$ can be extended to include additional parameter pairs, which differ by integers. In the extended identities, which involve hypergeometric functions of arbitrarily high order, the added parameters are nonlinearly constrained: in the quadratic case, they are the negated roots of certain orthogonal polynomials of a discrete argument (dual Hahn and Racah ones). Specializations and applications of the extended identities are given, including an extension of Whipple's identity relating very well poised ${}_7F_6(1)$ series and balanced ${}_4F_3(1)$ series, and extensions of other summation identities.