On transformations of hypergeometric functions with integer parameter differences

K. E. Bakhtin^ab, E. G. Prilepkina<sup>a

a</sup> Institute for Applied Mathematics, Far Eastern Branch, Russian Academy of Sciences, Vladivostok
^b Far Eastern Federal University, Vladivostok

Abstract: Several facts concerning transformations and summations of hypergeometric functions with integral parametric differences have been proven. A new summation formula complementing the well-known Karlsson – Minton summation formula has been established. A new three – term relation has been derived from the first Miller – Paris transformation. It is shown how the second Miller – Paris transformation can be obtained by induction from the Euler – Pfaff transformation, and a recursive formula for the representing polynomial is provided. An integral representation of Meijer's $G$-function, which underlies the second Miller – Paris transformation, has been established.

Key words: generalized hypergeometric function, summation formulas, hypergeometric identity, Miller – Paris transformations.

UDC: 517.588+517.44

MSC: Primary 33C20; Secondary 33C60

Received: 29.07.2025
Accepted: 07.11.2025

DOI: 10.47910/FEMJ202520