Differentiation identities for hypergeometric functions

TL;DR

This paper provides a straightforward proof of differentiation identities for hypergeometric functions, which are widely used in applied mathematics and sciences, using only their power series definitions.

Contribution

It introduces a simple proof method for hypergeometric differentiation identities based solely on power series coefficients, simplifying existing proofs.

Findings

Proof based only on power series coefficients

Simplifies understanding of hypergeometric differentiation identities

Applicable to various fields in applied mathematics and sciences

Abstract

It is well-known that differentiation of hypergeometric function multiplied by a certain power function yields another hypergeometric function with a different set of parameters. Such differentiation identities for hypergeometric functions have been used widely in various fields of applied mathematics and natural sciences. In this expository note, we provide a simple proof of the differentiation identities, which is based only on the definition of the coefficients for the power series expansion of the hypergeometric functions.