Quadratic Transformations of Hypergeometric Function and Series with Harmonic Numbers

TL;DR

This paper explores quadratic transformations of hypergeometric functions to derive new series transformation and summation formulas involving harmonic numbers, including a generating function expressed via hypergeometric and elementary functions.

Contribution

It introduces novel transformation and summation formulas for series with harmonic numbers using quadratic transformations of hypergeometric functions.

Findings

Derived new transformation formulas for series with harmonic numbers.

Provided a generating function involving hypergeometric and elementary functions.

Connected quadratic transformations to series with harmonic numbers.

Abstract

In this brief note, we show how to apply Kummer's and other quadratic transformation formulas for Gauss' and generalized hypergeometric functions in order to obtain transformation and summation formulas for series with harmonic numbers that contain one or two continuous parameters. We also give a generating function of the sequence $\frac{(a)_n (1-a)_n}{(n!)^2}H_n$ as a combination of Gauss hypergeometric function and elementary functions.