Quadratic Transformations of Hypergeometric Function and Series with Harmonic Numbers

2018/01/08 by Martin Nicholson, Nicholson, Martin
Mathematics · #Advanced Mathematical Identities #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1801.02428

openalex publication_date 2018/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ((a)_n (1-a)_n)/((n!)²)H_n as a combination of Gauss hypergeometric function and elementary functions.

Citations

• The integrals in Gradshteyn and Ryzhik. Part 16: Complete elliptic integrals
• Watson-type 3F2-series and summation formulae involving generalized harmonic numbers

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