Introduction

Abstract

In the 18th and 19th centuries there appeared a great number of types of special functions to solve the differential equations of mathematical physics and to calculate integrals. Many of them turned out to be special or limiting cases of the hypergeometric function F(α, β; γ; x) introduced in 1769 by L. Euler and scrutinized at the beginning of the 19th century by Gauss. Gauss’ work triggered a flow of investigations which established different recurrence relations, differential equations, integral representations, generating functions, addition and multiplication theorems, asymptotic expansions for the hypergeometric function and its associates (Legendre, Gegenbauer, Hermite, Laguerre, Chebyshev polynomials; Bessel, Neumann, Macdonald, Whittaker functions, etc.), sought for relations between these functions, and calculated puzzling integrals involving them, etc.