1 - Introduction and Preliminaries

Abstract

This chapter elaborates the definitions and notations of some special functions, polynomials, and numbers. The special functions include the gamma, beta, the polygamma functions, the multiple gamma functions, the Gaussian hypergeometric function, and the generalized hypergeometric function. It also reviews the two kinds of Stirling numbers, the Bernoulli and Euler, and their generalized forms. Stirling formula offers a simple way to compute n!, which for a large positive integer n is a very tedious task to perform. For a long time, double gamma and multiple gamma functions did not come in the limelight, but in the course of time these were used to prove many classical formula, such as the Integral formulas. The hypergeometric equation is the most celebrated equation of the Fuchsian class, which consists of differential equations with regular singular points. Its importance stems, in part, from a known theorem that every homogeneous linear differential equation of the second order, whose singularities are regular and at most three in number, can be transformed into the hypergeometric equation.