The Three-Term Recurrence Relation and the Differentiation Formulas for Hypergeometric-type Functions

Abstract

The functions of hypergeometric type are the solutions y ≡ yν(x) of the differential equation σ(z)y" + τ(z)y′ + λy = 0 where σ, τ are polynomials of degrees not higher than 2 and 1, respectively and λ is a constant. Here we consider a class of functions of hypergeometric type with the additional condition that λ + ντ′ + 12ν(ν − 1)σ" = 0, ν being a complex number, in general. Moreover, we assume that the coefficients of the polynomials σ and τ have no dependence on ν. To this class of functions belong Gauss, Kummer, and Hermite functions, the classical orthogonal polynomials, and many other functions encountered in linear and non-linear physics. We obtain two important structural properties of these functions: (i) the so-called three-term recurrence relation which correlates three functions of successive orders, and (ii) the differentiation formulas (also called ladder or structure relations or, even, differential-recurrence relations) which relate the first derivative y′ν(z) with the functions yν(z) and yν+1(z) or yν+1(z). Finally, these three relationships are applied to the polynomials of hypergeometric type which form a broad subclass of functions yν, where ν is a positive integer number and the associated contour is closed. For completeness, the explicit expressions corresponding to all classical orthogonal polynomials (Jacobi, Laguerre, Hermite, and Bessel) are tabulated.