THE 3-TERM RECURRENCE RELATION AND THE DIFFERENTIATION FORMULAS FOR HYPERGEOMETRIC-TYPE FUNCTIONS

The functions of hypergeometric type are the solutions y = y(nu)(x) of the differential equation sigma(z)y'' + tau(z)y' + lambda y = 0 where sigma, tau are polynomials of degrees not higher than 2 and 1, respec tively and lambda is a constant. Here we consider a class of functions of hypergeometric type with the additional condition that lambda + nu tau' + 1/2 nu(nu - 1)sigma'' = 0, nu being a complex number, in gener al. Moreover, we assume that the coefficients of the polynomials sigma and tau have no dependence on nu. To this class of functions belong G auss, Kummer, and Hermite functions, the classical orthogonal polynomi als, and many other functions encountered in linear and non-linear phy sics. We obtain two important structural properties of these functions : (i) the so-called three-term recurrence relation which correlates th ree functions of successive orders, and (ii) the differentiation formu las (also called ladder or structure relations or, even, differential- recurrence relations) which relate the first derivative y(nu)'(z) with the functions y(nu)(z) and y(nu+1)(z) or y(nu-1)(z). Finally, these t hree relationships are applied to the polynomials of hypergeometric ty pe which form a broad subclass of functions y(nu), where nu is a posit ive integer number and the associated contour is closed. For completen ess, the explicit expressions corresponding to all classical orthogona l polynomials (Jacobi, Laguerre, Hermite, and Bessel) are tabulated. ( C) 1994 Academic Press, Inc.