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

[摘要] 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, respectively 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 general. Moreover, we assume that the coefficients of the polynomials sigma and tau have no dependence on nu. 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(nu)'(z) with the functions y(nu)(z) and y(nu+1)(z) or y(nu-1)(z). Finally, these three relationships are applied to the polynomials of hypergeometric type which form a broad subclass of functions y(nu), where nu 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. (C) 1994 Academic Press, Inc.

[发布日期] 1994-12-15