Fundamental recurrence relations of functions of hypergeometric type and their derivatives of any order

Abstract

Letyν(k)(z), k≥0, denote thek-th derivative of the function of hydrogeometric typey ≡ yν(z). The latter name is given to the solutions of the differential equationσ(z)y″+τ(z)y′+λy=0, where σ and τ are polynomials of degree not higher than 2 and 1, respectively, and λ is a constant. The functions of hypergeometric type form a broad class of special functions in Mathematical Physics, to which Bessel functions, hypergeometric functions, the classical orthogonal polynomials and many other functions found in several branches of physics belong. Here the fundamental three-term recurrence and differential-recurrence relations of these functions and their derivatives are explicity shown. Firstly, we obtain the relation among thek-th derivatives of three functions of successive orders. Then, two generalized structure relations are found; these relations involveyν(k+1)(z), yν±1(k)(z) andyν(k)(z). In so doing, we generalize the known recurrence relation and differentation formulae of the functions of hypergeometric type. Finally, we apply all these relationships to the polynomials of hypergeometric type, which form a large subclass of functionsyν(z), where ν is a positive integer and the associated contour is closed. For illustration, the explicit expressions corresponding to all classical orthogonal polynomials (Jacobi, Laguerre, Hermite, Bessel) are tabulated.