Non-linear Characterizations for Functions of Hypergeometric Type and Their Derivatives of Any Order

To find characterizations for special functions is an important question in applied mathematics and mathematical physics. Up to now, most effort has been invested for polynomial sets which are orthogonal on the real line. In this paper two non-linear characterizations for a class of functions, the so-called functions of hypergeometric type, are found not only for the functions themselves but also for their derivatives of any order. Members of this class are the Bessel functions, the hypergeometric functions, the classical orthogonal polynomials as well as many other functions of mathematics and physics. Applications to the subclass of polynomials of hypergeometric type are made, with emphasis on the classical orthogonal polynomials for which the coefficients of the above-mentioned non-linear characterizations are tabulated.