Classical Orthogonal Polynomials of a Discrete Variable

Mathematical modelling of many physical processes involves rather complex dif­ ferential, integral, and integro-differential equations which can be solved directly only in a number of cases. Therefore, as a first step, an original problem has to be considerably simplified in order to get a preliminary knowledge of the most important qualitative features of the process under investigation and to estimate the effect of various factors. Sometimes a solution of the simplified problem can be obtained in the analytical form convenient for further investigation. At this stage of the mathematical modelling it is useful to apply various special functions. Many model problems of atomic, molecular, and nuclear physics, electrody­ namics, and acoustics may be reduced to equations of hypergeometric type, a(x)y" + r(x)y' + AY = 0 , (0.1) where a(x) and r(x) are polynomials of at most the second and first degree re­ spectively and A is a constant [E7, AI, N18]. Some solutions of (0.1) are functions extensively used in mathematical physics such as classical orthogonal polyno­ mials (the Jacobi, Laguerre, and Hermite polynomials) and hypergeometric and confluent hypergeometric functions.