Recurrences and explicit formulae for the expansion and connection coefficients in series of ordinary Bessel polynomials

Abstract

A formula expressing explicitly the derivatives of ordinary Bessel polynomials of any degree and for any order in terms of the ordinary Bessel polynomials themselves is proved. Another explicit formula, which expresses the ordinary Bessel expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of its original ordinary Bessel coefficients, is also given. A formula for the ordinary Bessel coefficients of the moments of one single ordinary Bessel polynomial of certain degree is proved. Formula for the ordinary Bessel coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its ordinary Bessel coefficients is also obtained. Application of these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica) in order to build and solve recursively for the connection coefficients between generalized Bessel and ordinary Bessel polynomials is described. Explicit formula for these coefficients between Jacobi and ordinary Bessel polynomials is given, of which the ultraspherical polynomials and its consequences are important special cases. Two analytical formulae for the connection coefficients between Laguerre–ordinary Bessel and Hermite–ordinary Bessel are also developed.