Orthogonal confluent hypergeometric lattice polynomials

摘要

Here is a method of solving the difference-differential equations of the confluent hypergeometric differential equation using a generalized Pochhammer matrix product. This method provides a convenient analytical way to relate various solutions of the confluent hypergeometric function to each other when their parameters fall on the same point lattice. These solutions also are of interest to the general classification of orthogonal polynomials and the metrics used to generate them. This method generates Laurent polynomials over the complex domain that are an orthogonal system utilizing a 2×2 matrix weight function where the weight matrix has elements that are products of a Kummer solution and its derivative. The index-incremented Pochhammer matrix polynomials obey a 4×4 system of differential equations with a Frobenius solution involving non-commuting matrices that also extends these results to non-integer values but with infinite Laurent series. The termination condition for a polynomial series in the midst of infinite series sheds light on solving general systems of regular linear differential equations. The differential equations generalize Heun's double confluent equation with matrix coefficients. For a radiative transfer flux integral there is a distinct advantage of using these lattice polynomials compared to an asymptotic series/power series combination. We conjecture similar convergence properties for evaluations of confluent hypergeometric functions of either kind and that these matrix methods can be extended to gauss hypergeometric functions and generalized hypergeometric functions.