Construction and properties of two sequences of orthogonal polynomials and the infinitely many, recursively generated sequences of associated orthogonal polynomials, directly related to Mathieu's differential equation and functions - Part I -

Abstract

When an attempt is made to construct an even particular solution of Mathieu's differential equation (written in a slightly modified notation), either with period 2π or with period 4π, by inserting into it an appropriate trigonometric series of the cosine type, it appears that the coefficients of this series satisfy a recurrence relation of a kind encountered in the theory of orthogonal polynomials. After splitting off a certain proportionality factor in each coefficient, there ultimately result two infinite sequences of parameter-dependent orthogonal polynomials constituting, respectively, a generalization of the Chebyshev polynomials of the first kind {T_n(x)} and a generalization of another special case of the Jacobi polynomials, namely {P^(1/2,−1/2)_n(x)}. The weight function corresponding to each of these orthogonal sequences consists of an infinite series of Dirac δ-peaks ("mass-points") whose locations and (positive) coefficients are given by quantities appearing in the standard theory of Mathieu functions. With the aid of the author's theory of recursive generation of systems of orthogonal polynomials, each of the two orthogonal sequences may be complemented with infinitely many systems of associated orthogonal polynomials whose recurrence relations present a positive integer shift on the discrete independent variable compared to the recursion of the orthogonal sequence from which they originate. The treatment of Mathieu's differential equation with trigonometric series of the sine type, in view of constructing an odd particular solution with 2π or 4π as period, does not produce any sequence of orthogonal polynomials which is in essence not already comprised in those previously obtained.