The orthogonality property of the Lommel polynomials and a twofold infinity of relations between Rayleigh's σ-sums

Abstract

Making use of a remarkable theorem which expresses a relationship between a certain type of infinite continued fractions and systems of orthogonal polynomials, it is proven that the known infinite continued fraction development of the ratio of Bessel functions J v−1 ( z)/ J v ( z) gives rise to an orthogonality property of the Lommel polynomials {R m,v( 1 z )|mϵ N} when v is real and positive. The corresponding weight function which appears to be non-negative in the interval of definition, is obtained by the application of two successive integral transforms. It consists of an infinite series of Dirac δ-functions whose singularities are distributed symmetrically around the origin on the real axis in such a manner that the origin is their limit point on both sides. For any positive v, the Lommel polynomials form a system of so-called orthogonal polynomials of a discrete variable. The orthogonality property may also be conveniently expressed by means of a Stieltjes integral. One of its corollaries is a twofold infinity of linear relations between the sums σ v ( r) defined by σ v ( r) = Σ n=1 +∞1/ j v, n 2 r , with v+1Σ R 0 +, rΣ N 0, in which j v,n represents the n th positive zero of J v(z) . Another by-product consists of a complement to a theorem of Hurwitz concerning the nature and the position of the zeros of the Lommel polynomials written as g m, v ( z) in the modified notation of the mentioned author. From this study also result two interesting approximations of j v,1 applicable for vϵ]−1, +1].