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

摘要

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 Jv−1(z)/Jv(z) gives rise to an orthogonality property of the Lommel polynomials {Rm,v(1z)|mϵN} 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 σv(r) defined by σv(r)=Σn=1+∞1/jv,n2r, with v+1ΣR0+, rΣN0, in which jv,n represents the nth positive zero of Jv(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 gm,v(z) in the modified notation of the mentioned author. From this study also result two interesting approximations of jv,1 applicable for vϵ]−1, +1].