Search

Lommel's function s μ,ν(z) is a particular solution of the differential equation z 2y″ + zy′ + (z 2 - ν 2)y = z μ+1. Here we present estimates and monotonicity properties of the positive zeros of s μ-1/2,1/2(z) when μ ∈ (0, 1). The positivity of a closely related integral is also considered. © 2012 American Mathematical Society. 140 9 3091 3100 Cited By :8

Indefinite integrals containing Bessel functions and their representation as simple Neumann series or alternatively in terms of Lommel's functions of two variables have been noted in the literature in connection with physical problems [1; 2; 3]. It is observed here that by a simple generalization of a result noted by Watson [4, p. 23, footnote] expressing a generating function for Bessel's function as an indefinite integral, all of the previously noted examples may be obtained as particular cases of a more general result, which is then applied to the evaluation of an integral arising in connection with noise theory. If we define

In particular we shall first determine a radius of univalence for the normalized Bessel functions [J,(z) ]1Iv for values of v belonging to the region G defined by the inequalities (i v} >0, I arg v

hnk,(X) = Rn,(11X)2 we have a polynomial set that obeys 1n,,(X) = 2x(n + v1)In_1, (x) hn-2,,(X). It was noted by Hahn [5] that this is an orthogonal type recurrence relation. In this paper, we shall establish explicitly the orthogonality of the hn;,(x) and will present some properties of the Bessel polynomials yn(X), a polynomial set that is orthogonal in a special sense and whose theory in many ways parallels that of the modified Lommel polynomials. From some formulas involving Lommel polynomials (see Watson [10, ?9.6]), we may define the modified Lommel polynomials hn,,(x) in terms of Bessel functions by