Bounds for modified Lommel functions of the first kind and their ratios

Abstract

The modified Lommel function t µ,ν (x) is an important special function, but to date there has been little progress on the problem of obtaining functional inequalities for t µ,ν (x). In this paper, we advance the literature substantially by obtaining a simple two-sided inequality for the ratio t µ,ν (x)/t µ−1,ν−1 (x) in terms of the ratio I ν (x)/I ν−1 (x) of modified Bessel functions of the first kind, thereby allowing one to exploit the extensive literature on bounds for this ratio. We apply this result to obtain two-sided inequalities for the condition numbers xt ′ µ,ν (x)/t µ,ν (x), the ratio t µ,ν (x)/t µ,ν (y) and the modified Lommel function t µ,ν (x) itself that are given in terms of xI ′ ν (x)/I ν (x), I ν (x)/I ν (y) and I ν (x), respectively, which again allows one to exploit the substantial literature on bounds for these quantities. The bounds obtained in this paper are quite accurate and often tight in certain limits. As an important special case we deduce bounds for modified Struve functions of the first kind and their ratios, some of which are new, whilst others extend the range of validity of some results given in the recent literature.

Inequalities for Some Integrals Involving Modified Lommel Functions of the First Kind

“…As such, the bounds obtained in this paper generalise those of [11,13]. Modified Lommel functions are widely used special functions, arising in areas of the applied sciences as diverse as the theory of steady-state temperature distribution [16], scattering amplitudes in quantum optics [25] and stress distributions in cylindrical objects [22]; see [12] for a list of further applications. The modified Lommel function of the first kind t µ,ν (x) is defined by the hypergeometric series…”

“…In the literature different notation is used for the modified Lommel functions; we adopt that of [27]. The terminology modified Lommel function of the first kind is also not standard in the literature, but has recently been adopted by [12]. Also, [2] have used the terminology Lommel function of the first kind for the function s µ,ν (x), which is related to the modified Lommel function of the first kind by t µ,ν (x) = −i 1−µ s µ,ν (ix) (see [21,27]).…”

“…For the purposes of this paper, we follow [12] and use the following normalization which will remove a number of multiplicative constants from our calculations:…”

“…As already noted, the properties of the modified Struve function L ν (x) that were exploited in the proofs of [11,13] are shared by the modified Lommel functiont µ,ν (x), which we now list. With the exception of the differentiation formula (1.9) (see [21]), the following basic properties can all be found in [12]. For x > 0, the functiont µ,ν (x) is positive if µ − ν ≥ −3 and µ + ν ≥ −3.…”

“…) . We shall also need another differentation formula that is not given in [12] or [21]. With the aid of (1.7) and (1.8) we obtain…”