On a Hankel transform integral containing an exponential function and two Laguerre polynomials

Given an entire function f ( z ) f(z) that has only negative zeros, we shall prove that all the functions of type f ( m ) ( x ) / f ( n ) ( x ) , m > n f^{(m)}(x)/f^{(n)}(x),\ m>n are completely monotonic. Examples of this type are given for Laguerre polynomials, ultraspherical polynomials, Jacobi polynomials, Stieltjes-Wigert polynomials, q q -Laguerre polynomials, Askey-Wilson polynomials, Ramanujan function, q q -exponential functions, q q -Bessel functions, Euler’s gamma function, Airy function, modified Bessel functions of the first and the second kind, and the confluent basic hypergeometric series.

In the numerical integration of nonlinear autonomous initial value problems, the computational process depends on the step size scaled vector field hf as a distinct entity. This paper considers a parameterized transformation hf↦hf∘(I-γhf)-1,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} hf \\mapsto hf \\circ (I-\\gamma hf)^{-1}, \\end{aligned}$$\\end{document}and its role in the finite step size stability of singly diagonally implicit Runge—Kutta (SDIRK) methods. For a suitably chosen gamma > 0, the transformed map is Lipschitz continuous with a reasonably small constant, whenever hf is negative monotone. With this transformation, an SDIRK method is equivalent to an explicit Runge–Kutta (ERK) method applied to the transformed vector field. Through this mapping, the SDIRK methods’ A-stability, and linear order conditions are investigated. The latter are closely related to approximations of the exponential function textrm{e}^z that are polynomial in z, when considering ERK methods, and polynomial in terms of the transformed variable z(1-gamma z)^{-1}, in case of SDIRK methods. Considering the second family of methods, and expanding the exponential function in terms of this transformed variable, the coefficients can be expressed in terms of Laguerre polynomials. Lastly, a family of methods is constructed using the transformed vector field, and its order conditions, A-stability, and B-stability are investigated.