Lee Lorch

Completely monotonic functions associated with the gamma function and its q-analogues

Several functions involving the gamma function Γ(x) and the q-gamma function Γq(x) are proved to be completely monotonic. Some of these results are used to establish the infinite divisibility of a number of probability distributions defined via their moment generating functions.

Inequalities for ultraspherical polynomials and the gamma function

Abstract Let M n (λ) = (n + λ) 1 − λ max 0⩽θ⩽π ( sin θ) λ ¦P n (λ) ( cos θ)¦, where P n (λ) (x) is the ultraspherical polynomial of degree n and parameter λ. It is shown that M n (λ) 2 1 − λ Γ(λ) , for 0 and n = 0, 1, 2… When λ = 0 and when λ = 1 , this inequality becomes an equality. It refines inequality (7.33.5) of G. Szegos “Orthogonal Polynomials” (4th edition 1975, p. 171), wherein the factor ( n + λ ) 1 − λ is replaced by n 1 − λ . The method of proof requires sharpening some inequalities for the ratio Γ(n + λ) Γ(n + 1) , n = 0, 1, 2,… .

Turánians and Wronskians for the Zeros of Bessel Functions

Paul Turan [On the zeros of the polynomials of Legendre, Casopis pro Peěstovani Mat. a Fys., 75 (1950), pp. 113–122] proved that the Legendre polynomials satisfy the inequality

Some inequalities for the first positive zeros of Bessel functions

P_n (x)P_{n + 2} (x) - [P_{n + 1} (x)]^2 < 0, - 1 < x < 1

MONOTONIC SEQUENCES RELATED TO ZEROS OF BESSEL FUNCTIONS

. Here it is shown that the positive zeros of arbitrary real Bessel functions satisfy sjmilar inequalities, even in a more general form. An analogous result is established for the corresponding Wronskian. In § 8, Remark 3, the monotonicity results established in the course of the proofs here are used to complement those derived by Sturm methods in [LEE LORCH, Elementary comparison techniques for certain classes of Sturm–Liouville equations, Proc. Uppsala 1977 Inter. Conf. Dill. Equations, Symposia Univ. Upsaliensis Annum Quingentesimum Celebrantis 7, Acta Univ. Upsaliensis, Uppsala 1977, pp. 125–133].

“Best possible” upper bounds for the first positive zeros of Bessel functions—the finite part

For the first positive zero

Monotonicity of the differences of zeros of Bessel functions as a function of order

j = j_{\nu 1}

Some Monotonicity Properties of Bessel Functions

of the Bessel function

A CHARACTERIZATION OF TOTALLY REGULAR (J, f(x)) TRANSFORMS

J_\nu (x)

The Gibbs phenomenon for Borel means

, it is shown for