Various integral transforms have been extended to various spaces of Boehmians. In this article, we discuss the Stieltjes transform in a class of Boehmians. The presented transform preserves many properties of the classical transform in the space of Boehmians.

The Stieltjes constants γk(a) are the expansion coefficients in the Laurent series for the Hurwitz zeta function about s = 1. We present new asymptotic, summatory, and other exact expressions for these and related constants.

where lnk z = ln z+2πik, and ln z is the principal branch of natural logarithm [14]. This paper considers only the principal branch k = 0, which is the branch that maps the positive real axis onto itself, and therefore we shall usually abbreviate W0 as W herein. Many functions of W are members of a number of function classes, specifically, the classes of Stieltjes functions, Pick functions and ...

A theorem of Stieltjes proves that, given any sequence {pn}n=0 of orthogonal polynomials, there is at least one zero of pn between any two consecutive zeros of pk if k < n, a property called Stieltjes interlacing. We show that Stieltjes interlacing extends, under certain conditions, to the zeros of Jacobi polynomials from different sequences. In particular, we prove that the zeros of P n+1 inte...

We express the generalized Cauchy-Stieltjes transforms (GCST) of some particular Beta distributions depending on a positive parameter λ as λ-powered Cauchy-Stieltjes transforms (CST) of some probability measures. The CST of the latter measures are shown to be the geometric mean of the CST of the Wigner law together with another one. Moreover, they are absolutely continuous and we derive their d...

Let wλ(x) := (1−x2)λ−1/2 and P (λ) n be the ultraspherical polynomials with respect to wλ(x). Then we denote by E (λ) n+1 the Stieltjes polynomials with respect to wλ(x) satisfying ∫ 1 −1 wλ(x)P (λ) n (x)E (λ) n+1(x)x dx { = 0, 0 ≤ m < n+ 1, = 0, m = n+ 1. In this paper, we show uniform convergence of the Hermite–Fejér interpolation polynomials Hn+1[·] and H2n+1[·] based on the zeros of the Sti...

Abstract We investigate the existence of solutions for a nonlinear integral inclusion Urysohn–Stieltjes type. As applications, we give Chandrasekhar quadratic equation and inclusion.

We obtain a sharp estimate of the speed convergence in Boolean central limit theorem for measures with finite sixth moment. The main tool is quantitative version Stieltjes-Perron inversion formula.