It appears that the only known representations for the Riemann zeta function ζ(z) in terms of continued fractions are those for z = 2 and 3. Here we give a rapidly converging continued-fraction expansion of ζ(n) for any integer n ≥ 2. This is a special case of a more general expansion which we have derived for the polylogarithms of order n, n ≥ 1, by using the classical Stieltjes technique. Our...

This paper is concerned with the existence of monotone positive solution for the following third-order nonlocal boundary value problem u′′′ (t)+f (t, u (t) , u′ (t)) = 0, 0 < t < 1; u (0) = 0, au′ (0) − bu′′ (0) = α[u], cu′ (1) + du′′ (1) = β[u], where f ∈ C([0, 1] × R+ × R+, R+), α[u] = ∫ 1 0 u(t)dA(t) and β[u] = ∫ 1 0 u(t)dB(t) are linear functionals on C[0, 1] given by Riemann-Stieltjes inte...

Abstract. Both analytic and geometric forms of an optimal monotone principle for Lp-integral of the Green function of a simply-connected planar domain Ω with rectifiable simple curve as boundary are established through a sharp one-dimensional power integral estimate of Riemann-Stieltjes type and the Huber analytic and geometric isoperimetric inequalities under finiteness of the positive part of...

Polynomial solutions to the generalized Lamé equation, the Stieltjes polynomials, and the associated Van Vleck polynomials have been studied since the 1830’s, beginning with Lamé in his studies of the Laplace equation on an ellipsoid, and in an ever widening variety of applications since. In this paper we show how the zeros of Stieltjes polynomials are distributed and present two new interlacin...

In Machine Learning (ML) algorithms, one of the crucial issues is the representation of the data. As the data become heterogeneous and large-scale, single kernel methods become insufficient to classify nonlinear data. The finite combinations of kernels are limited up to a finite choice. In order to overcome this discrepancy, we propose a novel method of ”infinite” kernel combinations for learni...

Large-scale problems in scientific and engineering computing often require solutions involving large-scale matrices. In this paper, we survey numerical techniques for solving a variety of large-scale matrix computation problems, such as computing the entries and trace of the inverse of a matrix, computing the determinant of a matrix, and computing the transfer function of a linear dynamical sys...

Lower and upper bounds of the μ Cebyšev functional for the RiemannStieltjes integral are given. Applications for the three point quadrature rules of functions that are n time di¤erentiable are also provided. 1. Introduction In 1975, P.R. Beesack [1] showed that, if y; v; w are real valued functions de…ned on a compact interval [a; b] ; where w is of bounded variation with total variation Wb a (...