Related papers: Some applications of the Stieltjes constants
The Stieltjes constants $\gamma_n$ appear in the coefficients in the Laurent expansion of the Riemann zeta function $\zeta(s)$ about the simple pole $s=1$. We present an asymptotic expansion for $\gamma_n$ as $n\rightarrow \infty$ based on…
The Stieltjes constants are the coefficients of the Laurent expansion of the Hurwitz zeta function and surprisingly little is known about them. In this paper we derive some relations for the difference between two Stieltjes constants…
The Stieltjes constants $\gamma_k(a)$ appear in the regular part of the Laurent expansion for the Hurwitz zeta function $\zeta(s,a)$. We present summatory results for these constants $\gamma_k(a)$ in terms of fundamental mathematical…
A closed form of the multi-peakon solutions of the Camassa-Holm equation is found using a theorem of Stieltjes on continued fractions. An explicit formula is obtained for the scattering shifts.
We present a simple but efficient method of calculating Stieltjes constants at a very high level of precision, up to about 80000 significant digits. This method is based on the hypergeometric-like expansion for the Riemann zeta function…
The Stieltjes constants $\gamma_k(a)$ appear in the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function $\zeta(s,a)$ about its only pole at $s=1$. We generalize a technique of Addison for the Euler…
In our previous work we found sufficient conditions to be imposed on the parameters of the generalized hypergeometric function in order that it be completely monotonic or of Stieltjes class. In this paper we collect a number of consequences…
In this work, we extend the concept of the Stieltjes derivative to encompass left-continuous derivators with bounded variation, thereby relaxing the monotonicity constraint. This generalization necessitates a refined definition of the…
We define integrals for functions on finite-dimensional algebras, adapting methods from Leinster's research. This paper discusses the relationships between the integrals of functions defined on subsets $\mathbb{I}_1 \subseteq…
Sharp inequalitieis of Gruss type for Stieltjes integrals with application in numerical integration are provided.
In this paper, we consider meromorphic extension of the function \[ \zeta_{h^{\left( r\right) }}\left( s\right) =\sum_{k=1}^{\infty} \frac{h_{k}^{\left( r\right) }}{k^{s}},\text{ }\operatorname{Re}\left( s\right) >r, \] (which we call…
Stieltjes boundary problems generalize the customary class of well-posed two-point boundary value problems in three independent directions, regarding the specification of the boundary conditions: (1) They allow more than two evaluation…
The Stieltjes constants 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.
In this paper new series for the first and second Stieltjes constants (also known as generalized Euler's constant), as well as for some closely related constants are obtained. These series contain rational terms only and involve the…
The Stieltjes constants $\gamma_k$ appear in the regular part of the Laurent expansion of the Riemman and Hurwitz zeta functions. We demonstrate that these coefficients may be written as certain summations over mathematical constants and…
We obtain a variety of series and integral representations of the digamma function $\psi(a)$. These in turn provide representations of the evaluations $\psi(p/q)$ at rational argument and for the polygamma function $\psi^{(j)}$. The…
Monotonicity properties of the ratio $$ \log \frac{f(x+a_1)\cdots f(x+a_n)}{f(x+b_1)\cdots f(x+b_n)}, $$ where $f$ is an entire function are investigated. Earlier results for Euler's gamma function and other entire functions of genus 1 are…
The present book gives a systematic overview of function theory and the theory of Stieltjes integral. In particular, we give a detailed account of the theory of functions of bounded variation and of the theory of regulated functions (=…
This work deals with the obtaining of solutions of first and second order Stieltjes differential equations. We define the notions of Stieltjes derivative on the whole domain of the functions involved, provide a notion of n-times…
In this paper, motivated by physical considerations, we introduce the notion of modified Riemann sums of Riemann-Stieltjes integrable functions, show that they converge, and compute them explicitely under various assumptions.