The Stieltjes constants γ_k(a) appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function ζ(s,a) about s=1. We present series representations of these constants of interest to theoretical and computational analytic number theory. A particular result gives an addition formula for the Stieltjes constants. As a byproduct, expressions for derivatives of all orders of the Stieltjes coefficients are given. Many other results are obtained, including instances of an exponentially fast converging series representation for γ_k=γ_k(1). Some extensions are briefly described, as well as the relevance to expansions of Dirichlet L functions.
We present a variety of series representations of the Stieltjes and related constants, the Stieltjes constants being the coefficients of the Laurent expansion of the Hurwitz zeta function zeta(s,a) about s=1. Additionally we obtain series and integral representations of a sum S_γ(n) formed as an alternating binomial series from the Stieltjes constants. The slowly varying sum S_γ(n)+n is an important subsum in application of the Li criterion for the Riemann hypothesis.
We present expressions in terms of a double infinite series for the Stieltjes constants $γ_k(a)$. These constants appear in the regular part of the Laurent expansion for the Hurwitz zeta function. We show that the case $γ_k(1)=γ$ corresponds to a series representation for the Riemann zeta function given much earlier by Brun. As a byproduct, we obtain a parameterized double series representation of the Hurwitz zeta function.
The Stieltjes constants $γ_k(a)$ appear in the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function $ζ(s,a)$ about its only pole at $s=1$. We generalize a technique of Addison for the Euler constant $γ=γ_0(1)$ to show its application to finding series representations for these constants. Other generalizations of representations of $γ$ are given.
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.
The Stieltjes constants $γ_k(a)$ appear in the regular part of the Laurent expansion of the Hurwitz zeta function about its only polar singularity at $s=1$. We present multi-parameter summation relations for these constants that result from identities for the Hurwitz zeta function. We also present multi-parameter summation relations for functions $A_k(x)$ that may be expressed as sums over the Stieltjes constants. Integral representations, especially including Mellin transforms, play an important role. As a byproduct, reciprocity and other summatory relations for polygamma functions and Bernou…
We develop series representations for the Hurwitz and Riemann zeta functions in terms of generalized Bernoulli numbers (Nörlund polynomials), that give the analytic continuation of these functions to the entire complex plane. Special cases yield series representations of a wide variety of special functions and numbers, including log Gamma, the digamma, and polygamma functions. A further byproduct is that $ζ(n)$ values emerge as nonlinear Euler sums in terms of generalized harmonic numbers. We additionally obtain series and integral representations of the first Stieltjes constant $γ_1(a)$. The …
The Stieltjes coefficients $γ_k(a)$ arise in the expansion of the Hurwitz zeta function $ζ(s,a)$ about its single simple pole at $s=1$ and are of fundamental and long-standing importance in analytic number theory and other disciplines. We present an array of exact results for the Stieltjes coefficients, including series representations and summatory relations. Other integral representations provide the difference of Stieltjes coefficients at rational arguments. The presentation serves to link a variety of topics in analysis and special function and special number theory, including logarithmic …
Recently, several studies of non-Hermitian Hamiltonians having $\mathcal{PT}$ symmetry have been conducted. Most striking about these complex Hamiltonians is how closely their properties resemble those of conventional Hermitian Hamiltonians. This paper presents further evidence of the similarity of these Hamiltonians to Hermitian Hamiltonians by examining the summation of the divergent weak-coupling perturbation series for the ground-state energy of the $\mathcal{PT}$-symmetric Hamiltonian $H=p^2+{1/4}x^2+iλx^3$ recently studied by Bender and Dunne. For this purpose the first 193 (nonzero) coe…
The Stieltjes constants gamma_k(a) are the expansion coefficients in the Laurent series for the Hurwitz zeta function about its only pole at s=1. We present the relation of gamma_k(1) to the eta_j coefficients that appear in the Laurent expansion of the logarithmic derivative of the Riemann zeta function about its pole at s=1. We obtain novel integral representations of the Stieltjes constants and new decompositions such as S_2(n) = S_gamma(n) + S_Lambda(n) for the crucial oscillatory subsum of the Li criterion for the Riemann hypothesis. The sum S_γ(n) is O(n) and we present various integral …
The Stieltjes constants $γ_k$ 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 specialized hypergeometric functions $_pF_{p+1}$. This family of results generalizes a representation of the Euler constant in terms of a summation over values of the trigonometric integrals Si or Ci. The series representations are suitable for acceleration. As byproducts, we evaluate certain sine-logarithm integrals and present the leading asymptotic form of the p…
A series of physically motivated operations appearing in the study of composite materials are interpreted in terms of elementary continued fraction transforms of matrix valued, rational Stieltjes functions.
We solve problem x proposed by O. Oloa, AMM xxx 2012 {\bf 119?} (to appear), p. yyy for certain definite logarithmic integrals. A number of generating functions are developed with certain coefficients $p_n$, and some extensions are presented. The explicit relation of $p_n$ to Nörlund numbers $B_n^{(n)}$ is discussed. Certain inequalities are conjectured for the $\{p_n\}$ sequence of coefficients, including its convexity, and an upper bound is demonstrated. It is shown that $p_n$ values may be used to express the Stieltjes constants for the Hurwitz and Riemann zeta functions, as well as values …
Dating back to Euler, in classical analysis and number theory, the Hurwitz zeta function $$ ζ(z,q)=\sum_{n=0}^{\infty}\frac{1}{(n+q)^{z}}, $$ the Riemann zeta function $ζ(z)$, the generalized Stieltjes constants $γ_k(q)$, the Euler constant $γ$, Euler's gamma function $Γ(q)$ and the digamma function $ψ(q)$ have many close connections on their definitions and properties. There are also many integrals, series or infinite product representations of them along the history. In this note, we try to provide a parallel story for the alternating Hurwitz zeta function (also known as the Hurwitz-type Eul…
The paper addresses the exact evaluation of the generalized Stieltjes transform $S_{n}[f]=\int_0^{\infty} f(x) (ω+x)^{-n}\mathrm{d}x$ of integral order $n=1,2, 3,\dots$ about $ω=0$ from which the asymptotic behavior of $S_{n}[f]$ for small parameters $ω$ is directly extracted. An attempt to evaluate the integral by expanding the integrand $(ω+x)^{-n}$ about $ω=0$ and then naively integrating the resulting infinite series term by term lead to an infinite series whose terms are divergent integrals. Assigning values to the divergent integrals, say, by analytic continuation or by Hadamard's finite…
We obtain a variety of series and integral representations of the digamma function $ψ(a)$. These in turn provide representations of the evaluations $ψ(p/q)$ at rational argument and for the polygamma function $ψ^{(j)}$. The approach is through a limit definition of the zeroth Stieltjes constant $γ_0(a)=-ψ(a)$. Several other results are obtained, including product representations for $\exp[γ_0(a)]$ and for the Gamma function $Γ(a)$. In addition, we present series representations in terms of trigonometric integrals Ci and Si for $ψ(a)$ and the Euler constant $γ=-ψ(1)$.
We present analytic properties and extensions of the constants ck appearing in the Baez-Duarte criterion for the Riemann hypothesis. These constants are the coefficients of Pochhammer polynomials in a series representation of the reciprocal of the Riemann zeta function. We present generalizations of this representation to the Hurwitz zeta and many other special functions. We relate the corresponding coefficients to other known constants including the Stieltjes constants and present summatory relations. In addition, we generalize the Maslanka hypergeometric-like representation for the zeta func…
We supplement a very recent paper of R. Crandall concerned with the multiprecision computation of several important special functions and numbers. We show an alternative series representation for the Riemann and Hurwitz zeta functions providing analytic continuation through out the whole complex plane. Additionally we demonstrate some series representations for the initial Stieltjes constants appearing in the Laurent expansion of the Hurwitz zeta function. A particular point of elaboration in these developments is the hypergeometric form and its equivalents for certain derivatives of the incom…
In this work we study the Plancherel-Rotach type asymptotics for selected $q$-series and $q$-orthogonal polynomials with complex scalings. The $q$-series we cover are Euler's $q$-exponential, Ramanujan function, Jackson's $q$-Bessel function of second kind, Ismail-Masson orthogonal polynomials, Stieltjes-Wigert polynomials and $q$-Laguerre polynomials. For a fixed $q$ with $0<q<1$, in each case the main term of the asymptotic formulas may contain Ramanujan function or theta function depending on the value of scaling parameter. Furthemore, when the scaling parameter is in certain strip of the c…
We find sets of solutions to the generalized spheroidal wave equation (GSWE) or, equivalently, to the confluent Heun equation. Each set is constituted by three solutions, one given by a series of ascending powers of the independent variable, and the others by series of regular and irregular confluent hypergeometric functions. For a fixed set, the solutions converge over different regions of the complex plane but present series coefficients proportional to each other. These solutions for the GSWE afford solutions to a double-confluent Heun equation by a taking-limit process due to Leaver. Anoth…
We generalize techniques of Addison to a vastly larger context. We obtain integral representations in terms of the first periodic Bernoulli polynomial for a number of important special functions including the Lerch zeta, polylogarithm, Dirichlet $L$- and Clausen functions. These results then enable a variety of Addison-type series representations of functions. Moreover, we obtain integral and Addison-type series for a variety of mathematical constants.
A new series representation of the Madelung constant is given. We represent Madelung constant as a sum of an exact term plus an exponentially fast converging series. The remarkable result is that even if the series part is discarded, one obtains Madelung constant correct up to ten good decimal figures. This, to the best of our knowledge, may be the fastest converging series representation of the Madelung constant. A few other important identities are also obtained.
For the almost Mathieu operator with a small coupling constant, for a series of spectral gaps, we describe the asymptotic locations of the gaps and get lower bounds for their lengths. The results are obtained by analysing a monodromy matrix.
In this note, using an idea from \cite{Amo-Carrillo-Sanchez} we derive some new series representations involving $ζ(2n)$ and Euler numbers. Using a well-known series representation for the Clausen function, we also provide some new representations of Apery's constant $ζ(3)$. In particular cases, we recover some well-known series representations of $π$.
