Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers

The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, are generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of double zeta values. A set of identities for the Ramanujan and generalized Ramanujan polynomials is presented. An alternative proof of Lettington's identity is provided, together with its generalizations to the Hurwitz and Lerch zeta functions, hence to Dirichlet $L$ series, to Eisenstein series, and to general Mellin transforms. The Hurwitz numbers $\tilde{H}_n$ occur in the Laurent expansion about the origin of a certain Weierstrass $\wp$ function for a square lattice, and are highly analogous to the Bernoulli numbers. An integral representation of the Laurent coefficients about the origin for general $\wp$ functions, and for these numbers in particular, is presented. As a Corollary, the asymptotic form of the Hurwitz numbers is determined. In addition, a series representation of the Hurwitz numbers is given, as well as a new recurrence.Comment: 40 pages, no figure

Parameterized summation relations for the Stieltjes constants

The Stieltjes constants $\gamma_k(a)$ 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 Bernoulli polynomials may be obtained.Comment: 15 pages, no figures, 2 new Propositions, Corollaries, and references adde

Special functions and the Mellin transforms of Laguerre and Hermite functions

We present explicit expressions for the Mellin transforms of Laguerre and Hermite functions in terms of a variety of special functions. We show that many of the properties of the resulting functions, including functional equations and reciprocity laws, are direct consequences of transformation formulae of hypergeometric functions. Interest in these results is reinforced by the fact that polynomial or other factors of the Mellin transforms have zeros only on the critical line Re s = 1/2. We additionally present a simple-zero Proposition for the Mellin transform of the wavefunction of the D-dimensional hydrogenic atom. These results are of interest to several areas including quantum mechanics and analytic number theory.Comment: 23 pages, no figures, to appear in Analysi

Generalizations of Russell-style integrals

First some definite integrals of W. H. L. Russell, almost all with trigonometric function integrands, are derived, and many generalized. Then a list is given in Russell-style of generalizations of integral identities of Amdeberhan and Moll. We conclude with a brief and noncomprehensive description of directions for further investigation, including the significant generalization to elliptic functions.Comment: 13 pages, no figure

Integral representations of functions and Addison-type series for mathematical constants

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.Comment: 36 pages, no figure

The Stieltjes constants, their relation to the eta_j coefficients, and representation of the Hurwitz zeta function

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_\gamma(n) is O(n) and we present various integral representations for it. We present novel series representations of S_2(n). We additionally present a rapidly convergent expression for \gamma_k= \gamma_k(1) and a variety of results pertinent to a parameterized representation of the Riemann and Hurwitz zeta functions.Comment: 37 pages, no figures Prop. 3(b) added and minor update

Some definite logarithmic integrals from Euler sums, and other integration results

We present explicit expressions for multi-fold logarithmic integrals that are equivalent to sums over polygamma functions at integer argument. Such relations find application in perturbative quantum field theory, quantum chemistry, analytic number theory, and elsewhere. The analysis includes the use of properties of a variety of special functions.Comment: 24 pages, no figure

Expected exit times of Brownian motion from planar domains: Complements to a paper of Markowsky

We supplement a very recent paper of G. Markowsky concerned with the expected exit times of Brownian motion from planar domains. Besides the use of conformal mapping, we apply results from potential theory. We treat the case of a wedge-shaped region exactly, subsuming an example of Markowsky. A number of other results are presented, including for a half disc, $n$-grams, and a variety of other regions.Comment: 30 pages, 7 figure

Integral and series representations of the digamma and polygamma functions

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 approach is through a limit definition of the zeroth Stieltjes constant $\gamma_0(a)=-\psi(a)$. Several other results are obtained, including product representations for $\exp[\gamma_0(a)]$ and for the Gamma function $\Gamma(a)$. In addition, we present series representations in terms of trigonometric integrals Ci and Si for $\psi(a)$ and the Euler constant $\gamma=-\psi(1)$.Comment: 27 pages, no figure

Series representations of the Riemann and Hurwitz zeta functions and series and integral representations of the first Stieltjes constant

We develop series representations for the Hurwitz and Riemann zeta functions in terms of generalized Bernoulli numbers (N\"{o}rlund 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 $\zeta(n)$ values emerge as nonlinear Euler sums in terms of generalized harmonic numbers. We additionally obtain series and integral representations of the first Stieltjes constant $\gamma_1(a)$. The presentation unifies some earlier results.Comment: 28 pages, no figure