arXiv subjects
Publications and source records attributed to Donal F. Connon.
In this paper we derive rapidly converging series for Catalan's constant and for Apéry's constant. The method may be easily generalised to produce new series representations for other values of the Riemann zeta function and the Dirichlet beta function.
In this paper we consider some possible approaches to the proof of the Riemann Hypothesis using the Li criterion.
In a rather straightforward manner, we develop the well-known formula for the Stirling numbers of the first kind in terms of the (exponential) complete Bell polynomials where the arguments include the generalised harmonic numbers. We also show how the (exponential) complete Bell polynomials feature in a number of other areas of mathematical interest.
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 together with various other relationships.
New proofs of the duplication formulae for the gamma and the Barnes double gamma functions are derived using the Hurwitz zeta function. Concise derivations of Gauss's multiplication theorem for the gamma function and a corresponding one for the double gamma function are also reported. This paper also refers to some connections with the Stieltjes constants.
Kummer's Fourier series for the log gamma function is well known, having been discovered in 1847. In this paper we develop a corresponding Fourier series for the logarithm of the Barnes double gamma function (and the method may be easily extended to higher order multiple gamma functions). Some applications of these Fourier series are explored.
Some new integrals involving the Stieltjes constants are developed in this paper.
A recurrence relation for the Li/Keiper constants in terms of the Stieltjes constants is derived in this paper. In addition, we also report a formula for the Stieltjes constants in terms of the higher derivatives of the Riemann zeta function. A formula for the Stieltjes constants in terms of the (exponential) complete Bell polynomials containing the eta constants as the arguments is also derived.
In this paper we present some applications of the Stieltjes constants including, for example, new derivations of Binet's formulae for the log gamma function and the evaluation of some integrals related to the Barnes multiple gamma functions.
Gencev has recently reported a closed form summation for an infinite series involving the harmonic numbers and the central binomial numbers. This note indicates a possible approach to the proof involving the dilogarithm function.
This short note deals with some applications of the Beta function
This note highlights an interesting connection between Euler sums of even weight and prime numbers.
This paper evaluates some generalised Euler sums involving the digamma function.
In this paper we derive two expressions for the Hurwitz zeta function involving the complete Bell polynomials in the restricted case where q is a positive integer greater than 1. The arguments of the complete Bell polynomials comprise the generalised harmonic number functions. These in turn give rise to Euler-Hurwitz series which may then be used to determine identities for combinations of both linear and non-linear Euler sums.
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.