arXiv subjects
Publications and source records attributed to Donal F. Connon.
This paper considers some integrals where the integrand comprises the log gamma function or the digamma function multiplied by exponential or trigonometric functions.
We present a new representation of the Stieltjes constants in the form of a limit of a Fourier series.
We provide a rigorous formulation of Entry 17(v) in Ramanujan's Notebooks and show how this relates to the first Stieltjes constant. A new proposition 4.5 is included to show the close relationship with some analysis presented by Candelpergher in 2017.
We show a direct approach to Kinkelin's integral involving the cotangent function.
We show that the generalised Stieltjes constants may be represented by infinite series involving logarithmic terms. Some relations involving the derivatives of the Hurwitz zeta function are also investigated
We derive an expression for the generalized Bernoulli numbers in terms of the Bernoulli numbers involving the (exponential) complete Bell polynomials.
Inter alia, we present a Fourier series involving the generalised Stieltjes constants.
The Stieltjes constants have attracted considerable attention in recent years and a number of authors, including the present one, have considered various ways in which these constants may be evaluated. The primary purpose of this paper is to belatedly highlight the fact that Deninger actually ascertained the first generalised Stieltjes constant at rational arguments as long ago as 1984 and that all of the higher constants (at rational arguments) were determined in principle by Chakraborty, Kanemitsu and Kuzumaki in 2009. Equivalent results were obtained by Musser in his 2011 thesis. The authors of the above papers simply referred to the constants as the Laurent coefficients which explains why various electronic searches conducted by this author for Stieltjes constants did not readily highlight these particular sources. In this paper the author has employed a slightly different argument to obtain a simpler expression for the results originally derived by Chakraborty et al. in 2009.
We show that the higher derivatives of the Riemann zeta function may be expressed in terms of integrals involving the digamma function. Related integrals for the Stieltjes constants are also shown. We also present a formula for the derivatives of the Riemann zeta function entirely in terms of the Lehmer constants.
A large number of the classical texts dealing with Fourier series more or less state that the hypothesis of periodicity is required for pointwise convergence. In this paper, we highlight the fact that this condition is not necessary.
We consider several possible approaches to evaluating an integral involving the digamma function and a related logarithmic series.
We show how the sine and cosine integrals may be usefully employed in the evaluation of some more complex integrals.
Using the Dirichlet integrals, which are employed in the theory of Fourier series, this paper develops a useful method for the summation of series and the evaluation of integrals.
We show that the formula recently derived by Coffey for the Stieltjes constants in terms of the Bernoulli numbers is mathematically equivalent to the much earlier representation derived by Briggs and Chowla.
Some new integrals involving the Stieltjes constants are developed in this paper.
This paper considers various integrals where the integrand includes the log gamma function (or its derivative, the digamma function) multiplied by a trigonometric or hyperbolic function. Some apparently new integrals and series are evaluated.
This paper considers some infinite series involving the Riemann zeta function.
We generalise the Bernoulli numbers to include the case where the index may be a continuous variable.