Related papers: On Some Sums of Digamma and Polygamma functions - …
We calculate some finite and infinite sums containing the digamma function in closed-form. For this purpose, we differentiate selected reduction formulas of the hypergeometric function with respect to the parameters applying some derivative…
Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known, however sums…
In this note, we derive a finite summation formula and an infinite summation formula involving Harmonic numbers of order up to some order by means of several definite integrals
We consider summations over digamma and polygamma functions, often with summands of the form (\pm 1)^n\psi(n+p/q)/n^r and (\pm 1)^n\psi^{(m)} (n+p/q)/n^r, where m, p, q, and r are positive integers. We develop novel general integral…
The Digamma and Polygamma functions are important tools in mathematical physics, not only for its many properties but also for the applications in statistical mechanics and stellar evolution. In many textbooks is found its develop almost by…
We present here a large collection of harmonic and quadratic harmonic sums, that can be useful in applied questions, e.g., probabilistic ones. We find closed-form formulae, that we were not able to locate in the literature.
We calculate some infinite sums containing the digamma function in closed-form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as…
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…
Motivated by open questions in the papers " Refinements and sharpenings of some double inequalities for bounding the gamma function" and "Complete monotonicity and monotonicity of two functions defined by two derivatives of a function…
This paper evaluates some generalised Euler sums involving the digamma function.
In the paper, the authors establish three kinds of double inequalities for the trigamma function in terms of the exponential function to powers of the digamma function. These newly established inequalities extend some known results. The…
In the paper, some lower bounds for polygamma functions are refined.
We consider several possible approaches to evaluating an integral involving the digamma function and a related logarithmic series.
We prove recursive formulas for sums of squares and sums of triangular numbers in terms of sums of divisors functions and we give a variety of consequences of these formulas. Intermediate applications include statements about positivity of…
Can any element in a sufficiently large finite field be represented as a sum of two $d$th powers in the field? In this article, we recount some of the history of this problem, touching on cyclotomy, Fermat's last theorem, and diagonal…
In this article we present certain formulas involving arithmetical functions. In the first part we study properties of sums and product formulas for general type of arithmetic functions. In the second part we apply these formulas to the…
We derive new infinite series involving Fibonacci numbers and Riemann zeta numbers. The calculations are facilitated by evaluating linear combinations of polygamma functions of the same order at certain arguments.
This paper considers some integrals where the integrand comprises the log gamma function or the digamma function multiplied by exponential or trigonometric functions.
Many integrals in the classical table by Gradshteyn and Ryzhik can be evaluated in terms of the digamma function (= the logarithmic derivative of the gamma function). Some of them are presented here.
In this paper we will give a proof of a certain summation formula for Gamma functions utilizing Gegenbauer polynomials.