Related papers: A summation by Gencev
The Fibonacci number is the residue of a rational function, from which follows that Fibonacci number summation identities can be derived with the integral representation method, a method also used to derive combinatorial identities. A…
Explicit determinations of several classes of trigonometric sums are given. These sums can be viewed as analogues or generalizations of Gauss sums. In a previous paper, two of the present authors considered primarily sine sums associated…
We propose an explicit representation of central $(2k+1)$-nomial coefficients in terms of finite sums over trigonometric constructs. The approach utilizes the diagonalization of circulant boolean matrices and is generalizable to all…
We present several sequences of Euler sums involving odd harmonic numbers. The calculational technique is based on proper two-valued integer functions, which allow to compute these sequences explicitly in terms of zeta values only.
We introduce a full solution to a problem considered by Wang and Chu concerning series involving the squares of finite sums of the form $1 + \frac{1}{3}+ \cdots + \frac{1}{2n-1}$. Our proof involves techniques from the theory of colored…
We construct the generating function for products of inverse central binomial coefficients with harmonic numbers.
In terms of the derivative operator, integral operator and Saalsch\"{u}tz's theorem, two families of summation formulae involving generalized harmonic numbers are established.
We study a general type of series and relate special cases of it to Stirling series, infinite series discussed by Choi and Hoffman, and also to special values of the Arakawa-Kaneko zeta function, complementing and generalizing earlier…
In this paper, using geometric polynomials, we obtain a generating function of p-Bernoulli numbers. As a consequences this generating function, we derive closed formulas for the finite summation of Bernoulli and harmonic numbers involving…
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
The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable Quantum Field Theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin…
We discuss the properties of the Hankel transformation of a sequence whose elements are the sums of consecutive generalized Catalan numbers and find their values in the closed form.
In this paper we use a contour integral method to derive a generating function in the form of a double series involving the product of two Chebyshev polynomials over generalized independent indices expressed in terms of the incomplete gamma…
In this paper we focus on two new families of polynomials which are connected with exponential polynomials and geometric polynomials. We discuss their generalizations and show that these new families of polynomials and their generalizations…
This paper builds on the research initiated by Boyadzhiev, but introduces generalized harmonic numbers, \[ H_n(\alpha)= \sum_{k=1}^n \frac{\alpha^{k}}{k}, \] which enable the derivation of new identities as well as the reformulation of…
Recent progress in analytical calculation of the multiple [inverse, binomial, harmonic] sums, related with epsilon-expansion of the hypergeometric function of one variable are discussed.
The skew-harmonic numbers are the partial sums of the alternating harmonic series, i.e. the expansion of log(2). We evaluate in closed form various power series and numerical series with skew-harmonic numbers. This provides a simultaneous…
We study two new classes of sums with inverse binomial coefficients and harmonic numbers. In addition we establish recursive solutions to the following power sums \begin{equation*} U_d(n) = \sum_{k=1}^n \frac{2^{2k}}{\binom{2k}{k}} \cdot…
A number of new terminating series involving $\sin(n^2/k)$ and $\cos(n^2/k)$ are presented and connected to Gauss quadratic sums. Several new closed forms of generic Gauss quadratic sums are obtained and previously known results are…
In a previous work (arXiv:2505.05574), a summation formula for harmonic Maass forms of polynomial growth was established. In this note, we use the theory of $L$-series of harmonic Maass forms to state and prove a summation formula for such…