Related papers: Evaluations and relations for finite trigonometric…
In this paper we investigate congruence relationships of particular finite generalized harmonic numbers sums. We suggest more transparent and simpler method to analyse these sums and present several additional results for certain special…
A finite sum of exponential functions may be expressed by a linear combination of powers of the independent variable and by successive integrals of the sum. This is proved for the general case and the connection between the parameters in…
In this paper, we present a general framework for the derivation of interesting finite combinatorial sums starting with certain classes of polynomial identities. The sums that can be derived involve products of binomial coefficients and…
We introduce a natural definition for sums of the form \[ \sum_{\nu=1}^x f(\nu) \] when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the…
A generating function for reciprocal binomial coefficients is written down, integral representations of this function are obtained, generating functions for sums of reciprocal binomial coefficients are derived, new identities are obtained,…
We construct an analytic approach to evaluate odd Euler sums, multiple zeta value $\zeta(3,2,\ldots,2)$ and multiple $t$-value $t\left(3,2,\ldots,2\right)$. Moreover, we also conjecture a closed expression for multiple $t$-value…
A new type of polynomial analogue of the Rogers-Ramanujan identities is proven. Here the product-side of the Rogers-Ramanujan identities is replaced by a partial theta sum and the sum-side by a weighted sum over Schur polynomials.
An operatorial method, already employed to formulate a generalization of the Ramanujan master theorem, is applied to the evaluation of integrals of various type. This technique provide a very flexible and powerful tool yielding new results…
In this work we consider sums of primes that converging very slow. We set as a base, a reformulation of analytic prime number theorem and we use the values of Riemann Zeta function for the approximation. We also give the truncation error of…
We study three classes of combinatorial sums involving central binomial coefficients and harmonic numbers, odd harmonic numbers, and even indexed harmonic numbers, respectively. In each case we use summation by parts to derive recursive…
We study matrix three term relations for orthogonal polynomials in two variables constructed from orthogonal polynomials in one variable. Using the three term recurrence relation for the involved univariate orthogonal polynomials, the…
In this paper, we evaluate in closed forms two families of infinite integrals containing hyperbolic and trigonometric functions in their integrands. We call them Berndt-type integrals since he initiated the study of similar integrals. We…
We introduce and prove evaluations for families of multiple elliptic integrals by solving special types of ordinary and partial differential equations. As an application, we obtain new expressions of Ramanujan-type series of level 4 and…
In this paper, we consider infinite sums derived from the reciprocals of the generalized Fibonacci numbers. We obtain some new and interesting identities for the generalized Fibonacci numbers.
This paper presents the equality of finite index sums of Bessel func- tions containing arbitrary numbers of terms. These reduce to the familiar three term recursion formulas in simple cases.
The authors assume the Deep Riemann Hypothesis to prove that a weighted sum of Ramanujan's $\tau$-function has a bias to being positive. This phenomenon is an analogue of Chebyshev's bias.
Two-point Feynman parameter integrals, with at most one mass and containing local operator insertions in $4+\ep$-dimensional Minkowski space, can be transformed to multi-integrals or multi-sums over hyperexponential and/or hypergeometric…
In this paper, we introduce vast generalizations of the Hardy-Berndt sums. They involve higher-order Euler and/or Bernoulli functions, in which the variables are affected by certain linear shifts. By employing the Fourier series technique…
We offer some summation formulas that appear to have great utility in probability theory. The proofs require some recent results from analysis that have thus far been applied to basic hypergeometric functions.
We consider a special class of binomial sums involving harmonic numbers and we prove three identities by using the elementary method of the partial fraction decomposition. Some applications to infinite series and congruences are given.