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The aim of this paper is to derive a summation formula for the alternating infinite series and an expression for zeta function by using hyperbolic secant random variables. These identities involve Euler numbers and are obtained by computing…
We demonstrate that certain class of infinite sums can be calculated analytically starting from a specific quantum mechanical problem and using principles of quantum mechanics. For simplicity we illustrate the method by exploring the…
In the present article, we study Bell based Euler polynomial of order {\alpha} and investigate some useful correlation formula, summation formula and derivative formula. Also, we introduce some relation of string number of the second kind.…
We find an explicit formula for the gamma vector in terms of the input polynomial in a way that extends it to arbitrary polynomials. More specifically, we find explicit linear combination in terms of coefficients of the input polynomial…
We derive and prove an explicit formula for the sum of the fractional parts of certain geometric series. Although the proof is straightforward, we have been unable to locate any reference to this result. This summation formula allows us to…
We use a method, first developed for the Riemann zeta-function by Masser in ["Rational values of the Riemann zeta function", Journ. Num. Th. 131 (2011), 2037-2046], to prove a new zero estimate for polynomials in z and 1/Gamma(z). This…
We produce formulas for $$\sum_{j=1}^{2^{n-2}}\frac{1}{\sin^s\left(\frac{(2j-1)\pi}{2^n}\right)}$$ in terms of Generalized Bernoulli and Euler polynomials and use one of the formulas to produce a nice integral representation of the Riemann…
We prove an Euler-Maclaurin formula for double polygonal sums and, as a corollary, we obtain approximate quadrature formulas for integrals of smooth functions over polygons with integer vertices. Our Euler-Maclaurin formula is in the spirit…
We show how the combined use of the generating function method and of the theory of multivariable Hermite polynomials is naturally suited to evaluate integrals of gaussian functions and of multiple products of Hermite polynomials.
We give a number new examples analytically and numerically to confirm the Kohler conjecture. It turned out that for a rather large class of nonnegative functions the equality (A) hold.
We derive special forms of the Poisson summation formula for even and odd functions, which are applied to obtain representations for Euler-type numbers and to sum various series related to elliptic functions.
We introduce a new factorial function which agrees with the usual Euler gamma function at both the positive integers and at all half-integers, but which is also entire. We describe the basic features of this function.
A proof of a formula, discovered by V.Fateev, on some Gamma products connected with root systems is given.
A novel general framework for the study of $\Gamma$-convergence of functionals defined over pairs of measures and energy-measures is introduced. This theory allows us to identify the $\Gamma$-limit of these kind of functionals by knowing…
A new expansion for integral powers of the hypergeometric function corresponding to a special case of the incomplete beta function is summarized, and consequences, including two new sums involving digamma (psi) functions are presented.
The main aim of this paper is to derive some new summation theorems for terminating and truncated Clausen's hypergeometric series with unit argument, when one numerator parameter and one denominator parameter are negative integers. Further,…
Using three basic facts concerning Hurwitz zeta function,we give new natural proofs of the known results on Bernoulli polynomials,gamma function and also obtain Gauss' expression for Psi function at a rational point,all in a unified…
A new general and unified method of summation, which is both regular and consistent, is invented. It is based on the idea concerning a way of integers reordering. The resulting theory includes a number of explicit and closed form summation…
We prove a combinatorial formula for Macdonald cumulants which generalizes the celebrated formula of Haglund for Macdonald polynomials. We provide several applications of our formula. Firstly, it gives a new, constructive proof of a strong…
In this article, we present some specific aspects of symmetric Gamma process mixtures for use in regression models. We propose a new Gibbs sampler for simulating the posterior and we establish adaptive posterior rates of convergence related…