Related papers: An Infinite Product for e^gamma via Hypergeometric…

For an arbitrary positive integer $p$, Landen's formula is extended to express theta function with modulus $p\tau$ by $p$ product of theta functions with $\tau$, which is applied to several examples. Next it is shown that double product of…

Finite Euler product is known to be one of the classical zeta functions in number theory. In [1], [2] and [3], we have introduced some multivariable zeta functions and studied their definable probability distributions on R^d. They include…

We show that indefinite theta series on cones converge and provide an explicit modular completion. Our completion rests on a convolution of the Gaussian with a piecewise constant function supported on the cone. Our main innovation is to…

We introduce an algorithm to compute the functions belonging to a suitable set ${\mathscr F}$ defined as follows: $f\in {\mathscr F}$ means that $f(s,x)$, $s\in A\subset {\mathbb R}$ being fixed and $x>0$, has a power series expansion…

Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function $\eta(s)$, and hence Riemann's function $\zeta(s)$, is obtained in terms of the Exponential Integral function $E_{s}(i\kappa)$ of…

In this paper, we obtain some formulas for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. By using these formulas, we give new closed form sums of several quadratic Euler series through Riemann zeta…

This paper deals with the evaluation of some definite Euler-type integrals in terms of the Wright hypergeometric function. We obtain a theorem on the Wright hypergeometric function and then use this theorem to evaluate some definite…

We aim to introduce a new extension of beta function and to study its important properties. Using this definition, we introduce and investigate new extended hypergeometric and confluent hypergeometric functions. Further, some hybrid…

We define the adelic hypergeometric function of special Gaussian type by means of a tower of hypergeometric curves. This function takes values in an adelic completed group ring and interpolates all the hypergeometric functions of the same…

We introduce a family of regularized functionals $g_n(x)$ that generalize the Euler--Mascheroni constant $\gamma$. They arise from a weighted regularization of Clausen-type trigonometric sums, and admit explicit integral representations,…

We show the modular properties of the multiple 'elliptic' gamma functions, which are an extension of those of the theta function and the elliptic gamma function. The modular property of the theta function is known as Jacobi's…

We aim to achieve the following three goals. First of all, we collect all known definitions, transformation properties and functional identities of Barnes double gamma function $G(z;\tau)$. Second, we derive an algorithm for numerically…

We show that each member of a doubly infinite sequence of highly nonlinear expressions of Bernoulli polynomials, which can be seen as linear combinations of certain higher-order convolutions, is a multiple of a specific product of linear…

A recurrent formula is presented, for the enumeration of the compositions of positive integers as sums over multisets of positive integers, that closely resembles Euler's recurrence based on the pentagonal numbers, but where the…

The classical modular equations involve bivariate polynomials that can be seen to be univariate with coefficients in the modular invariant $j$. Kiepert found modular equations relating some $\eta$-quotients and the Weber functions…

Several infinite products are studied that satisfy the transformation relation of the type $f(\alpha)=f(1/\alpha)$. For certain values of the parameters these infinite products reduce to modular forms. Finite counterparts of these infinite…

An asymptotic expansion of a ratio of products of gamma functions is derived. It generalizes a formula which was stated by Dingle, first proved by Paris, and recently reconsidered by Olver.

We present several results related to statistics for elliptic curves over a finite field $\mathbb{F}_p$ as corollaries of a general theorem about averages of Euler products that we demonstrate. In this general framework, we can reprove…

In this paper we show how to apply various techniques and theorems (including Pincherle's theorem, an extension of Euler's formula equating infinite series and continued fractions, an extension of the corresponding transformation that…

We present a large number of analytic evaluations of Euler sums, namely sums such as \begin{align} M(m,n_0,n_1,n_2, \ldots, n_t) &= \sum_{k=1}^\infty \frac{H(k)^m}{k^{n_0} (k+1)^{n_1} (k+2)^{n_2} \cdots (k+t)^{n_t}}, \nonumber \end{align}…