Related papers: On integral representations of q-gamma and q-beta …
We construct the new q-extension of Bernoulli numbers and polynomials in this paper. Finally we consider the q-zeta functions which interpolate the new q-extension of Bernoulli numbers and polynomials.
In this paper, we study a family of single variable integral representations for some products of $\zeta(2n+1)$, where $\zeta(z)$ is Riemann zeta function and $n$ is positive integer. Such representation involves the integral…
Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a_1,a_2,\ldots,a_k,n\in\Bbb N$ let $N(a_1,a_2,\ldots,a_k;n)$ be the number of representations of $n$ by…
We obtain a variety of series and integral representations of the digamma function $\psi(a)$. These in turn provide representations of the evaluations $\psi(p/q)$ at rational argument and for the polygamma function $\psi^{(j)}$. The…
Simple asymptotic expansions for the Jacobi functions $P_\nu^{(\alpha, \beta)}(z)$ and $Q_\nu^{(\alpha, \beta)}(z)$ for large degree $\nu$, with fixed parameters $\alpha$ and $\beta$, are surprisingly rare in the literature, with only a few…
The representation theory of the group U(1,q) is discussed in detail because of its possible application in a quaternion version of the Salam-Weinberg theory. As a consequence, from purely group theoretical arguments we demonstrate that the…
We derive two general transformations for certain basic hypergeometric series from the recurrence formulae for the partial numerators and denominators of two $q$-continued fractions previously investigated by the authors. By then…
We consider multiple sums and multi-integrals as tau functions of the BKP hierarchy using neutral fermions as the simplest tool for deriving these. The sums are over projective Schur functions $Q_\alpha$ for strict partitions $\alpha$. We…
A quantum integrable model is considered which describes a quantization of affine hyper-elliptic Jacobian. This model is shown to possess the property of duality: a dual model with inverse Planck constant exists such that the…
The variable change w=exp(u) is applied to establish novel integral representations of the incomplete gamma-function, hypergeometric F-function,confluent hypergeometric /Phi-function and beta-function, and to analyze these functionsas as…
In this paper, we give a purely algebraic proof of an identity coming directly from Euler's reflection formula for the gamma function. Our proof uses Hoffman's harmonic algebra and some binomial identities.
We use the Jacobi theta function to give a representation of the modulus of the Riemann $\xi$ function. Based on this modulus representation, we show that the Riemann hypothesis is equivalent to the validity of a family of polynomial…
We consider different pentagon identities realized by the hyperbolic hypergeometric functions and investigate their degenerations to the level of complex hypergeometric functions. In particular, we show that one of the degenerations yields…
We prove polynomial identities for the N=1 superconformal model SM(2,4\nu) which generalize and extend the known Fermi/Bose character identities. Our proof uses the q-trinomial coefficients of Andrews and Baxter on the bosonic side and a…
In the paper, the authors establish integral representations of some functions related to the remainder of Burnside's formula for the gamma function and find the (logarithmically) complete monotonicity of these and related functions. These…
This paper investigates a class of special Berndt-type integral calculations where the integrand contains only hyperbolic cosine functions. The research approach proceeds as follows: Firstly, through contour integration methods, we…
Infinite dimensional representations of the real form U_q(u_{n,1}) of the Drinfeld--Jimbo algebra U_q(gl_{n+1}) are defined. The principal series of representations of U_q(u_{n,1}) is studied. Intertwining operators for pairs of the…
The tensor product of a positive and a negative discrete series representation of the quantum algebra U_q(su(1,1)) decomposes as a direct integral over the principal unitary series representations. Discrete terms can appear, and these terms…
We present a constructive proof of Jacobi's identity for the sum of two squares. We present a combinatorial proof of the Jacobi Triple Product and combine with a proof of Hirschhorn to define an algorithm. The input is a factorization…
This paper explores the calculus of dual-valued functions and investigates the gamma function, beta function and generalized hypergeometric functions by incorporating dual numbers as parameters and variables. We examine its fundamental…