Related papers: A symmetric formula for hypergeometric series
In this note, we shall provide several properties of hypergeometric Bernoulli numbers and polynomials, including sums of products identity, differential equations and recurrence formulas.
In a recent paper, Chu and Zhou [Advances in Combinatorics, I.S. Kotsireas and E.V. Zima(eds.), 139-159 (2013)] established in all 40 closed formulae for terminating Watson-like hypergeometric $_{3}F_2$- series by investigating through…
Following a previous article we continue our study on non-terminating hypergeometric series with one free parameter, which aims to find arithmetical constraints for a given hypergeometric series to admit a gamma product formula. In this…
In 1990, Daubechies proved a fundamental identity for Weyl-Heisenberg systems which is now called the Weyl-Heisenberg Frame Identity. WH-Frame Identity: If $g\in W(L^{\infty},L^{1})$, then for all continuous, compactly supported functions f…
Explicit expressions for the hypergeometric series ${}_2F_1(-n, a; 2a\pm j;2)$ and ${}_2F_1(-n, a; -2n\pm j;2)$ for positive integer $n$ and arbitrary integer $j$ are obtained with the help of generalizations of Kummer's second and third…
We study the Borwein--Bailey--Girgensohn sinusoidal series S_BBG = sum_{n=1}^\infty (1/n) * ((2+sin n)/3)^n, originally posed as an open problem by Borwein, Bailey, and Girgensohn, whose convergence was established by Boppana using the…
The Teichm\"uller space $\mathcal{T}_S(\mathbf{b})$ of hyperbolic metrics on a surface $S$ with fixed lengths at the boundary components is symplectic. We prove that any sum of infinitesimal earthquakes on $S$ that is tangent to…
Using the framework relating hypergeometric motives to modular forms, we define an explicit family of weight 2 Hecke eigenforms with complex multiplication. We use the theory of ${}_2F_1(1)$ hypergeometric series and Ramanujan's theory of…
We obtain a formula which reduces the evaluation of a $_2\psi_2$ series to two $_2\phi_1$ series. In some sense, this identity may be considered as a companion of Slater's formulas. We also find that a two-term ${}_2\psi_2$ summation…
We prove hypergeometric type summation identities for a function defined in terms of quotients of the $p$-adic gamma function by counting points on certain families of hyperelliptic curves over $\mathbb{F}_{q}$. We also find certain special…
It is shown that the well known Racah sum rule and Biedenharn-Elliott identity satisfied by the recoupling coefficients or by the $6-j$ symbols of the usual rotation $SO(3)$ algebra can be extended to the corresponding features of the…
We consider time-inhomogeneous, second order linear parabolic partial differential equations of the non-divergence type, and assume the ellipticity and the continuity on the coefficient of the second order derivatives and the boundedness on…
In this paper we first investigate the Ansatz of one of the present authors for K(\Psi,\bar\Psi), the adimensional modular invariant non-holomorphic correction to the Wilsonian effective Lagrangian of an N=2 globally supersymmetric gauge…
The univariate elliptic beta integral is represented as a bilinear combination of infinite $_{10}V_9$ very-well-poised elliptic hypergeometric series representing the sum of residues of the integrand poles. Convergence of this combination…
We formalize some basic properties of Fourier series in the logic of ACL2(r), which is a variant of ACL2 that supports reasoning about the real and complex numbers by way of non-standard analysis. More specifically, we extend a framework…
In this paper, we investigate solutions of the hyperbolic Poisson equation $\Delta_{h}u(x)=\psi(x)$, where $\psi\in L^{\infty}(\mathbb{B}^{n}, \mathbb{R}^n)$ and \[ \Delta_{h}u(x)= (1-|x|^2)^2\Delta u(x)+2(n-2)(1-|x|^2)\sum_{i=1}^{n} x_{i}…
Adapting a method used by Cauchy, Bailey, Slater, and more recently, the second author, we give a new proof of Bailey's celebrated 6-psi-6 summation formula.
We give a proof of two identities involving binomial sums at infinity conjectured by Z-W Sun. In order to prove these identities, we use a recently presented method i.e. we view the series as specializations of generating series and derive…
We study a nonlinear system of partial differential equations which describe rotating shallow water with an arbitrary constant polytropic index $\gamma $ for the fluid. In our analysis we apply the theory of symmetries for differential…
We obtain an explicit classification of all $2\times 2$ real hypergeometric Bochner pairs, ie. pairs $(W(x),\mathfrak{D})$ consisting of a $2\times 2$ real hypergeometric differential operator $\mathfrak{D}$ and a $2\times 2$ weight matrix…