Related papers: The Clausenian hypergeometric function $_3F_2$ wit…
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 construct a class of representations of the quadratic R-matrix algebra, given by the reflection equation with the spectral parameter, in terms of certain ordinary difference operators. These operators turn out to act as parameter…
We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions ${}_0F_1$, ${}_1F_1$, ${}_2F_1$ and ${}_2F_0$ (or the Kummer $U$-function) are supported for unrestricted complex…
By means of inversion techniques and several known hypergeometric series identities, summation formulas for Fox-Wright function are explored. They give some new hypergeometric series identities when the parameters are specified.
Various miscellaneous functional inequalities are deduced for the so-called generalized inverse trigonometric and hyperbolic functions. For instance, functional inequalities for sums, difference and quotient of generalized inverse…
Present and future high-precision tests of the Standard Model and beyond for the fundamental constituents and interactions in Nature are demanding complex perturbative calculations involving multi-leg and multi-loop Feynman diagrams.…
A product of two hypergeometric series is generally not hypergeometric. However, there are a few cases when such product does reduce to a single hypergeometric series. The oldest result of this type, beyond the obvious…
We consider the integral operator $\mathcal{I}^{a,b,c}_{d,e}(f)(z)$ involving Clausen's Hypergeometric Function by means of convolution introduced by Chandrasekran and Prabhakaran for investigation. The conditions on the parameters $ a,b,…
In this paper, we present a new method for finding identities for hypergeoemtric series, such as the (Gauss) hypergeometric series, the generalized hypergeometric series and the Appell-Lauricella hypergeometric series. Furthermore, using…
A simple proof of a new summation formula for a terminating r+3Fr+2(1) hypergeometric series, representing an extension of Saalschutz's formula for a 3F2(1) series, is given for the case of r pairs of numeratorial and denominatorial…
The aim in this note is to provide a generalization of an interesting entry in Ramanujan's Notebooks that relate sums involving the derivatives of a function Phi(t) evaluated at 0 and 1. The generalization obtained is derived with the help…
Perturbative quantum field theory for the Ising model at the three-loop level yields a tetrahedral Feynman diagram C(a,b) with masses a and b and four other lines with unit mass. The completely symmetric tetrahedron C^Tet \equiv C(1,1) has…
In this paper, using similar symbolical method of Burchnall and Chaundy formulas of expansion for the generalized hypergeometric function were constructed. By means of the found formulas of expansion the formulas of an analytic continuation…
By some hypergeometric summation theorems, the authors establish a series of new infinite summation formulas involving generalized harmonic numbers related to Riemann-Zeta function, with three different patterns.
Kummer's function, also known as the confluent hypergeometric function (CHF), is an important mathematical function, in particular due to its many special cases, which include the Bessel function, the incomplete Gamma function and the error…
Using contiguous relations we construct an infinite number of continued fraction expansions for ratios of generalized hypergeometric series 3F2(1). We establish exact error term estimates for their approximants and prove their rapid…
We study multivariable (bilateral) basic hypergeometric series associated with (type $A$) Macdonald polynomials. We derive several transformation and summation properties for such series including analogues of Heine's ${}_2\phi_1$…
The main goal of this paper is to derive a number of identities for the generalized hypergeometric function evaluated at unity and for certain terminating multivariate hypergeometric functions from the symmetries and other properties of…
An exact formula for partition functions in 3d field theories was recently suggested by Jafferis, and Hama, Hosomichi, and Lee. These functions are expressed in terms of specific $q$-hypergeometric integrals whose key building block is the…
Using Bailey's very-well-poised $_6\psi_6$ summation, we show that a specific sequence of well-poised bilateral basic hypergeometric $_3\psi_3$ series form a family of orthogonal functions on the unit circle. We further extract a bilateral…