Related papers: On the adelic Gaussian hypergeometric function
Two exact evaluation formulae for multiple rarefied elliptic beta integrals related to the simplest lens space are proved. They generalize evaluations of the type I and II elliptic beta integrals attached to the root system $C_n$. In a…
The well-known fact that all elliptic curves are modular, proven by Wiles, Taylor, Breuil, Conrad and Diamond, leaves open the question whether there exists a 'nice' representation of the modular form associated to each elliptic curve. Here…
On the basis of analysis on the adele ring of any algebraic numbers field (Tate's formula) a regularization for divergent adelic products of gamma- and beta-functions for all completions of this field are proposed, and corresponding…
Integral representations of hypergeometric functions proved to be a very useful tool for studying their properties. The purpose of this paper is twofold. First, we extend the known representations to arbitrary values of the parameters and…
Finite hypergeometric functions are functions of a finite field ${\bf F}_q$ to ${\bf C}$. They arise as Fourier expansions of certain twisted exponential sums and were introduced independently by John Greene and Nick Katz in the 1980's.…
Hypergeometric class equations are given by second order differential operators in one variable whose coefficient at the second derivative is a polynomial of degree $\leq2$, at the first derivative of degree $\leq1$ and the free term is a…
The hypergeometric distribution is a popular distribution, whose properties have been extensively investigated. Generating functions of this distribution, such as the probability-generating function, the moment-generating function, and the…
We define a finite-field version of Appell-Lauricella hypergeometric functions built from period functions in several variables, paralleling the development by Fuselier, et. al in the single variable case. We develop geometric connections…
The authors survey recent results in special functions, particularly the gamma function and the Gaussian hypergeometric function.
We have shown that in some region where the Euler integral of the first kind diverges, the Euler formula defines a generalized function. The connected of this generalized function with the Dirac delta function is found.
In this paper we explore special values of Gaussian hypergeometric functions in terms of products of Euler $\Gamma$-functions and exponential functions of linear functions of the hypergeometric parameters. They include some classical…
By a non-Gaussian integral we mean integral of the product of an arbitrary function and exponent of a polynomial. We develop a theory of such integrals, which generalizes and simplifies the theory of general hypergeometric functions in the…
The main object of this work is to show how some rather elementary techniques based upon certain inverse pairs of symbolic operators would lead us easily to several decomposition formulas associated with confluent hypergeometric functions…
We establish asymptotic formulae for volumes of height balls in analytic varieties over local fields and in adelic points of algebraic varieties over number fields, relating the Mellin transforms of height functions to Igusa integrals and…
We give an overview of some of the main results from the theories of hypergeometric and basic hypergeometric series and integrals associated with root systems. In particular, we list a number of summations, transformations and explicit…
We review Aomoto's generalized hypergeometric functions on Grassmannian spaces Gr(k +1, n+1). Particularly, we clarify integral representations of the generalized hypergeometric functions in terms of twisted homology and cohomology. With an…
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.
The paper is a survey of recent results in analysis of additive functions over function fields motivated by applications to various classes of special functions including Thakur's hypergeometric function. We consider basic notions and…
We prove an asymptotic formula for a special case of the Gauss hypergeometric function which arises in explicit formulas for moments of Maass form symmetric square L-functions. The resulting formula is uniform in several variables, which is…
The Wright function, which arises in the theory of the space-time fractional diffusion equation, is an interesting mathematical object which has diverse connections with other special and elementary functions. The Wright function provides a…