Related papers: New transformations for elliptic hypergeometric se…
In this paper we consider a transformation $L_a$ of sequences of complex numbers. We find the inverse transformation of $L_a$ as well as the inverse of a related transformation $\tilde{L}_a$. We explore a connection to the binomial…
A new multiple-integral representation of a general family of very-well-poised hypergeometric series is proved. Inspite of an analytic character of the result, it is motivated by the recent arithmetic progress for the values of the Riemann…
A complete classification of Belyi functions for transforming certain hypergeometric equations to Heun equations is given. The considered hypergeometric equations have the local exponent differences 1/k,1/l,1/m that satisfy k,l,m in N and…
We give a closed form for $quotients$ of truncated basic hypergeometric series where the base $q$ is evaluated at roots of unity.
In this paper we extend several results about root systems of Kac-Moody algebras to superalgebra context. In particular, we describe the root bases and the sets of imaginary roots.
Recently, there emerges different versions of beta function and hypergeometric functions containing extra parameters. Gaining enlightenment from these ideas, we will first introduce a new extension of generalized hypergeometric function and…
It was recently understood that from the point of view of automorphic Lorentzian Kac-Moody algebras and some aspects of Mirror Symmetry, interesting hyperbolic root systems should have restricted arithmetic type and a generalized lattice…
One of spectacular results in mathematical physics is the expression of Racah matrices for symmetric representations of the quantum group $SU_q(2)$ through the Askey-Wilson polynomials, associated with the $q$-hypergeometric functions…
We express explicitly the Heckman-Opdam hypergeometric function for the root system of type A with a certain degenerate parameter in terms of the Lauricella hypergeometric function.
We give a new procedure for generalized factorization and construction of the complete solution of strictly hyperbolic linear partial differential equations or strictly hyperbolic systems of such equations in the plane. This procedure…
Certain many-particle Hardy inequalities are derived in a simple and systematic way using the so-called ground state representation for the Laplacian on a subdomain of $\mathbb{R}^n$. This includes geometric extensions of the standard Hardy…
In these lecture notes I give an elementary introduction to elliptic hypergeometric functions. I focus on motivating the main ideas and constructions, rather than giving a comprehensive survey. The lectures include a brief explanation of…
B.C. Berndt evaluated special values of the cotangent Dirichlet series. T. Arakawa studied a generalization of the series, or generalized cotangent Dirichlet series, and gave its transformation formulae. In this paper, we establish an…
We start from an interpretation of the $BC_2$-symmetric "Type I" (elliptic Dixon) elliptic hypergeometric integral evaluation as a formula for a Casoratian of the elliptic hypergeometric equation, and give an extension to higher-dimensional…
We derive finite difference equations of infinite order for theta hypergeometric series and investigate the space of their solutions. In general, such infinite series diverge, we describe some constraints on the parameters when they do…
Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose…
We give a new hypergeometric construction of rational approximations to $\zeta(4)$, which absorbs the earlier one from 2003 based on Bailey's ${}_9F_8$ hypergeometric integrals. With the novel ingredients we are able to get a better control…
We introduce a polyadic analog of supersymmetry by considering the polyadization procedure (proposed by the author) applied to the toy model of one-dimensional supersymmetric quantum mechanics. The supercharges are generalized to polyadic…
We describe the multiplicative invariant algebras of the root lattices of all irreducible root systems under the action of the Weyl group. In each case, a finite system of fundamental invariants is determined and the class group of the…
In this article we give evaluations of certain series of hyperbolic functions using Jacobi elliptic functions theory. We also define some new functions that enable us to give characterization of not solvable class of series.