Related papers: Elliptic Hypergeometric Summations by Taylor Serie…
After reviewing some fundamental facts from the theory of theta hypergeometric series we derive, using indefinite summation, several summation, transformation, and expansion formulas for multibasic theta hypergeometric series. Some of the…
Using matrix inversion and determinant evaluation techniques we prove several summation and transformation formulas for terminating, balanced, very-well-poised, elliptic hypergeometric series.
Several new identities for elliptic hypergeometric series are proved. Remarkably, some of these are elliptic analogues of identities for basic hypergeometric series that are balanced but not very-well-poised.
We prove a multivariable elliptic extension of Jackson's summation formula conjectured by Spiridonov. The trigonometric limit case of this result is due to Gustafson and Rakha. As applications, we obtain two further multivariable elliptic…
We derive a number of summation and transformation formulas for elliptic hypergeometric series on the root systems A_n, C_n and D_n. In the special cases of classical and q-series, our approach leads to new elementary proofs of the…
An analogue of Taylor's formula, which arises by substituting the classical derivative by a divided difference operator of Askey-Wilson type, is developed here. We study the convergence of the associated Taylor series. Our results…
Using Krattenthaler's operator method, we give a new proof of Warnaar's recent elliptic extension of Krattenthaler's matrix inversion. Further, using a theta function identity closely related to Warnaar's inversion, we derive summation and…
In this paper, we introduce the so-called elliptic Askey-Wilson polynomials which are homogeneous polynomials in two special theta functions. With regard to the significance of polynomials of such kind, we establish some general elliptic…
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…
We give a new theta-function identity, a special case of which is utilised to prove Kawanaka's Macdonald polynomial conjecture. The theta-function identity further yields a transformation formula for multivariable elliptic hypergeometric…
We provide an alternate approach to obtaining expansion formulas on the lines of the well-poised Bailey lemma. We recover results due to Spiridonov and Warnaar and one new formula of this type. These formulas contain an arbitrary sequence…
Motivated by the substantial development of the special functions, we contribute to establish some rigorous results on the general series identities with bounded sequences and hypergeometric functions with different arguments, which are…
Using multiple q-integrals and a determinant evaluation, we establish a multivariable extension of Bailey's nonterminating 10-phi-9 transformation. From this result, we deduce new multivariable terminating 10-phi-9 transformations, 8-phi-7…
We establish new transformation formulas involving theta functions and certain bilateral basic hypergeometric series. From these formulas, we construct companion $q$-series for a class of $q$-series such that the asymptotic expansion of…
Multiple elliptic polylogarithms can be written as (multiple) integrals of products of basic hypergeometric functions. The latter are computable, to arbitrary precision, using a q-difference equation and q-contiguous relations.
Eric Rains conjectured several quadratic transformations between multivariate elliptic hypergeometric functions in "Elliptic Littlewood Identities", with the integrand multiplied by interpolation functions. In this article two of these…
By using the theory of the elliptic integrals a new method of summation is proposed for a certain class of series and their derivatives involving hyperbolic functions. It is based on the termwise differentiation of the series with respect…
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…
In this article we present certain formulas involving arithmetical functions. In the first part we study properties of sums and product formulas for general type of arithmetic functions. In the second part we apply these formulas to the…
We prove analogues for elliptic interpolation functions of Macdonald's version of the Littlewood identity for (skew) Macdonald polynomials, in the process developing an interpretation of general elliptic "hypergeometric" sums as skew…