Bilateral Basic Hypergeometric Series Research Articles

Inversion of Bilateral Basic Hypergeometric Series

We present a new matrix inverse with applications in the theory of bilateral basic hypergeometric series. Our matrix inversion result is directly extracted from an instance of Bailey's very-well-poised ${}_6\psi_6$ summation theorem, and involves two infinite matrices which are not lower-triangular. We combine our bilateral matrix inverse with known basic hypergeometric summation theorems to derive, via inverse relations, several new identities for bilateral basic hypergeometric series.

The Saalschütz chain reactions and bilateral basic hypergeometric series

By iterating recursively the q-Saalschutz summation formula, we introduce the Saalschutz chain reactions. A general series transform, which expresses a nonterminating bilateral series in terms of a finite multiple unilateral sum, will be established. As applications we derive, by means of Bailey’s 6ψ6 -series identity, several bilateral transformations including one due to Milne [12]. These transformations further yield a number of closed formulas of very well-poised bilateral basic hypergeometric series; which are closely related to the identities obtained by Minton [13], Karlsson [11], Gasper [8], and Chu [5], [6], [7] through the partial fraction method and divided differences.

A New $A_n$ Extension of Ramanujan's ${}_1\psi_1$ Summation with Applications to Multilateral An Series

In this article, we derive some identities for multilateral basic hypergeometric series associated to the root system A_n. First, we apply Ismail's argument to an A_n q-binomial theorem of Milne and derive a new A_n generalization of Ramanujan's 1-psi-1 summation theorem. From this new A_n 1-psi-1 summation and from an A_n 1-psi-1 summation of Gustafson we deduce two lemmas for deriving simple A_n generalizations of bilateral basic hypergeometric series identities. These lemmas are closely related to the Macdonald identities for A_n. As samples for possible applications of these lemmas, we provide several A_n extensions of Bailey's 2-psi-2 transformations, and several A_n extensions of a particular 2-psi-2 summation.

Multiparameter ramanujan-type q-beta integrals

We consider general multiparameter Ramanujan-type q-beta integrals and derive integral analogs of Slater's transformation formulas for bilateral basic hypergeometric series. Evaluation of some Ramanujan-type q-beta integrals is discussed in a unified form.

Coherent states of theq-Weyl algebra

New coherent states of theq-Weyl algebraAA† −qA†A = 1,0 <q < 1, are constructed. They are defined as eigenstates of the operatorA† which is the lowering operator for nonhighest weight representations describing positive energy states. Depending on whether the positive spectrum is discrete or continuous, these coherent states are related either to the bilateral basic hypergeometric series or to some integrals over them. The free particle realization of theq-Weyl algebra whenA†A ∞ d2/dx2 is used for illustrations.

TheC ? Bailey transform and Bailey lemma

TheCl nonterminating Cl summation theorem is derived by appropriately specializing Gustafson's6ψ6 summation theorem for bilateral basic hypergeometric series very well-poised on symplecticCl groups. From this, the terminating6ϕ5 and, hence, terminating4ϕ3 summation theorem is obtained. A suitably modified4ϕ3 is then used to derive theCl generalization of the Bailey transform. The transform is then interpreted as a matrix inversion result for two infinite, lower-triangular matrices. This result is used to motivate the definition of theCl Bailey pair. TheCl generalization of Bailey's lemma is then proved. This result is inverted, and the concept of the bilateral Bailey chain is discussed. TheCl Bailey lemma is then used to obtain a connection coefficient result for generalCl littleq-Jacobi polynomials. All of this work is a natural extension of the unitaryAl, or equivalentlyU(l+1), case. The classical case, corresponding toA1 or equivalentlyU(2), contains an immense amount of the theory and application of one-variable basic hypergeometric series, including elegant proofs of the Rogers-Ramanujan-Schur identities. TheCl nonterminating6ϕ5 summation theorem is also used to recover C. Krattenthaler's multivariable summation which he utilized in deriving his refinement of the Bender-Knuth and MacMahon generating functions for certain sets of plane partitions.

Some Transformations of Basic Hypergeometric Series and Their Applications

Using Bailey’s transformation, relations between basic and basic bilateral hypergeometric series are obtained. Some interesting special cases, like identities of Rogers-Ramanujan type, summation theorems for particular basic bilateral hypergeometric series $_2{\psi _2}$, are also discussed.

Some transformations of basic hypergeometric series and their applications

Using Bailey’s transformation, relations between basic and basic bilateral hypergeometric series are obtained. Some interesting special cases, like identities of Rogers-Ramanujan type, summation theorems for particular basic bilateral hypergeometric series 2 ψ 2 _2{\psi _2} , are also discussed.

On a transformation of bilateral series with applications

In the cases we shall treat all series xvill be absolutely convergent, in particular for Zm Zn Olmn ZEn Em Olmn it is sufficient to have both double series absolutely convergent. This simple transformation has beeni used in one forrn or another by several authors [2], [3], [6], [11]. Our object here is to systematically explore several special cases of this result. In particular, it was hoped that it might be possible to sum the basic bilateral hypergeometric series 2V/2. Such a summation formula was originally sought by Bailey [4]; Agarwal and Verma [1] state that they wATere motivated to develop the theory of bi-basic hypergeometric series in the hope of finding such a formula. An incorrect summation of the 2V/2 is given in [10]; this may be seen by substituting b-+bt, c--ct, d-sdt, e->et in [10, Equation (1)] and verifying that the series satisfies a first order q-difference equation in t wrhich is inconsistent with a first order q-difference equation satisfied by the product. Unfor-

Certain Transformations and Summation Formulae for Basic Bilateral Hypergeometric Series 2 Ψ 2

Certain Transformations and Summation Formulae for Basic Bilateral Hypergeometric Series 2 Ψ 2

Generalized basic hypergeometric series with unconnected bases

In a series of recent papers Verma and Upadhyay (7,8,9) developed the theory of basic hypergeometric series with two bases q and q½. These investigations were made in an attempt to discover a summation formula for a bilateral basic hypergeometric series 2Ψ2 analogous to that for a 2H2 (cf. Bailey (2,3)) and in finding relations between certain q-infinite products. In one of their papers they mentioned that it did not seem possible to develop the corresponding general theory for basic series with two unconnected bases q and q1. A recent paper by Andrews (1) indicates that transformations between basic hypergeometric series with two unconnected bases can be very interesting and useful in the study of ‘mock’ theta functions and their extensions. Besides this interest, such a theory also enables one to extend the entire existing transformation theory of the generalized basic hypergeometric series.

A note on the partial sums of certain basic bilateral hypergeometric series

A note on the partial sums of certain basic bilateral hypergeometric series

On Lerch's Transcendant and the Basic Bilateral Hypergeometric Series 2 ψ2

On Lerch's Transcendant and the Basic Bilateral Hypergeometric Series ₂ ψ₂

On some Formulae in Partition Theory, and Bilateral Basic Hypergeometric Series

On some Formulae in Partition Theory, and Bilateral Basic Hypergeometric Series