Summation formulae for 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.

     

Abel's lemma on summation by parts and basic hypergeometric series

Basic hypergeometric series identities are revisited systematically by means of Abel's lemma on summation by parts. Several new formulae and transformations are also established. The author is convinced that Abel's lemma on summation by parts is a natural choice in dealing with basic hypergeometric series.     

Summations and transformations for multiple basic and elliptic hypergeometric series by determinant evaluations

Using multiple q-integrals and a determinant evaluation, we establish a multivariable extension of Bailey's nonterminating 1009 transformation. From this result, we deduce new multivariable terminating ₁₀φ₉ transformations, ₈φ₇ summations and other identities. We also use similar methods to derive new multivariable l ₁ψ₁ summations. Some of our results are extended to the case of elliptic hypergeometric series.     

Semi-finite forms of bilateral basic hypergeometric series

We show that several classical bilateral summation and transformation formulas have semi-finite forms. We obtain these semi-finite forms from unilateral summation and transformation formulas. Our method can be applied to derive Ramanujan's summation, Bailey's transformations, and Bailey's summation.

     

One-parameter orthogonality relations for basic hypergeometric series

The second order hypergeometric q-difference operator is studied for the value c = −q. For certain parameter regimes the corresponding recurrence relation can be related to a symmetric operator on the Hilbert space ℓ²( ). The operator has deficiency indices (1, 1) and we describe as explicitly as possible the spectral resolutions of the self-adjoint extensions. This gives rise to one-parameter orthogonality relations for sums of two ₂₁-series. In particular, we find that the Ismail-Zhang q-analogue of the exponential function satisfies certain orthogonality relations.     

Convergence of Magnus integral addition theorems for confluent hypergeometric functions

In 1946, Magnus presented an addition theorem for the confluent hypergeometric function of the second kind U with argument x+y expressed as an integral of a product of two U's, one with argument x and another with argument y. We take advantage of recently obtained asymptotics for U with large complex first parameter to determine a domain of convergence for Magnus' result. Using well-known specializations of U, we obtain corresponding integral addition theorems with precise domains of convergence for modified parabolic cylinder functions, and Hankel, Macdonald, and Bessel functions of the first and second kind with order zero and one.     

The Cauchy operator for basic hypergeometric series

We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine's transformation formula and Sears' transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it can be considered as a generalization of the operator T(bD_q). Using this operator, we obtain extensions of the Askey–Wilson integral, the Askey–Roy integral, Sears' two-term summation formula, as well as the q-analogs of Barnes' lemmas. Finally, we find that the Cauchy operator is also suitable for the study of the bivariate Rogers–Szegö polynomials, or the continuous big q-Hermite polynomials.     

Some identities between basic hypergeometric series deriving from a new Bailey-type transformation

We prove a new Bailey-type transformation relating WP-Bailey pairs. We then use this transformation to derive a number of new 3- and 4-term transformation formulae between basic hypergeometric series.     

Reduction and transformation formulae for bivariate basic hypergeometric series

The main object of this paper is to establish several bivariate basic hypergeometric series identities by means of elementary series manipulation. Some of them can be applied to yield transformation and reduction formulae for q-Kampé de Fériet functions.     

Two operator identities and their applications to terminating basic hypergeometric series and q-integrals

In this paper, we first give two interesting operator identities, and then, using them and the q-exponential operator technique to some terminating summation formulas of basic hypergeometric series and q-integrals, we obtain some q-series identities and q-integrals involving ₃?₂.     

Integro-local theorems for sums of independent random vectors in the series scheme

Let S(n) = ξ(1)+?+ξ(n) be a sum of independent random vectors ξ(i) = ξ _(n)(i) with general distribution depending on a parameter n. We find sufficient conditions for the uniform version of the integro-local Stone theorem to hold for the asymptotics of the probability P(S(n) ∈ Δ[x), where Δ[x) is a cube with edge Δ and vertex at a point x.     

New transformations for elliptic hypergeometric series on the root system A n

Recently, Kajihara gave a Bailey-type transformation relating basic hypergeometric series on the root system A _n , with different dimensions n. We give, with a new, elementary proof, an elliptic extension of this transformation. We also obtain further Bailey-type transformations as consequences of our result, some of which are new also in the case of basic and classical hypergeometric series. 2000 Mathematics Subject Classification Primary—33D67; Secondary—11F50     

Summation formulae for twisted cubic q‐series

A new class of twisted cubic q‐series is investigated by means of the modified Abel lemma on summation by parts. Several remarkable summation and transformation formulae are established for both terminating and nonterminating series.     

Bilateral inversions and terminating basic hypergeometric series identities

The q-analogue of Legendre inversions is established and generalized to bilateral sequences. They are employed to investigate the dual relations of three basic formulae due to Jackson and Bailey, on balanced ₃?₂-series, well-poised ₈?₇-series and bilateral ₆ψ₆-series. Several terminating well-poised series identities are consequently derived, including the q-Dixon formulae on terminating ₃ψ₃-series and two terminating well-poised ₅ψ₅-series identities due to [F.H. Jackson, Certain q-identities, Quart. J. Math. (Oxford) 12 (1941) 167-172; W.N. Bailey, On the analogue of Dixon’s theorem for bilateral basic hypergeometric series, Quart. J. Math. (Oxford) 1 (1950) 318-320].     

Two new transformation formulas of basic hypergeometric series

By means of a modified version of Cauchy's method for obtaining bilateral series identities, two new transformation formulas for bilateral basic hypergeometric series are derived. These contain several important identities for basic hypergeometric series as special cases, including the nonterminating q-Saalschütz summation, Bailey's very well-poised summation and the nonterminating Watson transformation.     

Short proofs of summation and transformation formulas for basic hypergeometric series

We show that several terminating summation and transformation formulas for basic hypergeometric series can be proved in a straightforward way. Along the same line, new finite forms of Jacobi's triple product identity and Watson's quintuple product identity are also proved.     

Transformation and reduction formulae for double q‐Clausen hypergeometric series

By means of the Sears transformations, we establish eight general transformation theorems on bivariate basic hypergeometric series. Several transformation, reduction and summation formulae on the double q‐Clausen hypergeometric series are derived as consequences. Copyright © 2007 John Wiley & Sons, Ltd.     

Abel's method on summation by parts and theta hypergeometric series

Abel's lemma on summation by parts is reformulated to investigate systematically terminating theta hypergeometric series. Most of the known identities are reviewed and several new transformation and summation formulae are established. The authors are convinced by the exhibited examples that the iterating machinery based on the modified Abel lemma is powerful and a natural choice for dealing with terminating theta hypergeometric series.     

C n and D n Very-Well-Poised 10 φ 9 Transformations

In this paper we derive multivariable generalizations of Bailey's classical terminating balanced very-well-poised _{10 9} transformation. We work in the setting of multiple basic hypergeometric series very-well-poised on the root systems A _n , C _n , and D _n . Following the distillation of Bailey's ideas by Gasper and Rahman [11], we use a suitable interchange of multisums. We obtain C _n and D _{n 10 9} transformations combined with A _n , C _n , and D _n extensions of Jackson's _{8 7} summation. Milne and Newcomb have previously obtained an analogous formula for A _n series. Special cases of our _{10 9} transformations include several new multivariable generalizations of Watson's transformation of an _{8 7} into a multiple of a _{4 3} series. We also deduce multidimensional extensions of Sears' _{4 3} transformation formula, the second iterate of Heine's transformation, the q -Gauss summation theorem, and of the q -binomial theorem. August 28, 1996. Date revised: September 8, 1997.