Balanced Hypergeometric Series Research Articles

Multiplicate inverse forms of terminating hypergeometric series

The multiplicate form of Gould–Hsu's inverse series relations enables to investigate the dual relations of the Chu–Vandermonde–Gauß’s, the Pfaff–Saalschütz's summation theorems and the binomial convolution formula due to Hagen and Rothe. Several identity and reciprocal relations are thus established for terminating hypergeometric series. By virtue of the duplicate inversions, we establish several dual formulae of Chu–Vandermonde–Gauß’s and Pfaff–Saalschütz's summation theorems in Section 3 and 4, respectively. Finally, the last section is devoted to deriving several identities and reciprocal relations for terminating balanced hypergeometric series from Hagen–Rothe's convolution identity in accordance with the duplicate, triplicate and multiplicate inversions.

Legendre inversions and balanced hypergeometric series identities

By means of Legendre inverse series relations, we prove two terminating balanced hypergeometric series formulae. Their reversals and linear combinations yield several known and new hypergeometric series identities.

Expansion formulas for terminating balanced 4F3-series from the Biedenharn–Elliot identity for [formula omitted

In a recent paper, George Gasper (Contemp. Math. 254 (2000) 187) proved some expansion formulas for terminating balanced hypergeometric series of type 4 F 3 with unit argument. In this article we show how one easily derives such expansion formulas from the Biedenharn–Elliot identity for the Lie algebra su(1,1) . Furthermore, we give a rather systematic method for determining when two apparently different expansion formulas are the same up to transformation formulas. This is a rather nice application of the so-called invariance groups of hypergeometric series. The method extends to other cases; we briefly indicate how it works in the case of expansion formulas for 3 F 2-series. We conclude with some basic analogues and show their relation with the Askey–Wilson polynomials.

Inversion Techniques and Combinatorial Identities: Balanced Hypergeometric Series

Following the earlier works on Inversion Techniques and Combinatorial Identities, the duplicate form of the Gould-Hsu (1973) inversion theorem is constructed. As applications, several terminating balanced hypergeometric formulas are demonstrated, including those due to Andrews (1995), which have been the primary stimulation to the present research. Encouraged by the recent work of Stanton (1998), we establish two higher hypergeometric evaluations with three additional parameters, which specialize further to over two hundreds hypergeometric identities.

Spectral Transformation Chains and Some New Biorthogonal Rational Functions

A discrete-time chain, associated with the generalized eigenvalue problem for two Jacobi matrices, is derived. Various discrete and continuous symmetries of this integrable equation are revealed. A class of its rational, elementary and elliptic functions solutions, appearing from a similarity reduction, are constructed. The latter lead to large families of biorthogonal rational functions based upon the very-well-poised balanced hypergeometric series of three types: the standard hypergeometric series 9 F 8, basic series 10ϕ9 and its elliptic analogue 10 E 9. For an important subclass of the elliptic biorthogonal rational functions the weight function and normalization constants are determined explicitly.

Towards the canonical tensor operators of uq(3). II. The denominator function problem

The explicit denominator (normalization) function of the canonical tensor operators of the quantum algebra uq(3), corresponding to the maximal null space case is derived ab initio in terms of double basic hypergeometric series, which cannot be obtained as any q-extension of the SU(3) denominator polynomial Gb″1(Δ,x) in terms of multiple (double or triple) balanced hypergeometric series, introduced by Biedenharn, Louck, and their collaborators (although their q=1 versions are shown being equivalent). The corresponding orthonormal seed isoscalar factors of the coupling (Wigner–Clebsch–Gordan) coefficients of uq(3) and SU(3) with multiple irreducible representations are presented. Conjectured expression of the q-polynomials [which ratios appear in the uq(3) and (new) SU(3) denominator functions for an arbitrary value of the canonical multiplicity label t of the repeating irreducible representations] in terms of multiple partition dependent q-series (extension of the maximal and minimal null space versions) is presented and considered.

Product and Addition Formulas for the Continuous q-Ultraspherical Polynomials

Using Askey and Wilson’s orthogonality relation and Rahman’s product formula for Askey–Wilson polynomials a Gegenbauer-type product formula is obtained for continuous q-ultraspherical polynomials. Summation and transformation formulas for balanced hypergeometric series are then employed to derive a q-analogue of Gegenbauer’s addition formula.