Transformation formulas for double hypergeometric series related to 9-j coefficients and their basic analogs

Abstract

In a recent paper, Ališauskas deduced different triple sum expressions for the 9-j coefficient of su(2) and su q (2). For a singly stretched 9-j coefficient, these reduce to different double sum series. Using these distinct series, we deduce a set of new transformation formulas for double hypergeometric series of Kampé de Fériet type and their basic analogues. These transformation formulas are valid for rather general parameters of the series, although a common feature is that all the series appearing here are terminating. It is also shown that the transformation formulas deduced here generate a group of transformation formulas, thus yielding an invariance group or symmetry group of particular double series.Running title : Transformations for double series.

Coupling coefficients of SO(n) and integrals involving Jacobi and Gegenbauer polynomials

“…Similarly as special terminating double-hypergeometric series of Kampé de Fériet-type [19][20][21]53] correspond to the stretched 9j coefficients of SU(2), the definite terminating triple-hypergeometric series correspond either to the semistretched isofactors of the second kind [14] of Sp(4), or to the isofactors of the symmetric irreps of the orthogonal group SO(n) in the canonical and semicanonical (tree type) bases. Our relation (2.6a)-(2.6c) (which is significant within the framework of Sp(4) isofactors) is a triple-sum generalization of transformation formula (9) of [21] for terminating F 1:2,2 1:1,1 Kampé de Fériet series with a fixed single-integer non-positive parameter, restricting all summation parameters. (This restriction is hidden in equation (2.7a)-(2.7b), rearranged for the aims of section 3.)…”

“…(This restriction is hidden in equation (2.7a)-(2.7b), rearranged for the aims of section 3.) Relations (2.6a)-(2.6c), with intermediate formula (2.6b), were derived using the transformation formulae [19,21] of the double series, treated as the stretched 9j coefficients. The relation (3.2c)-(3.2e) (important within the framework of SO(n) isofactors) cannot be associated with any transformation formula [21] for terminating F 1:2,2 1:1,1 Kampé de Fériet series with the same (single or double) parameters, restricting summation.…”

“…We transformed the double sum over x 1 , x 2 in (2.6a) into the sum over u, v in (2.6b) using relation (C1a)-(C1c) of [19] and later the double sum over v, x 3 in (2.6b) into the sum over x 1 , x 3 in (2.6c) using relation (C1f )-(C1b) of [19] and replacing u by x 2 . (The related transformations for the double hypergeometric series of Kampé de Fériet-type [20] are also considered by Lievens and Van der Jeugt [21]. )…”

“…(These triple sum series may be treated as extensions of the double-hypergeometric series of Kampé de Fériet [20] type, e.g. considered by Lievens and Van der Jeugt [21]. )…”

A Method for q-Calculus

“…These results were extended to double q-hypergeometric series in [69] and [98]. We are now ready for the first definitions in q-calculus.…”