Applications of 𝑞-Lagrange inversion to basic hypergeometric series

Abstract

Abstract. A family of g-Lagrange inversion formulas is given. Special cases include quadratic and cubic transformations for basic hypergeometric series. The g-analogs of the so-called "strange evaluations" are also corollaries. Some new RogersRamanujan identities are given. A connection between the work of Rogers and Andrews, and ^-Lagrange inversion is stated.

Abel’s lemma on summation by parts and terminating q-series identities

“…[3,Eq. 4.7] and [15,Eq. 4.32]): n k=0 (a; q 3 ) k [q −n , q n a; q] k (q; q) k [a, qa; q 2 ] k q k = χ(n = 3m) × a m q, q 2 qa, q 2 a q 3 m .…”

“…The objective of the present work is to explore further applications of the modified Abel lemma on summation by parts to terminating basic hypergeometric series identities. Numerous identities appeared in Chu [5][6][7], GasperRahman [11][12][13] and Gessel-Stanton [14,15] will be recovered and several new transformation and summation formulae will be established.…”

On an identity of Gessel and Stanton and the new little Göllnitz identities

“…Identity (1.1) was one of many derived by Gessel and Stanton in their 1983 paper on q-Lagrange inversion [11]. This identity came to our attention as a candidate for a symmetric version of a recently discovered variation of the little Göllnitz identities [9] in the same way that the Göllnitz-Gordon theorem is a symmetric version of the familiar little Göllnitz identities.…”

Abel’s lemma on summation by parts and partial q-series transformations

“…Reformulating the last equation as G n (a, c, e) = G n (q 4 a, q 4 c, q 4 e) (qe; q) 4 (1 − q 6 ce 2 /a 2 )(1 − q 6 e 3 /ac) (q 3 e 2 /a; q) 4 (1 − q 4 c) ( and then iterating this relation m-times, we get the following expression G n (a, c, e) = G n (q 4m a, q 4m c, q 4m e) (qe; q) 4m (q 3 e 2 /a; q) 4m q 6 ce 2 /a 2 , q 6 {1 − q 6+8k e 3 /a} (qe; q) 4k (q 6 e 2 /a; q) 4k q 6 ce 2 /a 2 , q 6 e 3 /ac q 4 c, q 4 ae/c |q 4 k q 4k (11) and then reformulating the R-function by singling out k-factorials R(q 4k a, q 4k c, q 4k e) = 1 − q 6+4n+8k e 3 /a 1 − q 6+8k e 3 /a a 2 /q 2 e 2 q 4+4k c, q 4+4k ae/c |q 4 n × (q 1+4k e; q) 3n [a/qe, a/q 2 e, a/q 3 e; q 2 ] n a/c, c/e q 6+4k e 2 /a |q n = 1 − q 6+4n+8k e 3 /a 1 − q 6+8k e 3 /a q 6 e 2 /a, q 1+3n e q 6+n e 2 /a, qe |q 4k q 4 c, q 4 ae/c q 4+4n c, q 4+4n ae/c |q 4 k × (qe; q) 3n [a/qe, a/q 2 e, a/q 3 e; q 2 ] n a/c, c/e q 6 e 2 /a |q n a 2 /q 2 e 2 q 4 c, q 4 ae/c |q 4 n , we establish another transformation formula.…”

“…We shall establish reciprocal relations for well-poised, quadratic, cubic and quartic series as well as transformation formulae from quadratic, cubic and quartic series to well-poised series. Several interesting known identities and transformations discovered by Chu [8] , Gapser [9] , Gapser-Rahman [10] , Gessel-Stanton [11] and Rahman [12] are exemplified as consequences. They will further convince that as classical analytic weapons, Abel's lemma on summation by parts is indeed a very natural and powerful method to investigate basic hypergeometric series.…”