2013/01/16 by Wang, Jin, Ma, Xinrong
#05A30 #33D15 #Combinatorics (math.CO) #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.1301.3582
With the use of the (f,g)-matrix inversion under specializations that f=1-xy,g=y-x, we establish an (1-xy,y-x)-expansion formula. When specialized to basic hypergeometric series, this (1-xy,y-x)-expansion formula leads us to some expansion formulas expressing any rϕs series in variable x~t in terms of a linear combination of r+2ϕs+1 series in t, as well as various specifications. All these results can be regarded as common generalizations of many konwn expansion formulas in the setting of q-series. As specific applications, some new transformation formulas of q-series including new approach to the Askey-Wilson polynomials, the Rogers-Fine identity, Andrews' four-parametric reciprocity theorem and Ramanujan's 1ψ1 summation formula, as well as a transformation for certain well-poised Bailey pairs, are presented.