Two q-difference equations and q-operator identities.pdf

Author Queries JOB NUMBER: MS 381208 JOURNAL: GDEA Q1 Reference Askey [2] is provided in the list but not cited in the text. Please supply appropriate citation details for the same. Q2 Please supply the page range for the reference Rogers [10]. Two q-difference equations and q-operator identities Zhi-Guo Liu* Department of Mathematics, East China Normal University, 500 Dongchuan Road, Shanghai 200241, P.R. China (Received 27 February 2008; final version received 9 February 2009) In this paper, we establish two general q-exponential operator identities by solving two simple q-difference equations, which contain two well-known operator identities as special cases. These operator identities allow us to derive naturally the q-Mehler formulas for the Rogers-Szego? polynomials and the q 21-Rogers-Szego? polynomials. The q-Mehler formulas are used to derive two q-exponential operator identities involving 3f2 series. We derive a q-integral formula which is an extension of the q-integral form of the Sears transformation. Finally, we set up a general identity for the bilateral q-series and from which a simple proof of Bailey’s 6c6 summation is given. Keywords: q-derivative; q-integral; q-difference equation; q-series; q-exponential operator; Rogers-Szego? polynomials; q-Mehler formula; Sears transformation; Bailey’s 6c6 summation 2000 Mathematics Subject Classification: 05A30; 33D15; 33D05; 33D45; 33D60 1. Introduction In this paper, we shall assume that 0 , q , 1. The q-shifted factorials are defined by ea; qT0 ? 1; ea; qTn ? Yn21 k?0 e1 2 aqkT; n $ 1; e1:1T ea; qT1 ? lim n!1 Yn21 k?0 e1 2 aqkT ? Y1 k?0 e1 2 aqkT: We also adopt the following compact notation for multiple q-shifted factorials: ea1; a2; . . . ; am; qTn ? ea1; qTnea2; qTn· · ·eam; qTn; e1:2T where n is an integer or 1. The q-derivative operator is defined by Dq{f eaT} ? f eaT2 f eaqT a ; Dnq{f eaT} ? Dq Dn21q {f eaT} n o : e1:3T Unless stated otherwise, it is understood that throughout this paper, Dq acts on th