q-Differential operator identities and applications

In this paper, we construct a new q-exponential operator and obtain some operator identities. Using these operator identities, we give a formal extension of Jackson’s 2Φ2 transformation formula. A formal extension of Bailey’s 3ψ3 summation and an extension of the Sears terminating balanced 4Φ3 transformation formula are also derived by our operator method. In addition, we also derive several interesting a formal extensions involving multiple sum about three terms of Sears 3Φ2 transformation formula and Heine’s 2Φ1 transformation formula. © 2006 Elsevier Inc. All rights reserved