A Taylor expansion theorem for an elliptic extension of the Askey-Wilson operator

We establish Taylor series expansions in rational (and elliptic) function bases using E. Rains' elliptic extension of the Askey-Wilson divided difference operator. The expansion theorem we consider extends M.E.H. Ismail's expansion for the Askey-Wilson monomial basis. Three immediate applications (essentially already due to Rains) include simple proofs of Frenkel and Turaev's elliptic extensions of Jackson's 8-phi-7 summation and of Bailey's 10-phi-9 transformation, and the computation of the connection coefficients of Spiridonov's elliptic extension of Rahman's biorthogonal rational functions. We adumbrate other examples including the nonterminating extension of Jackson's 8-phi-7 summation and a quadratic expansion.