  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.
