Michael J. Schlosser
A Taylor expansion theorem for an elliptic extension of
the Askey-Wilson operator
(13 pages)
Abstract.
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 8Φ7 summation and of Bailey's
10Φ9
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
8Φ7 summation and a quadratic expansion.
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