Michael J. Schlosser
Bilateral q-ultraspherical functions
(46 pages)
Abstract.
We introduce the bilateral q-ultraspherical functions, a
bilateral-series extension of the continuous q-ultraspherical
polynomials. They are defined by specific bilateral basic hypergeometric
2Ψ2 series, are analytic in the variable
x = cos θ, and depend on two parameters
β and γ and on a base q. We derive a product formula
for their bilateral generating function, a three-term recurrence relation,
their transformation under the Askey–Wilson divided difference operator,
three weight-based Rodrigues-type formulae, and explicit large-order
asymptotic expansions. The main results are full orthogonality relations
with respect to explicit orthogonality functionals involving analytic mass
aggregates. We also obtain shifted orthogonality relations and a bilateral
Chen–Liu type mixed orthogonality formula. In the limit γ→1
the construction and identities reduce to the classical results for the
continuous q-ultraspherical polynomials.
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