We develop a method for deriving integral representations of certain orthogonal polynomials as moments. These moment representations are applied to find linear and multilinear generating functions for $q$-orthogonal polynomials. As a byproduct we establish new transformation formulas for combinations of basic hypergeometric functions, including a new representation of the $q$-exponential function $\mathcal{E}_q$.

Keywords:

$q$-integral, $q$-orthogonal polynomials, associated polynomials, $q$-difference equations, generating functions, Al-Salam-Chihara polynomials, continuous $q$-ultraspherical polynomials $q$-integral, $q$-orthogonal polynomials, associated polynomials, $q$-difference equations, generating functions, Al-Salam-Chihara polynomials, continuous $q$-ultraspherical polynomials

MSC Classifications:

33D45, 33D20, 33C45, 30E05 show english descriptions Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
unknown classification 33D20
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]
Moment problems, interpolation problems 33D45 - Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
33D20 - unknown classification 33D20
33C45 - Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]
30E05 - Moment problems, interpolation problems

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