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Diagonalization of certain integral operators. (English) Zbl 0838.33012

The most general classical orthogonal polynomials are balanced \(_4\Phi_3\)’s. These have a divided difference operator which substitutes for the derivative applied to Jacobi polynomials. Brown and Ismail found an inverse for this operator. The present paper deals with this inverse. The authors start with the action of the derivative on the ultraspherical polynomials. One formula they use is the expansion of \(e^{ixt}\) as a series in ultraspherical polynomials. They then take the continuous \(q\)-ultraspherical polynomials times the \(_0\Phi_1\) \(q\)-Bessel functions and an appropriate extension of the remaining coefficients, and study the resulting function. It is a new extension of the exponential function. This exponential function is used to define \(q\)-Appell functions with respect to the divided difference operator mentioned above. In still unpublished work, Al-Salam has shown that the continuous \(q\)-Hermite polynomials of Rogers are the only such \(q\)-Appell polynomials which are orthogonal. \(q\)-Lommel polynomials, which had been studied more than 10 years ago by Ismail, arise in a very natural setting here.
Reviewer: R.Askey (Madison)

MSC:

39A70 Difference operators
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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