Linearization of the product of symmetric orthogonal polynomials

A new constructive approach is given to the linearization formulas of symmetric orthogonal polynomials. We use the monic three-term recurrence relation of an orthogonal polynomial system to set up a partial difference equation problem for the product of two polynomials and solve it in terms of the initial data. To this end, an auxiliary function of four integer variables is introduced, which may be seen as a discrete analogue of Riemann's function. As an application, we derive the linearization formulas for the associated Hermite polynomials and for their continuousq-analogues. The linearization coefficients are represented here in terms of3F2 and3Φ2 (basic) hypergeometric functions, respectively. We also give some partial results in the case of the associated continuousq-ultraspherical polynomials.

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