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Existence of a pair of new recurrence relations for the Meixner-Pollaczek polynomials. (English) Zbl 1435.33009

Summary: We report on existence of pair of new recurrence relations (or difference equations) for the Meixner-Pollaczek polynomials. Proof of the correctness of these difference equations is also presented. Next, we found that subtraction of the forward shift operator for the Meixner-Pollaczek polynomials from one of these recurrence relations leads to the difference equation for the Meixner-Pollaczek polynomials generated via \(\cosh\) difference differentiation operator. Then, we show that, under the limit \(\varphi \rightarrow 0\), new recurrence relations for the Meixner-Pollaczek polynomials recover pair of the known recurrence relations for the generalized Laguerre polynomials. At the end, we introduced differentiation formula, which expresses Meixner-Pollaczek polynomials with parameters \(\lambda>0\) and \(0 < \varphi < \pi\) via generalized Laguerre polynomials.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
39A10 Additive difference equations
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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