Existence of a pair of new recurrence relations for the Meixner-Pollaczek polynomials

E. I. Jafarov, A. M. Jafarova, S. M. Nagiyev

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 φ → 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 λ > 0 and 0 < φ < π via generalized Laguerre polynomials. 

Tbilisi Mathematical Journal, Vol. 11(3) (2018), pp. 29-39