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 $\varphi \to 0$, 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 $0 \lt \varphi \lt \pi$ via generalized Laguerre polynomials.
This work was supported by the Science Development Foundation under the President of the Republic of Azerbaijan Grant Nr EIF-KETPL-2-2015-1(25)-56/01/1 and Grant Nr EIF-KETPL-2-2015-1(25)-56/02/1. E.I. Jafarov kindly acknowledges support for visit to ICTP during July-September 2017, within the ICTP regular associateship scheme.
"Existence of a pair of new recurrence relations for the Meixner-Pollaczek polynomials." Tbilisi Math. J. 11 (3) 29 - 39, September 2018. https://doi.org/10.32513/tbilisi/1538532024