## Tbilisi Mathematical Journal

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

#### Abstract

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.

#### Note

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.

#### Article information

Source
Tbilisi Math. J., Volume 11, Issue 3 (2018), 29-39.

Dates
Accepted: 15 June 2018
First available in Project Euclid: 3 October 2018

https://projecteuclid.org/euclid.tbilisi/1538532024

Digital Object Identifier
doi:10.32513/tbilisi/1538532024

Mathematical Reviews number (MathSciNet)
MR3954192

#### Citation

Jafarov, E. I.; Jafarova, A. M.; Nagiyev, S. M. Existence of a pair of new recurrence relations for the Meixner-Pollaczek polynomials. Tbilisi Math. J. 11 (2018), no. 3, 29--39. doi:10.32513/tbilisi/1538532024. https://projecteuclid.org/euclid.tbilisi/1538532024

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