Q-Meixner–Pollaczek polynomials explained

Q-Meixner–Pollaczek polynomials should not be confused with q-Meixner polynomials.

In mathematics, the q-Meixner–Pollaczek polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.

Definition

The polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by ：^[1]

P_n(x;a\midq)=a^-ne^in\phi

	2;q)
(a
	n

(q;q)_n

{}_3\phi

	-n

	2(q

,ae^{i(\theta+2\phi)},ae^-i\theta;a^2,0\midq;q), x=\cos(\theta+\phi).

Notes and References

Roelof Koekoek, Hypergeometric Orthogonal Polynomials and its q-Analoques, p 460, Springer