Meixner–Pollaczek polynomials

In mathematics, the Meixner–Pollaczek polynomials are a family of orthogonal polynomials P(x,φ) introduced by , which up to elementary changes of variables are the same as the Pollaczek polynomials P(x,a,b) rediscovered by in the case λ=1/2, and later generalized by him.

They are defined by

Examples

The first few Meixner–Pollaczek polynomials are

Properties

Orthogonality

The Meixner–Pollaczek polynomials Pm(λ)(x;φ) are orthogonal on the real line with respect to the weight function

and the orthogonality relation is given byKoekoek, Lesky, & Swarttouw (2010), p. 213.

Recurrence relation

The sequence of Meixner–Pollaczek polynomials satisfies the recurrence relationKoekoek, Lesky, & Swarttouw (2010), p. 213.

Rodrigues formula

The Meixner–Pollaczek polynomials are given by the Rodrigues-like formulaKoekoek, Lesky, & Swarttouw (2010), p. 214.

where w(x;λ,φ) is the weight function given above.

Generating function

The Meixner–Pollaczek polynomials have the generating functionKoekoek, Lesky, & Swarttouw (2010), p. 215.

See also

Sieved Pollaczek polynomials

References