Al-Salam–Chihara polynomials
Encyclopedia
In mathematics, the Al-Salam–Chihara polynomials Qn(x;a,b;q) are a family of basic hypergeometric orthogonal polynomials
in the basic Askey scheme
, introduced by . give a detailed list of the properties of Al-Salam–Chihara polynomials.
by
where x = cos(θ).
Orthogonal polynomials
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...
in the basic Askey scheme
Askey scheme
In mathematics, the Askey scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed in , the Askey scheme was first drawn by and by , and has since been extended by and to cover basic...
, introduced by . give a detailed list of the properties of Al-Salam–Chihara polynomials.
Definition
The Al-Salam–Chihara polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbolPochhammer symbol
In mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation ', where is a non-negative integer. Depending on the context the Pochhammer symbol may represent either the rising factorial or the falling factorial as defined below. Care needs to be taken to check which...
by
where x = cos(θ).