Rahul Sharma (Editor)

Discrete q Hermite polynomials

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

In mathematics, the discrete q-Hermite polynomials are two closely related families hn(x;q) and ĥn(x;q) of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Al-Salam and Carlitz (1965). Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.

Definition

The discrete q-Hermite polynomials are given in terms of basic hypergeometric functions and the Al-Salam–Carlitz polynomials by

h n ( x ; q ) = q ( n 2 ) 2 ϕ 1 ( q n , x 1 ; 0 ; q , q x ) = x n 2 ϕ 0 ( q n , q n + 1 ; ; q 2 , q 2 n 1 / x 2 ) = U n ( 1 ) ( x ; q ) h ^ n ( x ; q ) = i n q ( n 2 ) 2 ϕ 0 ( q n , i x ; ; q , q n ) = x n 2 ϕ 1 ( q n , q n + 1 ; 0 ; q 2 , q 2 / x 2 ) = i n V n ( 1 ) ( i x ; q )

and are related by

h n ( i x ; q 1 ) = i n h ^ n ( x ; q )

References

Discrete q-Hermite polynomials Wikipedia