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Discrete q-Hermite polynomials

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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. hn(x;q) is also called discrete q-Hermite I polynomials and ĥn(x;q) is also called discrete q-Hermite II polynomials.

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 ) {\displaystyle \displaystyle h_{n}(x;q)=q^{\binom {n}{2}}{}_{2}\phi _{1}(q^{-n},x^{-1};0;q,-qx)=x^{n}{}_{2}\phi _{0}(q^{-n},q^{-n+1};;q^{2},q^{2n-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 ) {\displaystyle \displaystyle {\hat {h}}_{n}(x;q)=i^{-n}q^{-{\binom {n}{2}}}{}_{2}\phi _{0}(q^{-n},ix;;q,-q^{n})=x^{n}{}_{2}\phi _{1}(q^{-n},q^{-n+1};0;q^{2},-q^{2}/x^{2})=i^{-n}V_{n}^{(-1)}(ix;q)}

and are related by

h n ( i x ; q 1 ) = i n h ^ n ( x ; q ) {\displaystyle h_{n}(ix;q^{-1})=i^{n}{\hat {h}}_{n}(x;q)}


References

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