Difference between revisions of "Continuous q-Hermite polynomial"
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[http://arxiv.org/pdf/1101.2875v4.pdf On the q-Hermite polynomials and their relationship with some other families of orthogonal polynomials]<br /> | [http://arxiv.org/pdf/1101.2875v4.pdf On the q-Hermite polynomials and their relationship with some other families of orthogonal polynomials]<br /> | ||
Classical and quantum orthogonal polynomials in one variable by Ismail Mourad | Classical and quantum orthogonal polynomials in one variable by Ismail Mourad | ||
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| + | [[Category:SpecialFunction]] | ||
Latest revision as of 18:54, 24 May 2016
The continuous $q$-Hermite polynomials are defined by $$\left\{ \begin{array}{ll} H_0(x|q)=1 \\ H_1(x|q)=2x \\ H_{n+1}(x|q) = 2xH_n(x|q) - (1-q^n)H_{n-1}(x|q). \end{array} \right.$$
Properties
Theorem: The following formula holds: $$\dfrac{1}{(te^{i\theta};q)_{\infty}(te^{-i\theta};q)_{\infty}} = \displaystyle\sum_{k=0}^{\infty} H_k(\cos \theta|q) \dfrac{t^k}{(q;q)_k},$$ where $(\xi,q)_{\infty}$ denotes the $q$-Pochhammer symbol and $H_k(\xi|q)$ denotes a continuous $q$-Hermite polynomial.
Proof: █
Theorem: The following formula holds: $$H_n(-x|q)=(-1)^nH_n(x|q).$$
Proof: █
References
On the q-Hermite polynomials and their relationship with some other families of orthogonal polynomials
Classical and quantum orthogonal polynomials in one variable by Ismail Mourad
