Difference between revisions of "Continuous q-Hermite polynomial"
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| + | 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= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> 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|$q$-Pochhammer symbol]] and $H_k(\xi|q)$ denotes a [[continuous q-Hermite polynomial|continuous $q$-Hermite polynomial]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$H_n(-x|q)=(-1)^nH_n(x|q).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | |||
=References= | =References= | ||
| − | [http://arxiv.org/pdf/1101.2875v4.pdf On the q-Hermite polynomials and their relationship with some other families of orthogonal polynomials] | + | [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 | ||
| + | |||
| + | [[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
