Continuous q-Hermite polynomials

The polynomials are given in terms of basic hypergeometric functions by

:$H\_n(x|q)=e^\{in\backslash theta\}\{\}\_2\backslash phi\_0\backslash left[\backslash begin\{matrix\}$

q^{-n},0\\

-\end{matrix}

;q,q^n e^{-2i\theta}\right],\quad x=\cos\,\theta.

: $2x\; H\_n(x\backslash mid\; q)\; =\; H\_\{n+1\}\; (x\backslash mid\; q)\; +\; (1-q^n)\; H\_\{n-1\}\; (x\backslash mid\; q)$

with the initial conditions

: $H\_0\; (x\backslash mid\; q)\; =1,\; H\_\{-1\}\; (x\backslash mid\; q)\; =\; 0$

From the above, one can easily calculate:

: $$

\begin{align}

H_0 (x\mid q) & = 1 \\

H_1 (x\mid q) & = 2x \\

H_2 (x\mid q) & = 4x^2 - (1-q) \\

H_3 (x\mid q) & = 8x^3 - 2x(2-q-q^2) \\

H_4 (x\mid q) & = 16x^4 - 4x^2(3-q-q^2-q^3) + (1-q-q^3+q^4)

\end{align}

: $\backslash sum\_\{n=0\}^\backslash infty\; H\_n(x\; \backslash mid\; q)\; \backslash frac\{t^n\}\{(q;q)\_n\}\; =\; \backslash frac\{1\}$

{\left( t e^{i \theta},t e^{-i \theta};q \right)_\infty}

where $\backslash textstyle\; x=\backslash cos\; \backslash theta$.

• {{Citation | last1=Gasper | first1=George | last2=Rahman | first2=Mizan | title=Basic hypergeometric series | publisher=Cambridge University Press | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | mr=2128719 | year=2004 | volume=96}}
• {{Citation | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=Springer-Verlag | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010}}
• {{dlmf|id=18|title=Chapter 18: Orthogonal Polynomials|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
• {{cite thesis |last=Sadjang |first=Patrick Njionou |title=Moments of Classical Orthogonal Polynomials |type=Ph.D. |publisher=Universität Kassel |citeseerx=10.1.1.643.3896 }}

Category:Orthogonal polynomials

Category:Q-analogs

Category:Special hypergeometric functions