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Continuous q-Hermite polynomials
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<h2 id="definition">Definition</h2> <p>The polynomials are given in terms of <a href="/facts/Basic_hypergeometric_function/LuW9vAIs">basic hypergeometric functions</a> by </p> H n ( x | q ) = e i n θ 2 ϕ 0 [ q − n , 0 − ; q , q n e − 2 i θ ] , x = cos θ . {\displaystyle H_{n}(x|q)=e^{in\theta }{}_{2}\phi _{0}\left[{\begin{matrix}q^{-n},0\\-\end{matrix}};q,q^{n}e^{-2i\theta }\right],\quad x=\cos \,\theta .} <h2 id="recurrence-and-difference-relations">Recurrence and difference relations</h2> 2 x H n ( x ∣ q ) = H n + 1 ( x ∣ q ) + ( 1 − q n ) H n − 1 ( x ∣ q ) {\displaystyle 2xH_{n}(x\mid q)=H_{n+1}(x\mid q)+(1-q^{n})H_{n-1}(x\mid q)} <p>with the initial conditions </p> H 0 ( x ∣ q ) = 1 , H − 1 ( x ∣ q ) = 0 {\displaystyle H_{0}(x\mid q)=1,H_{-1}(x\mid q)=0} <p>From the above, one can easily calculate: </p> H 0 ( x ∣ q ) = 1 H 1 ( x ∣ q ) = 2 x H 2 ( x ∣ q ) = 4 x 2 − ( 1 − q ) H 3 ( x ∣ q ) = 8 x 3 − 2 x ( 2 − q − q 2 ) H 4 ( x ∣ q ) = 16 x 4 − 4 x 2 ( 3 − q − q 2 − q 3 ) + ( 1 − q − q 3 + q 4 ) {\displaystyle {\begin{aligned}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{aligned}}} <h2 id="generating-function">Generating function</h2> ∑ n = 0 ∞ H n ( x ∣ q ) t n ( q ; q ) n = 1 ( t e i θ , t e − i θ ; q ) ∞ {\displaystyle \sum _{n=0}^{\infty }H_{n}(x\mid q){\frac {t^{n}}{(q;q)_{n}}}={\frac {1}{\left(te^{i\theta },te^{-i\theta };q\right)_{\infty }}}} <p>where x = cos θ {\displaystyle \textstyle x=\cos \theta } . </p> <ul><li>Gasper, George; Rahman, Mizan (2004), <i>Basic hypergeometric series</i>, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-83357-8, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2128719">2128719</a></li> <li>Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), <i>Hypergeometric orthogonal polynomials and their q-analogues</i>, Springer Monographs in Mathematics, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-642-05014-5">10.1007/978-3-642-05014-5</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-05013-8, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2656096">2656096</a></li> <li>Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), <a href="http://dlmf.nist.gov/18">"Chapter 18: Orthogonal Polynomials"</a>, in <a href="/facts/Frank_W._J._Olver/nQYQVRHc">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <i><a href="/facts/Digital_Library_of_Mathematical_Functions/ymNQBbgK">NIST Handbook of Mathematical Functions</a></i>, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-19225-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a>.</li> <li>Sadjang, Patrick Njionou. <i>Moments of Classical Orthogonal Polynomials</i> (Ph.D.). Universität Kassel. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.643.3896">10.1.1.643.3896</a>.</li></ul>