Continuous Q-Hermite Polynomials

In mathematics, the continuous ''q''-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.

Definition

The polynomials are given in terms of basic hypergeometric functions by
:H_n(x, q)=e^_2\phi_0\left begin
q^,0\\
-\end
;q,q^n e^\right\quad x=\cos\,\theta.

Recurrence and difference relations

: $2x H_n(x\mid q) = H_ (x\mid q) + (1-q^n) H_ (x\mid q)$

with the initial conditions

: $H_0 (x\mid q) =1, H_ (x\mid q) = 0$

From the above, one can easily calculate:

: $\begin H_0 (x\mid q) & = 1 \\ H_1 (x\mid q) & = 2x \\ H_2 (x\mid q) & = 4x^2 - (1-q) \\ H_3 (x\mid q) & = 8x^3 - 2x(2-q-q^2) \\ H_4 (x\mid q) & = 16x^4 - 4x^2(3-q-q^2-q^3) + (1-q-q^3+q^4) \end$

Generating function

: $\sum_^\infty H_n(x \mid q) \frac = \frac$
where $\textstyle x=\cos \theta$.

References

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*{{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

Orthogonal polynomials
Q-analogs
Special hypergeometric functions