Continuous q-Hermite polynomials explained

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 ^{in\theta} _{n(x|q) = e} {} _{2\phi} _{0\left[\begin{matrix}} q ^{-n} , 0 \ - \end{matrix} ;q,q ^{n} e ^{-2i\theta} \right], x = \cos\theta.

Recurrence and difference relations

2 x H _{n(x\mid} q) = H _{n+1} (x\mid q) + ( 1- q ^{n)} H _{n-1} (x\mid q)

with the initial conditions

H _{0} (x\mid q) =1 , H _{-1} (x\mid q) = 0

From the above, one can easily calculate:

\begin{align} H _{0} (x\mid q) & = 1 \\ H _{1} (x\mid q) & = 2 x \\ H _{2} (x\mid q) & = 4 x ^{2} - ( 1- q) \\ H _{3} (x\mid q) & = 8 x ^{3} - 2 x( 2- q - q ^{2)} \\ H _{4} (x\mid q) & = 16 x ^{4} - 4 x ^{2(3-q-q} ^{2-q} ^{3)} + ( 1- q - q ^{3+q} ^{4) \end{align}}

Generating function

\sum ^{inf ty} _{n =0} H _{n(x} \mid q) (t ^{n})/((q;q) _{n}) = (1)/(\left( t e ^{i} ,t e ^{-i} ;q \right) _{infty}) where style x = \cos \theta .

References

• • • • Sadjang . Patrick Njionou . Moments of Classical Orthogonal Polynomials . Ph.D. . Universität Kassel . 10.1.1.643.3896 .