Neha Patil (Editor)

Continuous q Hermite polynomials

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In mathematics, the continuous q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.

Contents

Definition

The polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol.

Recurrence and difference relations

2 x H n ( x ∣ q ) = H n + 1 ( x ∣ q ) + ( 1 − q n ) H n − 1 ( x ∣ q )

with the initial conditions

H 0 ( x ∣ q ) = 1 , H − 1 ( x ∣ q ) = 0

From the above, one can easily calculate:

H 0 ( x ∣ q ) = 1 H 1 ( x ∣ q ) = 2 x H 2 ( x ∣ q ) = 4 x 2 − ( 1 − q n ) H 3 ( x ∣ q ) = 8 x 3 − 4 x ( 1 − q n ) H 4 ( x ∣ q ) = 16 x 4 − 12 x 2 ( 1 − q n ) + ( 1 − q n ) 2 H 5 ( x ∣ q ) = 32 x 5 − 32 x 3 ( 1 − q n ) + 6 x ( 1 − q n ) 2

Generating function

∑ n = 0 ∞ H n ( x ∣ q ) t n ( q ; q ) n = 1 ( t e i θ , t e − i θ ; q ) ∞

where x = cos ⁡ θ .

References

Continuous q-Hermite polynomials Wikipedia
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