Discrete Q-Hermite Polynomials

In mathematics, the discrete ''q''-Hermite polynomials are two closely related families ''h''_''n''(''x'';''q'') and ''ĥ''_''n''(''x'';''q'') of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of their properties. ''h''_''n''(''x'';''q'') is also called discrete q-Hermite I polynomials and ''ĥ''_''n''(''x'';''q'') is also called discrete q-Hermite II polynomials.

Definition

The discrete ''q''-Hermite polynomials are given in terms of basic hypergeometric functions and the Al-Salam–Carlitz polynomials by
:$\displaystyle h_n(x;q)=q^_2\phi_1(q^,x^;0;q,-qx) = x^n_2\phi_0(q^,q^;;q^2,q^/x^2) = U_n^(x;q)$
:$\displaystyle \hat h_n(x;q)=i^q^_2\phi_0(q^,ix;;q,-q^n) = x^n_2\phi_1(q^,q^;0;q^2,-q^/x^2) = i^V_n^(ix;q)$
and are related by
:$h_n(ix;q^) = i^n\hat h_n(x;q)$

References

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*{{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

Orthogonal polynomials
Q-analogs
Special hypergeometric functions