Discrete q-Hermite polynomials explained

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 ^{\binom{n} _{n(x;q) = q} { 2 }}{} _{2\phi} ^{- n} ^{1 (q} ,x ^{-1} ; 0 ;q, - qx) = x ^{n{}} _{2 \phi} ^{- n} ^{0 (q} ,q ^{-n+1} ;;q ^{2,q} ^{2n-1} /x ^{2)} = U ^{( -1 )} _{n} (x;q) \displaystyle \hat h ^{- n} _{n(x;q) = i} q ^{-\binom{n} { 2 }}{} _{2\phi} ^{- n} ^{0 (q} ,ix;;q, - q ^{n)} = x ^{n{}} _{2 \phi} ^{- n} ^{1 (q} ,q ^{-n+1} ; 0 ;q ^{2,-q} ^{2} /x ^{2)} = i ^{-n} V ^{( -1 )} _{n} (ix;q) and are related byh ^{-1} _{n(ix;q} ) = i ^{n\hat} h _{n(x;q)}