\(d\)-orthogonality of discrete \(q\)-Hermite type polynomials (Q387369)

Using the usual \(q\)-hypergeometric series notation, the author defines the discrete \(q\)-Hermite I type polynomials \(P_n^{(1)}(x | q)\) and the discrete \(q\)-Hermite II type polynomials \(P_n^{(2)}(x | q)\) by NEWLINE\[NEWLINE P_n^{(1)}(x | q) = x^n \;_{d+1}\phi_0 \Bigg( \begin{matrix} q^{-n},q^{-n+1}, \dots ,q^{-n+d} \\ - \end{matrix} \Bigg|q^{d+1}; \frac{q^{n(d+1)-1}}{x^{d+1}} \Bigg) NEWLINE\]NEWLINE and NEWLINE\[NEWLINE P_n^{(2)}(x | q) = x^n \;_{d+1}\phi_d \Bigg( \begin{matrix} q^{-n},q^{-n+1}, \dots ,q^{-n+d} \\ 0,\dots,0 \end{matrix} \Bigg|q^{d+1}; \frac{-q^{(d+1)}}{x^{d+1}} \Bigg). NEWLINE\]NEWLINE These are \(d\)-orthogonal polynomials arising as solutions to a certain characterization problem. They are \(q\)-analogues of the Gould-Hopper polynomials, and when \(d=1\), they are the classical discrete \(q\)-Hermite I and II polynomials.NEWLINENEWLINEThe author derives a number of properties of the discrete \(q\)-Hermite type polynomials which reduce to known properties of the discrete \(q\)-Hermite polynomials when \(d=1\). Among these are recurrence relations, difference formulas, generating functions, and inversion formulas.