On a \(q\)-extension of the Hermite polynomials \(H_n(x)\) with the continuous orthogonality property on \(\mathbb{R}\) (Q1394543)

The authors solve the following problem: To find \(q\)-extensions of the Hermite polynomials \(H_n\), \(n\in \mathbb{N}_0\) with the properties (i) they obey a three-term recurrence relation, (ii) they are orthogonal on the real line with respect to a continuous positive weight function, (iii) in the limit case \(q\to 1\) they coincide with the Hermite polynomials \(H_n\). It appears that these extensions can be expressed either in terms of the \(q\)-Laguerre polynomials of order \(\alpha-\pm 1/2\) or in terms of the discrete \(q\)-Hermite polynomials of order II.