It is well known that the q-Hermite polynomials defined by $$H_n(\theta; q)= \sum\limits_{k=0}^n \frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}e^{i(n-2k)\theta}$$ are orthogonal in $\theta \in [0, \pi]$ with respect to the measure $(e^{i2\theta},e^{-2i\theta}; q)_{\infty}d\theta=d\mu$.

Let me define now the modified q-Hermite polynomials $$H_n^{\lambda}(\theta; q)= \sum\limits_{k=0}^n \frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}\lambda^ke^{i(n-2k)\theta} $$. Is it still true that the polynomials $\{H_n^{\lambda}(\theta; q)\}_{n \in \mathbb{N}}$ are orthogonal with respect to some other measure $d\mu_{\lambda}$ in $[0, \pi]$? If Yes, which is their norm?