Characterizations of ultraspherical polynomials and their $q$-analogues

Abstract:

We investigate symmetric, suitably normalized orthogonal polynomial sequences $(P_n(x))_{n\in \mathbb {N}_0}$ and characterize the class of ultraspherical polynomials in terms of certain constancy properties of the Fourier coefficients which belong to $(P_{2n-1}^\prime (x))_{n\in \mathbb {N}}$. Similar characterizations are obtained for the discrete, resp. continuous, $q$-ultraspherical polynomials after replacing the derivative $\frac {\mathrm {d}}{\mathrm {d}x}$ by the $q$-difference operator $D_{q^{-1}}$, resp. Askey–Wilson operator $\mathcal {D}_q$. Our results extend earlier work of Lasser–Obermaier and Ismail–Obermaier where the whole sequences $(P_n^\prime (x))_{n\in \mathbb {N}}$, $(D_{q^{-1}}P_n(x))_{n\in \mathbb {N}}$ and $(\mathcal {D}_q P_n(x))_{n\in \mathbb {N}}$ had to be taken into account; we shall see that the characterizing properties concerning the even indices turn out to be redundant. We also characterize a large subclass of the continuous $q$-ultraspherical polynomials via the averaging operator $\mathcal {A}_q$, and we show that this characterization cannot be extended to the whole class.