Chebyshev polynomials of the second kind via raising operator preserving the orthogonality

Abstract

It is well known that monic orthogonal polynomial sequences $\{T_n\}_{n\ge 0}$ and $\{U_n\}_{n\ge 0}$, the Chebyshev polynomials of the first and second kind, satisfy the relation $DT_{n+1}=(n+1)U_n$ ($n\ge 0$). One can also easily check that the following “inverse” of the mentioned formula holds: ${\mathcal {U}}_{-1}(U_n)=(n+1)T_{n+1}$ ($n\ge 0$), where ${\mathcal {U}}_\xi =x(xD+{\mathbb {I}})+\xi D$ with $\xi$ being an arbitrary nonzero parameter and ${\mathbb {I}}$ representing the identity operator. Note that whereas the first expression involves the operator D which lowers the degree by one, the second one involves ${\mathcal {U}}_\xi$ which raises the degree by one (i.e. it is a “raising operator”). In this paper it is shown that the scaled Chebyshev polynomial sequence $\{a^{-n}U_n(ax)\}_{n\ge 0}$ where $a^2=-\xi ^{-1}$, is actually the only monic orthogonal polynomial sequence which is ${\mathcal {U}}_\xi$-classical (i.e. for which the application of the raising operator ${\mathcal {U}}_\xi$ turns the original sequence into another orthogonal one).