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).