Umbral ``classical'' polynomials
In the mid of 30s of the last century, W.~Hahn classified the orthogonal polynomials whose derivatives or \(q\)-derivatives are also orthogonal. The author of the present paper generalizes Hahn's problem considering the monic polynomials \(P_n(x)\) and their generalized derivatives \(Q_n(x)=\mu_{n+1}^{-1}{\mathcal D}P_{n+1}(x)\), where \({\mathcal D}x^n=\mu_nx^{n-1}\). He finds necessary and sufficient conditions that \(\{P_n(x)\}_{n=0}^{\infty}\) and \(\{Q_n(x)\}_{n=0}^{\infty}\) are both orthogonal with respect to some linear functionals \(\sigma\) and \(\tau\).NEWLINENEWLINEThe key ingredient of the author's approach is the so-called reciprocal operator \({\mathcal R}\) such that \({\mathcal R}Q_n(x)=\nu_{n+1}P_{n+1}(x)\). The necessary and sufficient condition of the author's interest has the form of an infinite system of algebraic equations involving the moments \(g_n=\langle\sigma,x^n\rangle\), \(\tilde g_n=\langle\tau,x^n\rangle\), the defining coefficients \(\mu_n\) and \(\nu_n\) and the matrix elements of the operator \({\mathcal R}\) in the monomial basis. The corresponding orthogonal polynomials appear as the solutions of the eigenvalue problems \(LP_n(x)=\lambda_nP_n(x)\), \(\tilde LQ_n(x)=\lambda_{n+1}Q_n(x)\) for the operators \(L={\mathcal R}{\mathcal D}\) and \(\tilde L={\mathcal D}{\mathcal R}\).NEWLINENEWLINEThe proposed approach is illustrated by two important particular cases. The first case corresponds to the finite-diagonal matrix of \({\mathcal R}\) when both \({\mathcal D}\) and \({\mathcal R}\) become linear combinations of \(q\)-difference and differential operators, moreover in the simplest two-diagonal case, the author reproduces Hahn's result. The second illustrative situation corresponds to the complete degeneration of the operator \(L\) (\(\lambda_1=\lambda_2=\dots=1\)). Now, assuming that \(\mu_n=\mu(n)\), \(g_n=g(n)\) are analytic in \(n\), the author finds these coefficients and orthogonal polynomials \(P_n(x)\), \(Q_n(x)\) in terms of Weierstraß' elliptic sigma function.
scientific article; zbMATH DE number 6334362
scientific article