Tridiagonalization of the hypergeometric operator and the Racah-Wilson algebra
This contribution develops the tridiagonalization of the hypergeometric operator NEWLINE\[NEWLINEL=x(1-x)\partial^2_x+(\alpha+1-(\alpha+\beta+2)x) \partial_xNEWLINE\]NEWLINE where \(\alpha,\beta\) are real constants such that \(\alpha,\beta>-1\), provided by a set of eigenfunctions of an operator \(M\) of the form NEWLINE\[NEWLINEM=\tau_1XL+\tau_2LX+\tau_3X+\tau_0,NEWLINE\]NEWLINE where \(Xf(x)=xf(x)\) and \(\tau_0,\tau_1,\tau_2,\tau_3\) are constants fulfilling \(\tau_1+\tau_2\neq 0\). The authors explore a connection between the tridiagonalization and the generic Wilson polynomials and uncover ``an embedding of the Racah algebra into the Jacobi algebra in their standard realization on the space of polynomials.'' The tridiagonalization in the case for which \(\tau_1+\tau_2=0\) is also treated being enlightened in this situation a connection with the generic Hahn polynomials.
scientific article; zbMATH DE number 6620009
scientific article