Recursively Generated Polynomials and Geronimus’ Version of Orthogonality on a Contour

Abstract

Let $\{ p_n \} $ be a sequence of polynomials generated by the three-term recurrence $p_0 = 1$, $p_1 = z + b_0 $, $p_{n + 1} = (z + b_n )p_n - c_n p_{n - 1} $, $n \geqq 1$. Using only the hypothesis that $\{ b_n \}$ and $\{ c_n \} $ are bounded complex sequences (with $c_n \ne 0$), we show by constructing a weight function as a convergent Laurent series—with coefficients given explicitly in terms of $b_n $ and $c_n $—that the $p_n $ are orthogonal on a contour in the sense that Geronimus describes (and that the Bessel polynomials exemplify). We thus obtain an elementary approach to this kind of orthogonality and provide a construction of the weight function which is an alternative to the continued fraction representation discussed by Askey and Ismail.