Orthogonal polynomials and continued fractions arising from contiguous relations and generalizations of Kummer and \(q\)-Kummer identities (Q2422143)

This paper considers particular types of continued fractions called \(R\)-fractions (\(R\)-fractions of type I and II, denoted by $R_I$ and $R_{II}$, respectively). In particular, starting from a ${}_{2}F_1$ contiguous relation, the author derives a family of orthogonal polynomials that satisfies a three-term recurrence relation of type $R_I$. The author shows that the measure of orthogonality is unique and derives the associated $R_I$ type fraction and the measure of orthogonality. \par Moreover, the author also derives a family of orthogonal polynomials and $R_{II}$ continued fractions starting from a ${}_2\phi_1$ contiguous relation (function ${}_2\phi_1$ is defined by the \(q\)-Kummer's identity).