Riordan arrays and Jacobi and Thron continued fractions (2109.07333v1)

Abstract: We show that certain Riordan arrays have generating functions that can be expressed as continued fractions of Jacobi and Thron type. We investigate the inverses of such arrays, which in certain circumstances can also have generating functions representable as continued fractions. Links to orthogonal polynomial moment sequences, and to Laurent biorthogonal polynomials are developed. We show that certain Riordan group involutions can be defined by continued fractions. We also show how simple transformations of the Jacobi continued fractions can lead to exponential Riordan arrays. Finally, by way of contrast, we look at the case of some non Riordan arrays that are of combinatorial significance, including the Narayana numbers.