Some Explicit Padé Approximants for the Function ${{\Phi '} / \Phi }$ and a Related Quadrature Formula Involving Bessel Functions

Abstract

In this paper we determined in closed form the $[n\mid n]$ Padé approximant for the logarithmic derivative of the confluent hypergeometric function of the first kind, and also an explicit formula for the error. We next show how the recurrence defining the numerators and denominators of the approximants can be used to deduce a certain discrete orthogonality relationship. A consequence of this is a discrete orthogonality relation for the Bessel function of the first kind and an exact quadrature formula involving this function.