Quadrature rules from a RII type recurrence relation and associated quadrature rules on the unit circle

Abstract

We consider the theoretical and numerical aspects of the quadrature rules associated with a sequence of polynomials generated by a special R II recurrence relation. We also look into some methods for generating the nodes (which lie on the real line) and the positive weights of these quadrature rules. With a simple transformation these quadrature rules on the real line also lead to certain positive quadrature rules of highest algebraic degree of precision on the unit circle. This way, we also introduce new approaches to evaluate the nodes and weights of these specific quadrature rules on the unit circle.

Chain sequences and Zeros of a perturbed $R_{II}$ type recurrence relation

“…Example 2.1. In the recurrence relation (1.4), following [5], allow the parameters to be constant sequences, e.g. let c n = 0, λ n = 1 4 , a n = −i and b n = i.…”

“…Also, these are useful in the study of Schrődinger equations, regular Coulomb wave functions and extended regular Coulomb wave functions [23,24]. Interested authors may look at [4,5,18] and references therein for some recent progress related to R II type recurrence relations and R II polynomials. The objective of this manuscript is to study some properties of polynomials which satisfy the recurrence relation (1.4) with new recurrence coefficients, perturbed in a (generalized) co-dilated and/or co-recursive way, i.e.,…”