Biorthogonality and para-orthogonality of $R_I$ polynomials

TL;DR

This paper explores biorthogonality and para-orthogonality properties of $R_I$ polynomials, constructing sequences with common zeros, and illustrates the findings through hypergeometric functions.

Contribution

It introduces a new sequence of $R_I$ polynomials with common zeros and establishes biorthogonality and para-orthogonality relations, extending the understanding of these polynomial classes.

Findings

Constructed a sequence of $R_I$ polynomials with a common zero.

Established biorthogonality relations for the sequence.

Derived para-orthogonal polynomials by removing the common zero.

Abstract

In this paper, a sequence of linear combination of $R_{I}$ type polynomials such that the terms in this sequence have a common zero is constructed. A biorthogonality relation arising from such a sequence is discussed. Besides a sequence of para-orthogonal polynomials by removing the common zero using suitable conditions is obtained. Finally, a case of hypergeometric functions is studied to illustrate the results obtained.