### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 075, 17 pages      arXiv:1601.06898      https://doi.org/10.3842/SIGMA.2016.075
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

### Orthogonal Polynomials Associated with Complementary Chain Sequences

Kiran Kumar Behera a, A. Sri Ranga b and A. Swaminathan a
a) Department of Mathematics, Indian Institute of Technology Roorkee, Uttarakhand-247667, India
b) Departamento de Matemática Aplicada, IBILCE, UNESP-Univ. Estadual Paulista, 15054-000, São José do Rio Preto, SP, Brazil

Received March 17, 2016, in final form July 22, 2016; Published online July 27, 2016

Abstract
Using the minimal parameter sequence of a given chain sequence, we introduce the concept of complementary chain sequences, which we view as perturbations of chain sequences. Using the relation between these complementary chain sequences and the corresponding Verblunsky coefficients, the para-orthogonal polynomials and the associated Szegő polynomials are analyzed. Two illustrations, one involving Gaussian hypergeometric functions and the other involving Carathéodory functions are also provided. A connection between these two illustrations by means of complementary chain sequences is also observed.

Key words: chain sequences; orthogonal polynomials; recurrence relation; Verblunsky coefficients; continued fractions; Carathéodory functions; hypergeometric functions.

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