Structure relations for orthogonal polynomials on the unit circle

• Published in: Linear algebra and its applications Vol. 436; no. 11; pp. 4296 - 4310
• Main Authors: Branquinho, A., Rebocho, M.N.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.06.2012; Elsevier
• Subjects: Algebra; Exact sciences and technology; Fourier analysis; Hermitian linear functionals; Linear and multilinear algebra, matrix theory; Mathematical analysis; Mathematics; Orthogonal polynomials on the unit circle; Recurrence relations; Sciences and techniques of general use; Semi-classical linear functionals; Special functions; Structure relations; 33C45; Hermitian linear functionals; Structure relations; Semi-classical linear functionals; Orthogonal polynomials on the unit circle; Recurrence relations; 42C05; Circle; Recurrence relation; Fourier analysis; Text; Orthogonal polynomial; Complex number; Orthogonal function; Hypergeometric function; Linear functional
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2012.01.034

Structure relations for orthogonal polynomials with respect to Hermitian linear functionals are studied. Firstly, we prove that semi-classical orthogonal polynomials satisfy structure relations of the following type: ∑k=0s1βn,kPn+s1-k+∑k=0s2γn,kzkPn-1-k∗=∑k=0r1αn,kPn+s1-k[1]+∑k=0r2ηn,kPn+r2-k∗′, where s1,s2,r1,r2 are integers (specified in the text), Pn∗ is the reversed polynomial of Pn, Pn[1]=Pn+1′/(n+1), and βn,k,γn,k,αn,k,ηn,k are complex numbers. Then, we study the semi-classical character of sequences of orthogonal polynomials {Rn},{Pn}, connected through a structure relation of the following type: ∑k=0s1βn,kRn+s1-k+∑k=0s2γn,kRn+s2-k∗=∑k=0r1αn,kPn+r1-k[1]+∑k=0r2ηn,kPn+r2-k∗′, where the integers s1,s2,r1,r2 satisfy some natural conditions specified in the text.