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We present a class of functions satisfying a certain orthogonality property for which there also exists a three term recurrence formula. This class of functions, which can be considered as an extension to the class of symmetric orthogonal polynomials on [-1, 1], has a complete connection to the orthogonal polynomials on the unit circle. Interpolatory properties, quadrature rules and other properties based on the zeros of these functions are also considered. [ABSTRACT FROM AUTHOR]

The principal objective here is to look at some algebraic properties of the orthogonal polynomials Ψn(b,s,t) n with respect to the Sobolev inner product on the unit circle... Some related bounds and asymptotic properties are also given. [ABSTRACT FROM AUTHOR]

Given a nontrivial positive measure μ on the unit circle T , the associated Christoffel–Darboux kernels are K n ( z , w ; μ ) = ∑ k = 0 n φ k ( w ; μ ) ¯ φ k ( z ; μ ) , n ≥ 0 , where φ k ( ⋅ ; μ ) are the orthonormal polynomials with respect to the measure μ . Let the positive measure ν on the unit circle be given by d ν ( z ) = | G 2 m ( z ) | d μ ( z ) , where G 2 m is a conjugate reciprocal polynomial of exact degree 2 m . We establish a determinantal formula expressing { K n ( z , w ; ν ) } n ≥ 0 directly in terms of { K n ( z , w ; μ ) } n ≥ 0 . Furthermore, we consider the special case of w = 1 ; it is known that appropriately normalized polynomials K n ( z , 1 ; μ ) satisfy a recurrence relation whose coefficients are given in terms of two sets of real parameters { c n ( μ ) } n = 1 ∞ and { g n ( μ ) } n = 1 ∞ , with 0 < g n < 1 for n ≥ 1 . The double sequence { ( c n ( μ ) , g n ( μ ) ) } n = 1 ∞ characterizes the measure μ . A natural question about the relation between the parameters c n ( μ ) , g n ( μ ) , associated with μ , and the sequences c n ( ν ) , g n ( ν ) , corresponding to ν , is also addressed. Finally, examples are considered, such as the Geronimus weight (a measure supported on an arc of T ), a measure for which the Christoffel–Darboux kernels, with w = 1 , are given by basic hypergeometric polynomials and a measure for which the orthogonal polynomials and the Christoffel–Darboux kernels, again with w = 1 , are given by hypergeometric polynomials. [ABSTRACT FROM AUTHOR]