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Para-orthogonal polynomials derived from orthogonal polynomials on the unit circle are known to have all their zeros on the unit circle. In this note we study the zeros of a family of hypergeometric para-orthogonal polynomials. As tools to study these polynomials, we obtain new results which can be considered as extensions of certain classical results associated with three term recurrence relations and differential equations satisfied by orthogonal polynomials on the real line. One of these results which might be considered as an extension of the classical Sturm comparison theorem, enables us to obtain monotonicity with respect to the parameters for the zeros of these para-orthogonal polynomials. Finally, a monotonicity of the zeros of Meixner-Pollaczek polynomials is proved. [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]