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It is known that given a pair of real sequences {{cn}n=1∞,{dn}n=1∞}, with {dn}n=1∞ a positive chain sequence, we can associate a unique nontrivial probability measure μ on the unit circle. Precisely, the measure is such that the corresponding Verblunsky coefficients {αn}n=0∞ are given by the relation αn−1=ρ‾n−1[1−2mn−icn1−icn],n≥1, where ρ0=1ρ0=1, ρn=∏k=1n(1−ick)/(1+ick), n≥1n≥1 and {mn}n=0∞ is the minimal parameter sequence of {dn}n=1∞. In this paper we consider the space, denoted by NpNp, of all nontrivial probability measures such that the associated real sequences {cn}n=1∞ and {mn}n=1∞ are periodic with period p , for p∈Np∈N. By assuming an appropriate metric on the space of all nontrivial probability measures on the unit circle, we show that there exists a homeomorphism gpgp between the metric subspaces NpNp and VpVp, where VpVp denotes the space of nontrivial probability measures with associated p -periodic Verblunsky coefficients. Moreover, it is shown that the set FpFp of fixed points of gpgp is exactly Vp∩NpVp∩Np and this set is characterized by a (p−1)(p−1)-dimensional submanifold of RpRp. We also prove that the study of probability measures in NpNp is equivalent to the study of probability measures in VpVp. Furthermore, it is shown that the pure points of measures in NpNp are, in fact, zeros of associated para-orthogonal polynomials of degree p . We also look at the essential support of probability measures in the limit periodic case, i.e., when the sequences {cn}n=1∞ and {mn}n=1∞ are limit periodic with period p. Finally, we give some examples to illustrate the results obtained.

The purpose of this paper is twofold. Firstly we investigate the distribution, simplicity and monotonicity of the zeros around the unit circle and real line of the real self-reciprocal polynomials R n ( λ ) ( z ) = 1 + λ ( z + z 2 + ⋯ + z n − 1 ) + z n , n ≥ 2 and λ ∈ R . Secondly, as an application of the first results we give necessary and sufficient conditions to guarantee that all zeros of the self-reciprocal polynomials S n ( λ ) ( z ) = ∑ k = 0 n s n , k ( λ ) z k , n ≥ 2 , with s n , 0 ( λ ) = s n , n ( λ ) = 1 , s n , n − k ( λ ) = s n , k ( λ ) = 1 + k λ , k = 1 , 2 , … , ⌊ n / 2 ⌋ when n is odd, and s n , n − k ( λ ) = s n , k ( λ ) = 1 + k λ , k = 1 , 2 , … , n / 2 − 1 , s n , n / 2 ( λ ) = ( n / 2 ) λ when n is even, lie on the unit circle, solving then an open problem given by Kim and Park in 2008.

We obtain the asymptotic behavior of the zeros of a class of generalized hypergeometric polynomials. For this purpose, we make use of a Mehler-Heine type formula for these polynomials. We illustrate these results with numerical experiments and some figures.

Sobolev orthogonal polynomials in two variables are defined via inner products involving gradients. Such a kind of inner product appears in connection with several physical and technical problems. Matrix second-order partial differential equations satisfied by Sobolev orthogonal polynomials are studied. In particular, we explore the connection between the coefficients of the second-order partial differential operator and the moment functionals defining the Sobolev inner product. Finally, some old and new examples are given.

Submitted by Guilherme Lemeszenski (guilherme@nead.unesp.br) on 2014-02-26T17:15:47Z No. of bitstreams: 1 WOS000181888700008.pdf: 150681 bytes, checksum: 728da2bf670384f99c93e88449528e28 (MD5) Made available in DSpace on 2014-02-26T17:15:47Z (GMT). No. of bitstreams: 1 WOS000181888700008.pdf: 150681 bytes, checksum: 728da2bf670384f99c93e88449528e28 (MD5) Previous issue date: 2003-04-01 Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-20T15:20:39Z No. of bitstreams: 1 WOS000181888700008.pdf: 150681 bytes, checksum: 728da2bf670384f99c93e88449528e28 (MD5) Made available in DSpace on 2014-05-20T15:20:39Z (GMT). No. of bitstreams: 1 WOS000181888700008.pdf: 150681 bytes, checksum: 728da2bf670384f99c93e88449528e28 (MD5) Previous issue date: 2003-04-01 We consider the real Szego polynomials and obtain some relations to certain self inversive orthogonal L-polynomials defined on the unit circle and corresponding symmetric orthogonal polynomials on real intervals. We also consider the polynomials obtained when the coefficients in the recurrence relations satisfied by the self inversive orthogonal L-polynomials are rotated. (C) 2002 Elsevier B.V. B.V. All rights reserved. Univ Estadual Paulista, Dept Ciências Comp & Estatist, Inst Biociencias Letras & Ciências Exatas, BR-15054000 Sao Jose do Rio Preto, SP, Brazil Univ Estadual Paulista, Dept Ciências Comp & Estatist, Inst Biociencias Letras & Ciências Exatas, BR-15054000 Sao Jose do Rio Preto, SP, Brazil

Submitted by Guilherme Lemeszenski (guilherme@nead.unesp.br) on 2014-02-26T17:13:36Z No. of bitstreams: 1 WOS000176146300007.pdf: 132901 bytes, checksum: fc1b950412460776d2f1efc9b5384394 (MD5) Made available in DSpace on 2014-02-26T17:13:36Z (GMT). No. of bitstreams: 1 WOS000176146300007.pdf: 132901 bytes, checksum: fc1b950412460776d2f1efc9b5384394 (MD5) Previous issue date: 2002-06-01 Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-20T15:26:54Z No. of bitstreams: 1 WOS000176146300007.pdf: 132901 bytes, checksum: fc1b950412460776d2f1efc9b5384394 (MD5) Made available in DSpace on 2014-05-20T15:26:54Z (GMT). No. of bitstreams: 1 WOS000176146300007.pdf: 132901 bytes, checksum: fc1b950412460776d2f1efc9b5384394 (MD5) Previous issue date: 2002-06-01 in this paper, we derive an explicit expression for the parameter sequences of a chain sequence in terms of the corresponding orthogonal polynomials and their associated polynomials. We use this to study the orthogonal polynomials K-n((lambda.,M,k)) associated with the probability measure dphi(lambda,M,k;x), which is the Gegenbauer measure of parameter lambda + 1 with two additional mass points at +/-k. When k = 1 we obtain information on the polynomials K-n((lambda.,M)) which are the symmetric Koornwinder polynomials. Monotonicity properties of the zeros of K-n((lambda,M,k)) in relation to M and k are also given. (C) 2002 Elsevier B.V. B.V. All rights reserved. Univ Estadual Paulista, Inst Biociencias Letras & Ciências Exatas, Dept Ciências Comp & Estat, BR-15054000 Sao Jose do Rio Preto, SP, Brazil Univ Estadual Paulista, Inst Biociencias Letras & Ciências Exatas, Dept Ciências Comp & Estat, BR-15054000 Sao Jose do Rio Preto, SP, Brazil

Submitted by Guilherme Lemeszenski (guilherme@nead.unesp.br) on 2014-02-26T17:03:01Z No. of bitstreams: 1 WOS000080681500014.pdf: 112775 bytes, checksum: e7120be31e58aa5eecc4f6787b38d593 (MD5) Made available in DSpace on 2014-02-26T17:03:01Z (GMT). No. of bitstreams: 1 WOS000080681500014.pdf: 112775 bytes, checksum: e7120be31e58aa5eecc4f6787b38d593 (MD5) Previous issue date: 1999-05-31 Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-20T14:01:35Z No. of bitstreams: 1 WOS000080681500014.pdf: 112775 bytes, checksum: e7120be31e58aa5eecc4f6787b38d593 (MD5) Made available in DSpace on 2014-05-20T14:01:35Z (GMT). No. of bitstreams: 1 WOS000080681500014.pdf: 112775 bytes, checksum: e7120be31e58aa5eecc4f6787b38d593 (MD5) Previous issue date: 1999-05-31 A strong Stieltjes distribution d psi(t) is called symmetric if it satisfies the propertyt(omega) d psi(beta(2)/t) = -(beta(2)/t)(omega) d psi(t), for t is an element of (a, b) subset of or equal to (0, infinity), 2 omega is an element of Z, and beta > 0.In this article some consequences of symmetry on the moments, the orthogonal L-polynomials and the quadrature formulae associated with the distribution are given. (C) 1999 Elsevier B.V. B.V. All rights reserved. UNESP, IBILCE, Dept Ciências Comp & Estatist, BR-15054000 Sao Jose Rio Preto, SP, Brazil Univ St Andrews, Sch Math & Computat Sci, St Andrews KY16 9SS, Fife, Scotland UNESP, IBILCE, Dept Ciências Comp & Estatist, BR-15054000 Sao Jose Rio Preto, SP, Brazil