Matrix exceptional Laguerre polynomials

We give an analog of exceptional polynomials in the matrix-valued setting by considering suitable factorizations of a given second-order differential operator and performing Darboux transformations. Orthogonality and density of the exceptional sequence are discussed in detail. We give an example of matrix-valued exceptional Laguerre polynomials of arbitrary size.     

Some generalized hypergeometric d-orthogonal polynomial sets

In this paper, we characterize the d-orthogonal polynomial sets given by their explicit expressions in a specific basis. As application, we consider the generalized hypergeometric case to characterize d-orthogonal polynomial sets of Laguerre type, Meixner type, Meixner-Pollaczek type, Krawtchouk type, continuous dual Hahn type, and dual Hahn type. For d=1, we obtain a unification of some characterization theorems in the orthogonal polynomials theory.     

Bivariate fibonacci like p-polynomials

In this article, we study the bivariate Fibonacci and Lucas p-polynomials (p ? 0 is integer) from which, specifying x, y and p, bivariate Fibonacci and Lucas polynomials, bivariate Pell and Pell-Lucas polynomials, Jacobsthal and Jacobsthal-Lucas polynomials, Fibonacci and Lucas p-polynomials, Fibonacci and Lucas p-numbers, Pell and Pell-Lucas p-numbers and Chebyshev polynomials of the first and second kind, are obtained. Afterwards, we obtain some properties of the bivariate Fibonacci and Lucas p-polynomials.     

Simultaneous Preservation of Orthogonality of Polynomials by Linear Operators Arising from Dilation of Orthogonal Polynomial Systems

For an orthogonal polynomial system and a sequence of nonzero numbers,let be the linear operator defined on the linear spaceof all polynomials via for all .We investigate conditions on and under which can simultaneously preserve the orthogonality ofdifferent polynomial systems. As an application, we get that for , a generalized Laguerre polynomial system, no can simultaneously preserve the orthogonality of twoadditional Laguerre systems, and , where and . On the other hand, for ,the Chebyshev polynomial system and , simultaneously preserves the orthogonality of uncountablymany kernel polynomial systems associated with p. We study manyother examples of this type.     

Extensions of homogeneous polynomials on c 0(l 2 i )

We show that a 2-homogeneous polynomial on the complex Banach space c ₀ l ₂ ⁱ ) is norm attaining if and only if it is finite (i.e, depends only on finite coordinates). As the consequence, we show that there exists a unique norm-preserving extension for norm-attaining 2-homogeneous polynomials on c ₀(l ₂ ⁱ ). The second author was supported by FAPESP, Brazil, Research Grant 01/04220-8.     

Permutation polynomials of the form

Recently, several classes of permutation polynomials of the form (x²+x+δ)^s+x over have been discovered. They are related to Kloosterman sums. In this paper, the permutation behavior of polynomials of the form (x^p−x+δ)^s+L(x) over is investigated, where L(x) is a linearized polynomial with coefficients in . Six classes of permutation polynomials on are derived. Three classes of permutation polynomials over are also presented.     

Asymptotics and weighted estimates of meixner polynomials orthogonal on the gird {0, δ, 2δ, ...}

Suppose that 0<δ≤1,N=1/δ, and α, ga≥0, is an integer. For the classical Meixner polynomials orthonormal on the gird {0, δ, 2δ, ...} with weight ρ(x)=(1-e ^−δ)^αг(Nx+α+ 1)/г(Nx+1), the following asymptotic formula is obtained: . The remainderv _n,N ^α (z) forn≤λN satisfies the estimate