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Register a new userKrall--type Orthogonal Polynomials in several variables
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For a bilinear form obtained by adding a Dirac mass to a positive definite moment functional in several variables, explicit formulas of orthogonal polynomials are derived from the orthogonal polynomials associated with the moment functional. Explicit formula for the reproducing kernel is also derived and used to establish certain inequalities for classical orthogonal polynomials.
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In this contribution we deal with sequences of monic polynomials orthogonal with respect to the Freud Sobolev-type inner product begin{equation*} leftlangle p,qrightrangle _{s}=int_{mathbb{R}}p(x)q(x)e^{-x^{4}}dx+M_{0}p(0)q(0)+M_{1}p^{prime }(0)q^{prime }(0), end{equation*}% where $p,q$ are polynomials, $M_{0}$ and $M_{1}$ are nonnegative real numbers. Connection formulas between these polynomials and Freud polynomials are deduced and, as a consequence, a five term recurrence relation for such polynomials is obtained. The location of their zeros as well as their asymptotic behavior is studied. Finally, an electrostatic interpretation of them in terms of a logarithmic interaction in the presence of an external field is given.
Sets of orthogonal martingales are importants because they can be used as stochastic integrators in a kind of chaotic representation property, see [20]. In this paper, we revisited the problem studied by W. Schoutens in [21], investigating how an inner product derived from an Uvarov transformation of the Laguerre weight function is used in the orthogonalization procedure of a sequence of martingales related to a certain Levy process, called Teugels Martingales. Since the Uvarov transformation depends by a c<0, we are able to provide infinite sets of strongly orthogonal martingales, each one for every c in (-infty,0). In a similar fashion of [21], we introduce a suitable isometry between the space of polynomials and the space of linear combinations of Teugels martingales as well as the general orthogonalization procedure. Finally, the new construction is applied to the Gamma process.
The aim of this paper is to apply generalized operators of fractional integration and differentiation involving Appells function $F_{3}(:)$ due to Marichev-Saigo-Maeda (MSM), to the Jacobi type orthogonal polynomials. The results are expressed in terms of generalized hypergeometric function. Some of the interesting special cases of the main results also established.
In this contribution we consider the sequence ${Q_{n}^{lambda}}_{ngeq 0} $ of monic polynomials orthogonal with respect to the following inner product involving differences begin{equation*} langle p,qrangle _{lambda}=int_{0}^{infty}pleft(xright) qleft(xright) dpsi ^{(a)}(x)+lambda ,Delta p(c)Delta q(c), end{equation*} where $lambda in mathbb{R}_{+}$, $Delta $ denotes the forward difference operator defined by $Delta fleft(xright) =fleft(x+1right) -fleft(xright) $, $psi ^{(a)}$ with $a>0$ is the well known Poisson distribution of probability theory% begin{equation*} dpsi ^{(a)}(x)=frac{e^{-a}a^{x}}{x!}quad text{at}x=0,1,2,ldots, end{equation*}% and $cin mathbb{R}$ is such that $psi ^{(a)}$ has no points of increase in the interval $(c,c+1)$. We derive its corresponding hypergeometric representation. The ladder operators and two differe
We analyze the effect of symmetrization in the theory of multiple orthogonal polynomials. For a symmetric sequence of type II multiple orthogonal polynomials satisfying a high-term recurrence relation, we fully characterize the Weyl function associated to the corresponding block Jacobi matrix as well as the Stieltjes matrix function. Next, from an arbitrary sequence of type II multiple orthogonal polynomials with respect to a set of d linear functionals, we obtain a total of d+1 sequences of type II multiple orthogonal polynomials, which can be used to construct a new sequence of symmetric type II multiple orthogonal polynomials. Finally, we prove a Favard-type result for certain sequences of matrix multiple orthogonal polynomials satisfying a matrix four-term recurrence relation with matrix coefficients.
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