The aim of this paper is to develop foundations of umbral calculus on the space D′ of distributions on Rd, which leads to a general theory of Sheffer polynomial sequences on D′. We define a sequence ...of monic polynomials on D′, a polynomial sequence of binomial type, and a Sheffer sequence. We present equivalent conditions for a sequence of monic polynomials on D′ to be of binomial type or a Sheffer sequence, respectively. We also construct a lifting of a sequence of monic polynomials on R of binomial type to a polynomial sequence of binomial type on D′, and a lifting of a Sheffer sequence on R to a Sheffer sequence on D′. Examples of lifted polynomial sequences include the falling and rising factorials on D′, Abel, Hermite, Charlier, and Laguerre polynomials on D′. Some of these polynomials have already appeared in different branches of infinite dimensional (stochastic) analysis and played there a fundamental role.
A matrix approach to Sheffer polynomials Aceto, Lidia; Cação, Isabel
Journal of mathematical analysis and applications,
02/2017, Letnik:
446, Številka:
1
Journal Article
Recenzirano
Odprti dostop
This paper deals with a unified matrix representation for the Sheffer polynomials. The core of the proposed approach is the so-called creation matrix, a special subdiagonal matrix having as nonzero ...entries positive integer numbers, whose exponential coincides with the well-known Pascal matrix. In fact, Sheffer polynomials may be expressed in terms of two matrices both connected to it. As we will show, one of them is strictly related to Appell polynomials, while the other is linked to a binomial type sequence. Consequently, different types of Sheffer polynomials correspond to different choices of these two matrices.
A class of orthogonal polynomials relative to special discrete weights is considered. These self-dual weights which are completely determined by their finite support appear in the polynomial ...approximation of a function over this set, in the barycentric form of classical Lagrange interpolation polynomial as well as in mathematical statistics and statistical physics. Orthogonal polynomials possess remarkable symmetry reflected in the properties of their coefficients in the three-term recurrence relations, some of which have an explicit form. Formulas for the expected values are derived.
•The orthogonal polynomials with regard to self-dual weights are known to satisfy a special three-term recurrence with a persymmetric Jacobi matrix, and this is a characteristic property of self-dual weights;•The corresponding moments satisfy the recurrence which translates into factorization formula for associate polynomials;•Explicit form of central orthogonal polynomials allows for asymptotic study when the weights are random. This study indicates importance of the maximal attraction domain of the underlying distribution;•A form of Central Limit Theorem for the coefficients of studied polynomials is suggested.
For certain Sheffer sequences (sn)n=0∞ on C, Grabiner (1988) proved that, for each α∈0,1, the corresponding Sheffer operator zn↦sn(z) extends to a linear self-homeomorphism of Eminα(C), the Fréchet ...topological space of entire functions of order at most α and minimal type (when the order is equal to α>0). In particular, every function f∈Eminα(C) admits a unique decomposition f(z)=∑n=0∞cnsn(z), and the series converges in the topology of Eminα(C). Within the context of a complex nuclear space Φ and its dual space Φ′, in this work we generalize Grabiner's result to the case of Sheffer operators corresponding to Sheffer sequences on Φ′. In particular, for Φ=Φ′=Cn with n≥2, we obtain the multivariate extension of Grabiner's theorem. Furthermore, for an Appell sequence on a general co-nuclear space Φ′, we find a sufficient condition for the corresponding Sheffer operator to extend to a linear self-homeomorphism of Eminα(Φ′) when α>1. The latter result is new even in the one-dimensional case.
Bivariate Gončarov polynomials are a basis of the solutions of the bivariate Gončarov Interpolation Problem in numerical analysis. A sequence of bivariate Gončarov polynomials is determined by a set ...of nodes Z={(xi,j,yi,j)∈R2} and is an affine sequence if Z is an affine transformation of the lattice grid N2, i.e., (xi,j,yi,j)T=A(i,j)T+(c1,c2)T for some 2×2 matrix A and constants c1,c2. In this paper we prove that a sequence of bivariate Gončarov polynomials is of binomial type if and only if it is an affine sequence with c1=c2=0. Such polynomials form a higher-dimensional analog of the Abel polynomial An(x;a)=x(x−an)n−1. We present explicit formulas for a general sequence of bivariate affine Gončarov polynomials and its exponential generating function, and use the algebraic properties of Gončarov polynomials to give some new two-dimensional generalizations of Abel identities.
A Family of Induced Distributions Koutras, Vasileios M.; Koutras, Markos V.; Dafnis, Spiros D.
Methodology and computing in applied probability,
09/2022, Letnik:
24, Številka:
3
Journal Article
Recenzirano
In the present paper a family of discrete distributions is introduced through the probability generating function of any discrete distribution (generator). The properties of the family are ...systematically studied when the generator belongs to well-known families of discrete distributions (power series distributions, Bernoulli mixtures, Panjer family, Phase-type distributions). Applications are also provided in problems arising from the areas of reliability theory and start-up demonstration testing, which highlight the beneficial use of the family in order statistics related models.
The exact distributions of random minimum and maximum of a random sample of continuous positive random variables are studied when the support of the sample size distribution contains zero providing a ...probability model that has not been systematically studied in the literature.
In this paper, we consider a weighted-r-within-consecutive-k-out-of-n : F system. The weighted system has in general n components, each one having a positive integer weight wi, i = 1, 2, . . . , n. ...The weighted-r-within-consecutive-k-out-of-n : F system fails if and only if the total weight of failed components among k consecutive components is at least r. We introduce a binomial-type weighted scan statistic and study the reliability, the Birnbaum, improvement potential importance and Bayesian reliability importance of the system taken into consideration. We develop an explicit closed-form formula for the evaluation of reliability and reliability importance measures of a weighted-r-within-consecutive-k-out-of-n : F system and demonstrate the results numerically. We present a study showing the effectiveness of the method in terms of CPU time.
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