Multivariate Sequences of Binomial Type

Classical Gončarov polynomials are polynomials which interpolate derivatives. Delta Gončarov polynomials are polynomials which interpolate delta operators, e.g., forward and backward difference operators. We extend fundamental aspects of the theory of classical bivariate Gončarov polynomials and univariate delta Gončarov polynomials to the multivariate setting using umbral calculus. After introducing systems of delta operators, we define multivariate delta Gončarov polynomials, show that the associated interpolation problem is always solvable, and derive a generating function (an Appell relation) for them. We show that systems of delta Gončarov polynomials on an interpolation grid Z ⊆ R d are of binomial type if and only if Z = AN d for some d × d matrix A. This motivates our definition of delta Abel polynomials to be exactly those delta Gončarov polynomials which are based on such a grid. Finally, compact formulas for delta Abel polynomials in all dimensions are given for separable systems of delta operators. This recovers a former result for classical bivariate Abel polynomials and extends previous partial results for classical trivariate Abel polynomials to all dimensions.

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“…Comparing this with the alternate Appell relation (19) and using the uniqueness of the coefficients, we have…”

Section: Theorem 42 Given a System Of Delta Operatorsmentioning

confidence: 99%

“…It was extended to multivariate polynomials and applied to combinatorial problems in [2,17,18,19]. In [3,7] and [23], higher dimensional umbral calculus is used to derive various versions of the Lagrange inversion formula.…”

Section: Introductionmentioning

confidence: 99%

Multivariate delta Gončarov and Abel polynomials

Lorentz

Tringali

Yan

2017

Journal of Mathematical Analysis and Applications

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“…Since Sheffer and cross sequences can be decomposed into sequences of binomial type [6], we have a combinatorial interpretation of these. THEOREM 57.…”

Section: =Npn-lm·mentioning

confidence: 99%

The Combinatorics of Polynomial Sequences

Reiner¹

1978

Stud Appl Math

Self Cite

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We present a combinatorial model for the several kinds of polynomial sequences of binomial type and develop many of the theorems about them from this model. In the first section, we present a prefab model for the binomial formula and the generating-function theorem. In Sec. 2, we introduce the notion of V-graph and give examples of binomial prefabs of V-graphs. The umbral composition of V-graphs provides an interpretation of umbral composition of polynomial sequences in Secs. 3 and 5. Rota's interpretation of the Stirling numbers 'Of the first kind as sums 'Of the Mobius functi'On in the partiti'On lattice inspired 'Our model f'Or inverse sequences of binomial type in Sec. 4. Section 6 c'Ontains combinatorial pro'Ofs of several 'Operator-the'Oretic results. The acti'Ons of shift operators and delta operat'Ors are explained in set-the'Oretic terms. Finally, in Sec. 6 we give a model f'Or cross sequences and Sheffer sequences which is c'Onsistent with their decomp'Osition int'O sequences 'Of bin'Omial type.This provides an interpretati'On 'Of shift-invariant operators. Of course, all 'Of these interpretati'Ons require that the coefficients inv'Olved be integer and usually n'On-negative as well.

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“…Many extensions of umbral calculus to the case of polynomials of several, or even infinitely many variables were discussed e.g. in [5,10,14,28,32,35,36,41,42], for a longer list of such papers see the introduction to [13]. Appell and Sheffer sequences of polynomials of several noncommutative variables arising in the context of free probability, Boolean probability, and conditionally free probability were discussed in [2][3][4], see also the references therein.…”

Section: Introductionmentioning

confidence: 99%

An infinite dimensional umbral calculus

Finkelshtein

Kondratiev

Lytvynov

et al. 2019

Journal of Functional Analysis

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The aim of this paper is to develop foundations of umbral calculus on the space D of distributions on R d , 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.

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Multivariate Sequences of Binomial Type

Abstract: Using the notion of convolution of binomial sequences it is possible to show that Sheffer sequences, cross sequences, and Steffensen sequences are only mild generalizations of ordinary sequences of binomial type.

Cited by 17 publications

References 4 publications

Multivariate delta Gončarov and Abel polynomials

Multivariate delta Gončarov and Abel polynomials

The Combinatorics of Polynomial Sequences

An infinite dimensional umbral calculus

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