A determinantal approach to Sheffer sequences

Wang, Wei Ping

doi:10.1016/j.laa.2014.09.009

Cited by 21 publications

(17 citation statements)

References 16 publications

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“…[10] has given a new approach to Bernoulli polynomials which was further extended to provide the determinantal definition of the Appell polynomials [11]. Recently, the determinantal definition of Appell sequences is extended to Sheffer sequences by using the theory of Riordan arrays [29].…”

Section: Discussionmentioning

confidence: 99%

“…The determinantal approach considered in [10,11,29] n (x, y, z)(n = 0, 1, 2, ..., ) are the Legendre-Gould Hopper polynomials defined by equation (1.2), S n×(n+1) = (a j−1,i−1 )1 ≤ i ≤ n, 1 ≤ j ≤ n+1 and α n,k is the (n, k) entry of the Riordan array (g(t), f (t)). are the Legendre-Gould Hopper polynomials defined by equation (1.1), S n×(n+1) = (α j−1,i−1 )1 ≤ i ≤ n, 1 ≤ j ≤ n + 1 and a n,k is the (n, k) entry of the Riordan array (g(t), f (t)).…”

Section: Discussionmentioning

confidence: 99%

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Legendre-Gould Hopper Based Sheffer Polynomials: Properties and Applications

Yasmin¹,

Muhyi²

2020

Preprint

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In this article, the Legendre-Gould Hopper polynomials are combined with Sheffer sequences to introduce certain mixed type special polynomials. Generating functions, differential equations and certain other properties of Legendre-Gould Hopper based Sheffer polynomials are derived. Further, operational and integral representations providing connections between these polynomials and known special polynomials are established. Certain identities and results for some members of these new mixed polynomials are also obtained. Finally, the determinantal definitions of Legendre-Gould Hopper based Sheffer polynomials are also given.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Legendre-Gould Hopper Based Sheffer Polynomials: Properties and Applications

Yasmin¹,

Muhyi²

2020

Preprint

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“…Using determinant representation of binomial coefficient ( n k ) [23], the above equations can be expressed as:…”

Section: Applicationsmentioning

confidence: 99%

Finding Determinant Forms of Certain Hybrid Sheffer Sequences

et al. 2019

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In this article, the integral transform is used to introduce a new family of extended hybrid Sheffer sequences via generating functions and operational rules. The determinant forms and other properties of these sequences are established using a matrix approach. The correspondingresults for the extended hybrid Appell sequences are also obtained. Certain examples in terms of the members of the extended hybrid Sheffer and Appell sequences are framed. By employing operational rules, the identities involving the Lah, Stirling and Pascal matrices are derived for the aforementioned sequences.

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“…Later on, their study, leaded by Rota and Roman, developed in what it is known today as Umbral Calculus [26][27][28]. In contrast, a more recent approach has been made using matrix and determinantal representations, see, e.g., [1,2,13,14,37,38]. Also, current research has focussed on special sequences [16] and other alternative descriptions of the theory, for instance, through random variables [3,33].…”

Section: Introductionmentioning

confidence: 99%

Appell and Sheffer sequences: on their characterizations through functionals and examples

Carrillo¹,

Hurtado²

2021

Comptes Rendus. Mathématique

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The aim of this paper is to present a new simple recurrence for Appell and Sheffer sequences in terms of the linear functional that defines them, and to explain how this is equivalent to several well-known characterizations appearing in the literature. We also give several examples, including integral representations of the inverse operators associated to Bernoulli and Euler polynomials, and a new integral representation of the re-scaled Hermite d-orthogonal polynomials generalizing the Weierstrass operator related to the Hermite polynomials. Résumé. L'objectif de cet article est de présenter une nouvelle récurrence simple pour les suites d'Appell et de Sheffer en termes de la fonctionnelle linéaire qui les définit, et d'expliquer comment cela équivaut à plusieurs caractérisations bien connues qui apparaissent dans la littérature. Nous donnons aussi plusieurs exemples, y compris des représentations intégrales des opérateurs inverses associés aux polynômes de Bernoulli et d'Euler, et une nouvelle représentation intégrale des polynômes d'Hermite d-orthogonaux remis à l'échelle, qui généralise l'opérateur de Weierstrass associé aux polynômes d'Hermite.

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A determinantal approach to Sheffer sequences

Cited by 21 publications

References 16 publications

Legendre-Gould Hopper Based Sheffer Polynomials: Properties and Applications

Legendre-Gould Hopper Based Sheffer Polynomials: Properties and Applications

Finding Determinant Forms of Certain Hybrid Sheffer Sequences

Appell and Sheffer sequences: on their characterizations through functionals and examples

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