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Open AccessFeature PaperArticle

Symbolic Methods Applied to a Class of Identities Involving Appell Polynomials and Stirling Numbers

by Tian-Xiao He and Emanuele Munarini

Mathematics 2025, 13(11), 1732; https://doi.org/10.3390/math13111732 - 24 May 2025

Viewed by 634

In this paper, we present two symbolic methods, in particular, the method starting from the source identity, umbra identity, for constructing identities of s-Appell polynomials related to Stirling numbers and binomial coefficients. We discuss some properties of s-Appell polynomial sequences related [...] Read more.

In this paper, we present two symbolic methods, in particular, the method starting from the source identity, umbra identity, for constructing identities of s-Appell polynomials related to Stirling numbers and binomial coefficients. We discuss some properties of s-Appell polynomial sequences related to Riordan arrays, Sheffer matrices, and their q analogs. Full article

(This article belongs to the Special Issue Combinatorics, Riordan Matrices and Umbral Calculus—in Memory of Prof. Emanuele Munarini)

18 pages, 4885 KB

Open AccessArticle

A Study of the q-Truncated Exponential–Appell Polynomials

by Francesco Aldo Costabile, Subuhi Khan and Hassan Ali

Mathematics 2024, 12(23), 3862; https://doi.org/10.3390/math12233862 - 8 Dec 2024

Cited by 7 | Viewed by 1344

Abstract

This article introduces the 2-variable q-truncated exponential–Appell (q-trunc. exp. Appell) polynomials and investigates their fundamental properties. Specific results are derived for the q-trunc. exp. Appell family along with their graphical representations which contribute to advancing the understanding of q [...] Read more.

This article introduces the 2-variable q-truncated exponential–Appell (q-trunc. exp. Appell) polynomials and investigates their fundamental properties. Specific results are derived for the q-trunc. exp. Appell family along with their graphical representations which contribute to advancing the understanding of q-series and q-special functions. Potential applications of these polynomials span various disciplines, including combinatorics (such as partition theory and combinatorial identities), number theory (such as q-analogues of classical number-theoretic functions), and mathematical physics (such as in quantum groups and statistical mechanics). This study concludes with the introduction of the 2-variable q-trunc. exp.

λ

-polynomials, thereby broadening the scope and relevance of this research. Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications, 2nd Edition)

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9 pages, 261 KB

Open AccessArticle

Quasi-Monomiality Principle and Certain Properties of Degenerate Hybrid Special Polynomials

by Rabab Alyusof

Symmetry 2023, 15(2), 407; https://doi.org/10.3390/sym15020407 - 3 Feb 2023

Cited by 3 | Viewed by 1513

Abstract

This article aims to introduce degenerate hybrid type Appell polynomials

_{H} Q_{m} (u, v, w; η)

and establishes their quasi-monomial characteristics. Additionally, a number of features of these polynomials are established, including symmetric identities, implicit summation formulae, differential equations, series definition and operational formalism. Full article

(This article belongs to the Special Issue Numerical Analysis, Approximation Theory, Differential Equations)

11 pages, 1044 KB

Open AccessArticle

A Note on the Laguerre-Type Appell and Hypergeometric Polynomials

by Paolo Emilio Ricci and Rekha Srivastava

Mathematics 2022, 10(11), 1951; https://doi.org/10.3390/math10111951 - 6 Jun 2022

Cited by 8 | Viewed by 2275

Abstract

The Laguerre derivative and its iterations have been used to define new sets of special functions, showing the possibility of generating a kind of parallel universe for mathematical entities of this kind. In this paper, we introduce the Laguerre-type Appell polynomials, in particular, [...] Read more.

The Laguerre derivative and its iterations have been used to define new sets of special functions, showing the possibility of generating a kind of parallel universe for mathematical entities of this kind. In this paper, we introduce the Laguerre-type Appell polynomials, in particular, the Bernoulli and Euler case, and we examine a set of hypergeometric Laguerre–Bernoulli polynomials. We show their main properties and indicate their possible extensions. Full article

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18 pages, 354 KB

Open AccessArticle

Truncated-Exponential-Based Appell-Type Changhee Polynomials

by Tabinda Nahid, Parvez Alam and Junesang Choi

Symmetry 2020, 12(10), 1588; https://doi.org/10.3390/sym12101588 - 24 Sep 2020

Cited by 13 | Viewed by 2601

Abstract

The truncated exponential polynomials

e_{m} (x)

(1), their extensions, and certain newly-introduced polynomials which combine the truncated exponential polynomials with other known polynomials have been investigated and applied in various ways. In this paper, by incorporating the Appell-type Changhee polynomials [...] Read more.

The truncated exponential polynomials

e_{m} (x)

(1), their extensions, and certain newly-introduced polynomials which combine the truncated exponential polynomials with other known polynomials have been investigated and applied in various ways. In this paper, by incorporating the Appell-type Changhee polynomials

C h_{n}^{*} (x)

(10) and the truncated exponential polynomials in a natural way, we aim to introduce so-called truncated-exponential-based Appell-type Changhee polynomials

_{e} C_{n}^{*} (x)

in Definition 1. Then, we investigate certain properties and identities for these new polynomials such as explicit representation, addition formulas, recurrence relations, differential and integral formulas, and some related inequalities. We also present some integral inequalities involving these polynomials

_{e} C_{n}^{*} (x)

. Further we discuss zero distributions of these polynomials by observing their graphs drawn by Mathematica. Lastly some open questions are suggested. Full article

(This article belongs to the Special Issue Symmetry and IoT Intelligence in the Post Pandemic Economy)

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18 pages, 247 KB

Open AccessArticle

On Two Bivariate Kinds of Poly-Bernoulli and Poly-Genocchi Polynomials

by Cheon Seoung Ryoo and Waseem A. Khan

Mathematics 2020, 8(3), 417; https://doi.org/10.3390/math8030417 - 14 Mar 2020

Cited by 19 | Viewed by 2146

Abstract

In this paper, we introduce two bivariate kinds of poly-Bernoulli and poly-Genocchi polynomials and study their basic properties. Finally, we consider some relationships for Stirling numbers of the second kind related to bivariate kinds of poly-Bernoulli and poly-Genocchi polynomials. Full article

20 pages, 741 KB

Open AccessArticle

A Numerical Computation of Zeros of q-Generalized Tangent-Appell Polynomials

by Ghazala Yasmin, Cheon Seoung Ryoo and Hibah Islahi

Mathematics 2020, 8(3), 383; https://doi.org/10.3390/math8030383 - 9 Mar 2020

Cited by 6 | Viewed by 2145

Abstract

The intended objective of this study is to define and investigate a new class of q-generalized tangent-based Appell polynomials by combining the families of 2-variable q-generalized tangent polynomials and q-Appell polynomials. The investigation includes derivations of generating functions, series definitions, [...] Read more.

The intended objective of this study is to define and investigate a new class of q-generalized tangent-based Appell polynomials by combining the families of 2-variable q-generalized tangent polynomials and q-Appell polynomials. The investigation includes derivations of generating functions, series definitions, and several important properties and identities of the hybrid q-special polynomials. Further, the analogous study for the members of this q-hybrid family are illustrated. The graphical representation of its members is shown, and the distributions of zeros are displayed. Full article

(This article belongs to the Special Issue Special Functions and Applications)

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14 pages, 251 KB

Open AccessArticle

A Note on the Degenerate Type of Complex Appell Polynomials

by Dojin Kim

Symmetry 2019, 11(11), 1339; https://doi.org/10.3390/sym11111339 - 31 Oct 2019

Cited by 22 | Viewed by 3072

Abstract

In this paper, complex Appell polynomials and their degenerate-type polynomials are considered as an extension of real-valued polynomials. By treating the real value part and imaginary part separately, we obtained useful identities and general properties by convolution of sequences. To justify the obtained [...] Read more.

In this paper, complex Appell polynomials and their degenerate-type polynomials are considered as an extension of real-valued polynomials. By treating the real value part and imaginary part separately, we obtained useful identities and general properties by convolution of sequences. To justify the obtained results, we show several examples based on famous Appell sequences such as Euler polynomials and Bernoulli polynomials. Further, we show that the degenerate types of the complex Appell polynomials are represented in terms of the Stirling numbers of the first kind. Full article

(This article belongs to the Special Issue The 32th Congress of The Jangjeon Mathematical Society (ICJMS2019) will be Held at Far Eastern Federal Universit, Vladivostok Russia)

19 pages, 894 KB

Open AccessArticle

Some Identities for Euler and Bernoulli Polynomials and Their Zeros

by Taekyun Kim and Cheon Seoung Ryoo

Axioms 2018, 7(3), 56; https://doi.org/10.3390/axioms7030056 - 14 Aug 2018

Cited by 48 | Viewed by 5560 | Correction

Abstract

In this paper, we study some special polynomials which are related to Euler and Bernoulli polynomials. In addition, we give some identities for these polynomials. Finally, we investigate the zeros of these polynomials by using the computer. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications)

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12 pages, 800 KB

Open AccessArticle

Special Numbers and Polynomials Including Their Generating Functions in Umbral Analysis Methods

by Yilmaz Simsek

Axioms 2018, 7(2), 22; https://doi.org/10.3390/axioms7020022 - 1 Apr 2018

Cited by 14 | Viewed by 4696

Abstract

In this paper, by applying umbral calculus methods to generating functions for the combinatorial numbers and the Apostol type polynomials and numbers of order k, we derive some identities and relations including the combinatorial numbers, the Apostol-Bernoulli polynomials and numbers of order [...] Read more.

In this paper, by applying umbral calculus methods to generating functions for the combinatorial numbers and the Apostol type polynomials and numbers of order k, we derive some identities and relations including the combinatorial numbers, the Apostol-Bernoulli polynomials and numbers of order k and the Apostol-Euler polynomials and numbers of order k. Moreover, by using p-adic integral technique, we also derive some combinatorial sums including the Bernoulli numbers, the Euler numbers, the Apostol-Euler numbers and the numbers

y_{1} (n, k; λ)

. Finally, we make some remarks and observations regarding these identities and relations. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications)

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