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26 pages, 397 KB

Open AccessArticle

Type B (p, q)-Stirling Numbers via Signed Restricted Growth Functions and Rook Theory

by Hasan Arslan, Mariam Zaarour, Nazmiye Alemdar and Hüseyin Altındiş

Mathematics 2026, 14(6), 1025; https://doi.org/10.3390/math14061025 - 18 Mar 2026

Viewed by 304

Stirling numbers are among the most classical objects in enumerative combinatorics, counting set partitions and permutations. In this paper, we study their

(p, q)

-analogues in type B from a rook-theoretic point of view. We introduce a type B Ferrers [...] Read more.

Stirling numbers are among the most classical objects in enumerative combinatorics, counting set partitions and permutations. In this paper, we study their

(p, q)

-analogues in type B from a rook-theoretic point of view. We introduce a type B Ferrers board and establish a bijection between signed restricted growth functions and type B rook placements. In addition, we defined the weighted statistics

{levLB}_{B} (w)

and

{levLS}_{B} (w)

over the set of signed restricted growth functions. The associated statistics yield a weighted enumeration that recovers the

(p, q)

-Stirling polynomials of type B, their recurrence relations and generating functions. We then introduce type B Laguerre boards and prove that their rook numbers coincide with the Lah numbers of type B. Full article

(This article belongs to the Section A: Algebra and Logic)

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17 pages, 296 KB

Open AccessArticle

Combinatorial Properties and Values of High-Order Eulerian Numbers

by Tian-Xiao He

Axioms 2026, 15(1), 16; https://doi.org/10.3390/axioms15010016 - 25 Dec 2025

Viewed by 399

Abstract

This paper studies higher-order Eulerian numbers based on Stirling permutations and utilizing Eulerian triangles. It primarily focuses on the chain of higher-order Eulerian numbers, higher-order Eulerian polynomials, and higher-order Eulerian fractions, especially their computation. Many results for Eulerian numbers and second-order Eulerian numbers [...] Read more.

This paper studies higher-order Eulerian numbers based on Stirling permutations and utilizing Eulerian triangles. It primarily focuses on the chain of higher-order Eulerian numbers, higher-order Eulerian polynomials, and higher-order Eulerian fractions, especially their computation. Many results for Eulerian numbers and second-order Eulerian numbers are generalized to higher-order Eulerian numbers. More specifically, we present recurrence relations of high-order Eulerian numbers, row-generating functions, and row sums of higher-order Eulerian triangles. Furthermore, we investigate the higher-order Eulerian fraction and its alternative form. Some properties of higher-order Eulerian fractions are expressed using differentiation and integration. We derive the inversion relations between second-order Eulerian numbers and Stirling numbers of the second and first kinds. Finally, we provide exact expressions and a computational method for higher-order Eulerian numbers. Full article

16 pages, 277 KB

Open AccessArticle

Identities Involving the Higher-Order Degenerate Type 2 ω-Daehee Polynomials

by Pengfei Zhang, Yonglin Yang and Huihui Wang

Symmetry 2025, 17(12), 2034; https://doi.org/10.3390/sym17122034 - 28 Nov 2025

Viewed by 363

Abstract

In this paper, based on previous study of type 2

ω

-Daehee polynomials and some of their properties, we further introduce the generating function definition for the higher-order degenerate type 2

ω

-Daehee polynomials. By employing the methods of generating functions and Riordan [...] Read more.

In this paper, based on previous study of type 2

ω

-Daehee polynomials and some of their properties, we further introduce the generating function definition for the higher-order degenerate type 2

ω

-Daehee polynomials. By employing the methods of generating functions and Riordan arrays, we investigate the properties of these higher-order degenerate polynomials in depth and establish identities that relate them to certain special combinatorial sequences. Full article

(This article belongs to the Section Mathematics)

21 pages, 441 KB

Open AccessArticle

Discovering New Recurrence Relations for Stirling Numbers: Leveraging a Poisson Expectation Identity for Higher-Order Moments

by Ying-Ying Zhang and Dong-Dong Pan

Axioms 2025, 14(10), 747; https://doi.org/10.3390/axioms14100747 - 1 Oct 2025

Viewed by 676

Abstract

This paper establishes two novel recurrence relations for Stirling numbers of the second kind—an L recurrence and a vertical recurrence—discovered through a probabilistic analysis of Poisson higher-order origin moments. While the link between these moments and Stirling numbers is known, our derivation via [...] Read more.

This paper establishes two novel recurrence relations for Stirling numbers of the second kind—an L recurrence and a vertical recurrence—discovered through a probabilistic analysis of Poisson higher-order origin moments. While the link between these moments and Stirling numbers is known, our derivation via a specific expectation identity provides a clear and efficient pathway to their computation, circumventing the need for infinite series. The primary theoretical contribution is the proof of these previously undocumented combinatorial recurrences, which are of independent mathematical interest. Furthermore, we demonstrate the severe practical inadequacy of high-order sample moments as estimators, highlighting the necessity of our analytical approach to obtaining reliable estimates in applied fields. Full article

(This article belongs to the Section Mathematical Analysis)

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36 pages, 437 KB

Open AccessArticle

Formulas Involving Cauchy Polynomials, Bernoulli Polynomials, and Generalized Stirling Numbers of Both Kinds

by José L. Cereceda

Axioms 2025, 14(10), 746; https://doi.org/10.3390/axioms14100746 - 1 Oct 2025

Viewed by 674

Abstract

In this paper, we derive novel formulas and identities connecting Cauchy numbers and polynomials with both ordinary and generalized Stirling numbers, binomial coefficients, central factorial numbers, Euler polynomials, r-Whitney numbers, and hyperharmonic polynomials, as well as Bernoulli numbers and polynomials. We also [...] Read more.

In this paper, we derive novel formulas and identities connecting Cauchy numbers and polynomials with both ordinary and generalized Stirling numbers, binomial coefficients, central factorial numbers, Euler polynomials, r-Whitney numbers, and hyperharmonic polynomials, as well as Bernoulli numbers and polynomials. We also provide formulas for the higher-order derivatives of Cauchy polynomials and obtain corresponding formulas and identities for poly-Cauchy polynomials. Furthermore, we introduce a multiparameter framework for poly-Cauchy polynomials, unifying earlier generalizations like shifted poly-Cauchy numbers and polynomials with a q parameter. Full article

27 pages, 357 KB

Open AccessArticle

New Families of Certain Special Polynomials: A Kaniadakis Calculus Viewpoint

by Ugur Duran, Mehmet Acikgoz and Serkan Araci

Symmetry 2025, 17(9), 1534; https://doi.org/10.3390/sym17091534 - 14 Sep 2025

Viewed by 692

Abstract

In this paper, we introduce a new family of Stirling polynomials of the second kind, Bell polynomials, bivariate Bell polynomials, Bernoulli polynomials of higher order, and Euler polynomials of higher order arising from the Kaniadakis calculus viewpoint. We refer to each of them [...] Read more.

In this paper, we introduce a new family of Stirling polynomials of the second kind, Bell polynomials, bivariate Bell polynomials, Bernoulli polynomials of higher order, and Euler polynomials of higher order arising from the Kaniadakis calculus viewpoint. We refer to each of them as

κ

-polynomials. Through the defined concepts of Kaniadakis calculus, we derive explicit formulas, summation formulas, and addition formulas for the polynomials discussed in the present paper. We also present the Volkenborn integral and the fermionic p-adic integral representations in terms of the

κ

-Stirling polynomials of the second kind, bivariate

κ

-Bell polynomials,

κ

-Bernoulli polynomials of higher order, and

κ

-Euler polynomials of higher order. We establish some formulae, including old and new polynomials. Finally, we investigate determinantal representations for the

κ

-Euler polynomials and the

κ

-Bernoulli polynomials. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials: 3rd Edition)

11 pages, 245 KB

Open AccessArticle

Formulae for Generalization of Touchard Polynomials with Their Generating Functions

by Ayse Yilmaz Ceylan and Yilmaz Simsek

Symmetry 2025, 17(7), 1126; https://doi.org/10.3390/sym17071126 - 14 Jul 2025

Cited by 1 | Viewed by 1140

Abstract

One of the main motivations of this paper is to construct generating functions for generalization of the Touchard polynomials (or generalization exponential functions) and certain special numbers. Many novel formulas and relations for these polynomials are found by using the Euler derivative operator [...] Read more.

One of the main motivations of this paper is to construct generating functions for generalization of the Touchard polynomials (or generalization exponential functions) and certain special numbers. Many novel formulas and relations for these polynomials are found by using the Euler derivative operator and functional equations of these functions. Some novel relations among these polynomials, beta polynomials, Bernstein polynomials, related to Binomial distribution from discrete probability distribution classes, are given. Full article

(This article belongs to the Section Mathematics)

27 pages, 341 KB

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 889

Abstract

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)

14 pages, 272 KB

Open AccessFeature PaperArticle

Elementary Operators with m-Null Symbols

by Isabel Marrero

Mathematics 2025, 13(5), 741; https://doi.org/10.3390/math13050741 - 25 Feb 2025

Cited by 2 | Viewed by 945

Abstract

Motivated by Botelho and Jamison’s seminal 2010 study on elementary operators that are m-isometries, in this paper, we introduce the concept of m-null pairs of operators and establish some structural properties and characterizations of the class of elementary operators whose symbols [...] Read more.

Motivated by Botelho and Jamison’s seminal 2010 study on elementary operators that are m-isometries, in this paper, we introduce the concept of m-null pairs of operators and establish some structural properties and characterizations of the class of elementary operators whose symbols are m-null (so-called m-null elementary operators). It is shown that if the symbols of an elementary operator L are, in turn, a p-null elementary operator and a q-null elementary operator, then L is a

(p + q - 1)

-null elementary operator. Some extant results on elementary m-isometries can be recovered from this renewed perspective, often providing added value. Full article

(This article belongs to the Section C: Mathematical Analysis)

16 pages, 947 KB

Open AccessFeature PaperArticle

The Rosencrantz Coin: Predictability and Structure in Non-Ergodic Dynamics—From Recurrence Times to Temporal Horizons

by Dimitri Volchenkov

Entropy 2025, 27(2), 147; https://doi.org/10.3390/e27020147 - 1 Feb 2025

Viewed by 1867

Abstract

We examine the Rosencrantz coin that can “stick” in states for extended periods. Non-ergodic dynamics is highlighted by logarithmically growing block lengths in sequences. Traditional entropy decomposition into predictable and unpredictable components fails due to the absence of stationary distributions. Instead, sequence structure [...] Read more.

We examine the Rosencrantz coin that can “stick” in states for extended periods. Non-ergodic dynamics is highlighted by logarithmically growing block lengths in sequences. Traditional entropy decomposition into predictable and unpredictable components fails due to the absence of stationary distributions. Instead, sequence structure is characterized by block probabilities and Stirling numbers of the second kind, peaking at block size

n / \log n

. For large n, combinatorial growth dominates probability decay, creating a deterministic-like structure. This approach shifts the focus from predicting states to predicting temporal horizons, providing insights into systems beyond traditional equilibrium frameworks. Full article

(This article belongs to the Section Statistical Physics)

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33 pages, 3753 KB

Open AccessFeature PaperArticle

Matching Polynomials of Symmetric, Semisymmetric, Double Group Graphs, Polyacenes, Wheels, Fans, and Symmetric Solids in Third and Higher Dimensions

by Krishnan Balasubramanian

Symmetry 2025, 17(1), 133; https://doi.org/10.3390/sym17010133 - 17 Jan 2025

Cited by 2 | Viewed by 3527

Abstract

The primary objective of this study is the computation of the matching polynomials of a number of symmetric, semisymmetric, double group graphs, and solids in third and higher dimensions. Such computations of matching polynomials are extremely challenging problems due to the computational and [...] Read more.

The primary objective of this study is the computation of the matching polynomials of a number of symmetric, semisymmetric, double group graphs, and solids in third and higher dimensions. Such computations of matching polynomials are extremely challenging problems due to the computational and combinatorial complexity of the problem. We also consider a series of recursive graphs possessing symmetries such as D_2h-polyacenes, wheels, and fans. The double group graphs of the Möbius types, which find applications in chemically interesting topologies and stereochemistry, are considered for the matching polynomials. Hence, the present study features a number of vertex- or edge-transitive regular graphs, Archimedean solids, truncated polyhedra, prisms, and 4D and 5D polyhedra. Such polyhedral and Möbius graphs present stereochemically and topologically interesting applications, including in chirality, isomerization reactions, and dynamic stereochemistry. The matching polynomials of these systems are shown to contain interesting combinatorics, including Stirling numbers of both kinds, Lucas polynomials, toroidal tree-rooted map sequences, and Hermite, Laguerre, Chebychev, and other orthogonal polynomials. Full article

(This article belongs to the Collection Feature Papers in Chemistry)

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52 pages, 869 KB

Open AccessReview

Series and Connections Among Central Factorial Numbers, Stirling Numbers, Inverse of Vandermonde Matrix, and Normalized Remainders of Maclaurin Series Expansions ^†

by Feng Qi

Mathematics 2025, 13(2), 223; https://doi.org/10.3390/math13020223 - 10 Jan 2025

Cited by 7 | Viewed by 1439

Abstract

This paper presents an extensive investigation into several interrelated topics in mathematical analysis and number theory. The author revisits and builds upon known results regarding the Maclaurin power series expansions for a variety of functions and their normalized remainders, explores connections among central [...] Read more.

This paper presents an extensive investigation into several interrelated topics in mathematical analysis and number theory. The author revisits and builds upon known results regarding the Maclaurin power series expansions for a variety of functions and their normalized remainders, explores connections among central factorial numbers, the Stirling numbers, and specific matrix inverses, and derives several closed-form formulas and inequalities. Additionally, this paper reveals new insights into the properties of these mathematical objects, including logarithmic convexity, explicit expressions for certain quantities, and identities involving the Bell polynomials of the second kind. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

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11 pages, 271 KB

Open AccessArticle

On Qi’s Normalized Remainder of Maclaurin Power Series Expansion of Logarithm of Secant Function

by Hong-Chao Zhang, Bai-Ni Guo and Wei-Shih Du

Axioms 2024, 13(12), 860; https://doi.org/10.3390/axioms13120860 - 8 Dec 2024

Cited by 7 | Viewed by 1502

Abstract

In the study, the authors introduce Qi’s normalized remainder of the Maclaurin power series expansion of the function

\ln \sec x = - \ln \cos x

; in view of a monotonicity rule for the ratio of two Maclaurin power series and by [...] Read more.

In the study, the authors introduce Qi’s normalized remainder of the Maclaurin power series expansion of the function

\ln \sec x = - \ln \cos x

; in view of a monotonicity rule for the ratio of two Maclaurin power series and by virtue of the logarithmic convexity of the function

(2^{x} - 1) ζ (x)

on

(1, \infty)

, they prove the logarithmic convexity of Qi’s normalized remainder; with the aid of a monotonicity rule for the ratio of two Maclaurin power series, the authors present the monotonic property of the ratio between two Qi’s normalized remainders. Full article

(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization, 2nd Edition)

12 pages, 299 KB

Open AccessArticle

Sălăgean Differential Operator in Connection with Stirling Numbers

by Basem Aref Frasin and Luminiţa-Ioana Cotîrlă

Axioms 2024, 13(9), 620; https://doi.org/10.3390/axioms13090620 - 12 Sep 2024

Viewed by 1076

Abstract

Sălăgean differential operator

D^{κ}

plays an important role in the geometric function theory, where many studies are using this operator to introduce new subclasses of analytic functions defined in the open unit disk. Studies of Sălăgean differential operator

D^{κ}

in connection [...] Read more.

Sălăgean differential operator

D^{κ}

plays an important role in the geometric function theory, where many studies are using this operator to introduce new subclasses of analytic functions defined in the open unit disk. Studies of Sălăgean differential operator

D^{κ}

in connection with Stirling numbers are relatively new. In this paper, the differential operator

D^{κ}

involving Stirling numbers is considered. A new sufficient condition involving Stirling numbers for the series

Υ_{θ}^{s} (ϰ)

written with the Pascal distribution are discussed for the subclass

T_{κ} (ϵ, ♭)

. Also, we provide a sufficient condition for the inclusion relation

I_{θ}^{s} (R^{ϖ} (E, D)) \subset T_{κ} (ϵ, ♭)

. Further, we consider the properties of an integral operator related to Pascal distribution series. New special cases as a consequences of the main results are also obtained. Full article

(This article belongs to the Special Issue Advances in Geometric Function Theory and Related Topics)

13 pages, 282 KB

Open AccessArticle

The Formulae and Symmetry Property of Bernstein Type Polynomials Related to Special Numbers and Functions

by Ayse Yilmaz Ceylan and Buket Simsek

Symmetry 2024, 16(9), 1159; https://doi.org/10.3390/sym16091159 - 5 Sep 2024

Cited by 2 | Viewed by 1311

Abstract

The aim of this paper is to derive formulae for the generating functions of the Bernstein type polynomials. We give a PDE equation for this generating function. By using this equation, we give recurrence relations for the Bernstein polynomials. Using generating functions, we [...] Read more.

The aim of this paper is to derive formulae for the generating functions of the Bernstein type polynomials. We give a PDE equation for this generating function. By using this equation, we give recurrence relations for the Bernstein polynomials. Using generating functions, we also derive some identities including a symmetry property for the Bernstein type polynomials. We give some relations among the Bernstein type polynomials, Bernoulli numbers, Stirling numbers, Dahee numbers, the Legendre polynomials, and the coefficients of the classical superoscillatory function associated with the weak measurements. We introduce some integral formulae for these polynomials. By using these integral formulae, we derive some new combinatorial sums involving the Bernoulli numbers and the combinatorial numbers. Moreover, we define Bezier type curves in terms of these polynomials. Full article

(This article belongs to the Section Mathematics)

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