MDPI - Publisher of Open Access Journals

11 pages, 245 KiB

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

Viewed by 189

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 KiB

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 310

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 KiB

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 557

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 KiB

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 1119

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)

► Show Figures

Figure 1

33 pages, 3753 KiB

Open AccessArticle

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 1 | Viewed by 1704

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)

► Show Figures

Figure 1

52 pages, 869 KiB

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 4 | Viewed by 886

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)

► Show Figures

Figure 1

11 pages, 271 KiB

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 5 | Viewed by 964

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 KiB

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 772

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 KiB

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 842

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)

► Show Figures

Figure 1

21 pages, 382 KiB

Open AccessArticle

Power Series Expansions of Real Powers of Inverse Cosine and Sine Functions, Closed-Form Formulas of Partial Bell Polynomials at Specific Arguments, and Series Representations of Real Powers of Circular Constant

by Feng Qi

Symmetry 2024, 16(9), 1145; https://doi.org/10.3390/sym16091145 - 3 Sep 2024

Cited by 2 | Viewed by 1910

Abstract

In this paper, by means of the Faà di Bruno formula, with the help of explicit formulas for partial Bell polynomials at specific arguments of two specific sequences generated by derivatives at the origin of the inverse sine and inverse cosine functions, and [...] Read more.

In this paper, by means of the Faà di Bruno formula, with the help of explicit formulas for partial Bell polynomials at specific arguments of two specific sequences generated by derivatives at the origin of the inverse sine and inverse cosine functions, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes power series expansions for real powers of the inverse cosine (sine) functions and the inverse hyperbolic cosine (sine) functions. By comparing different series expansions for the square of the inverse cosine function and for the positive integer power of the inverse sine function, the author not only finds infinite series representations of the circular constant

π

and its real powers, but also derives several combinatorial identities involving central binomial coefficients and the Stirling numbers of the first kind. Full article

(This article belongs to the Section Mathematics)

23 pages, 539 KiB

Open AccessArticle

On Convoluted Forms of Multivariate Legendre-Hermite Polynomials with Algebraic Matrix Based Approach

by Mumtaz Riyasat, Amal S. Alali, Shahid Ahmad Wani and Subuhi Khan

Mathematics 2024, 12(17), 2662; https://doi.org/10.3390/math12172662 - 27 Aug 2024

Viewed by 872

Abstract

The main purpose of this article is to construct a new class of multivariate Legendre-Hermite-Apostol type Frobenius-Euler polynomials. A number of significant analytical characterizations of these polynomials using various generating function techniques are provided in a methodical manner. These enactments involve explicit relations [...] Read more.

The main purpose of this article is to construct a new class of multivariate Legendre-Hermite-Apostol type Frobenius-Euler polynomials. A number of significant analytical characterizations of these polynomials using various generating function techniques are provided in a methodical manner. These enactments involve explicit relations comprising Hurwitz-Lerch zeta functions and

λ

-Stirling numbers of the second kind, recurrence relations, and summation formulae. The symmetry identities for these polynomials are established by connecting generalized integer power sums, double power sums and Hurwitz-Lerch zeta functions. In the end, these polynomials are also characterized Svia an algebraic matrix based approach. Full article

(This article belongs to the Section E: Applied Mathematics)

► Show Figures

Figure 1

15 pages, 351 KiB

Open AccessArticle

Linear Combination of Order Statistics Moments from Log-Extended Exponential Geometric Distribution with Applications to Entropy

by Fatimah E. Almuhayfith, Mahfooz Alam, Hassan S. Bakouch, Sudeep R. Bapat and Olayan Albalawi

Mathematics 2024, 12(11), 1744; https://doi.org/10.3390/math12111744 - 3 Jun 2024

Viewed by 1101

Abstract

Moments of order statistics (OSs) characterize the Weibull–geometric and half-logistic families of distributions, of which the extended exponential–geometric (EEG) distribution is a particular case. The EEG distribution is used to create the log-extended exponential–geometric (LEEG) distribution, which is bounded in the unit interval [...] Read more.

Moments of order statistics (OSs) characterize the Weibull–geometric and half-logistic families of distributions, of which the extended exponential–geometric (EEG) distribution is a particular case. The EEG distribution is used to create the log-extended exponential–geometric (LEEG) distribution, which is bounded in the unit interval (0, 1). In addition to the generalized Stirling numbers of the first kind, a few years ago, the polylogarithm function and the Lerch transcendent function were used to determine the moments of order statistics of the LEEG distributions. As an application based on the L-moments, we expand the features of the LEEG distribution in this work. In terms of the Gauss hypergeometric function, this work presents the precise equations and recurrence relations for the single moments of OSs from the LEEG distribution. Along with recurrence relations between the expectations of function of two OSs from the LEEG distribution, it also displays the truncated and conditional distribution of the OSs. Additionally, we use the L-moments to estimate the parameters of the LEEG distribution. We further fit the LEEG distribution on three practical data sets from medical and environmental sciences areas. It is seen that the estimated parameters through L-moments of the OSs give a superior fit. We finally determine the correspondence between the entropies and the OSs. Full article

(This article belongs to the Special Issue Advances in Applied Probability and Statistical Inference)

► Show Figures

Figure 1

21 pages, 826 KiB

Open AccessArticle

On Certain Properties of Parametric Kinds of Apostol-Type Frobenius–Euler–Fibonacci Polynomials

by Hao Guan, Waseem Ahmad Khan, Can Kızılateş and Cheon Seoung Ryoo

Axioms 2024, 13(6), 348; https://doi.org/10.3390/axioms13060348 - 24 May 2024

Viewed by 896

Abstract

This paper presents an overview of cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials, as well as several identities that are associated with these polynomials. By applying a partial derivative operator to the generating functions, the authors obtain derivative formulae and finite combinatorial sums involving [...] Read more.

This paper presents an overview of cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials, as well as several identities that are associated with these polynomials. By applying a partial derivative operator to the generating functions, the authors obtain derivative formulae and finite combinatorial sums involving these polynomials and numbers. Additionally, the paper establishes connections between cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials of order

α

and several other polynomial sequences, such as the Apostol-type Bernoulli–Fibonacci polynomials, the Apostol-type Euler–Fibonacci polynomials, the Apostol-type Genocchi–Fibonacci polynomials, and the Stirling–Fibonacci numbers of the second kind. The authors also provide computational formulae and graphical representations of these polynomials using the Mathematica program. Full article

(This article belongs to the Special Issue Fractional and Stochastic Differential Equations in Mathematics)

► Show Figures

Figure 1

16 pages, 306 KiB

Open AccessArticle

Summation Formulas for Certain Combinatorial Sequences

by Yulei Chen and Dongwei Guo

Mathematics 2024, 12(8), 1210; https://doi.org/10.3390/math12081210 - 17 Apr 2024

Cited by 2 | Viewed by 1508

Abstract

In this work, we establish some characteristics for a sequence,

A_{α} (n, k)

, including recurrence relations, generating function and inversion formula, etc. Based on the sequence, we derive, by means of the generating function approach, some transformation formulas [...] Read more.

In this work, we establish some characteristics for a sequence,

A_{α} (n, k)

, including recurrence relations, generating function and inversion formula, etc. Based on the sequence, we derive, by means of the generating function approach, some transformation formulas concerning certain combinatorial numbers named after Lah, Stirling, harmonic, Cauchy and Catalan, as well as several closed finite sums. In addition, the relationship between

A_{α} (n, k)

and r-Whitney–Lah numbers is established, and some formulas for the r-Whitney–Lah numbers are obtained. Full article

15 pages, 737 KiB

Open AccessArticle

Some Properties of Generalized Apostol-Type Frobenius–Euler–Fibonacci Polynomials

by Maryam Salem Alatawi, Waseem Ahmad Khan, Can Kızılateş and Cheon Seoung Ryoo

Mathematics 2024, 12(6), 800; https://doi.org/10.3390/math12060800 - 8 Mar 2024

Cited by 5 | Viewed by 1272

Abstract

In this paper, by using the Golden Calculus, we introduce the generalized Apostol-type Frobenius–Euler–Fibonacci polynomials and numbers; additionally, we obtain various fundamental identities and properties associated with these polynomials and numbers, such as summation theorems, difference equations, derivative properties, recurrence relations, and more. [...] Read more.

In this paper, by using the Golden Calculus, we introduce the generalized Apostol-type Frobenius–Euler–Fibonacci polynomials and numbers; additionally, we obtain various fundamental identities and properties associated with these polynomials and numbers, such as summation theorems, difference equations, derivative properties, recurrence relations, and more. Subsequently, we present summation formulas, Stirling–Fibonacci numbers of the second kind, and relationships for these polynomials and numbers. Finally, we define the new family of the generalized Apostol-type Frobenius–Euler–Fibonacci matrix and obtain some factorizations of this newly established matrix. Using Mathematica, the computational formulae and graphical representation for the mentioned polynomials are obtained. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions: Recent Trends and Their Applications)

► Show Figures

Figure 1

Search Results (103)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (103)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI