Guest Editors
Dr. Serkan Araci, Hasan Kalyoncu University, Turkey
Prof. Dr. Juan Luis García Guirao, Universidad Politécnica de Cartagena, Spain
Summary
The computers began to appear in the 1950s, and often incorrect, estimations were done related to the impact of these devices on applied mathematics, science and engineering. One of these estimations was that the need for special functions, or higher transcendental functions (as they are also known), would disappear entirely. This was based on the observation that the key use of these functions in those days was to approximate the solutions of classical differential (or partial differential) equations: with the mathematical software it would become possible to solve these equations by direct numerical methods. This observation is in fact correct; even so, a study of current computational journals in the sciences reveals a continuous need for numerical algorithms to generate Airy functions, Bessel functions, Coulomb wave functions, error functions and exponential integrals, etc.
This special issue focuses on the applications and computer modeling of the special functions and polynomials to various areas of mathematics. Thorough knowledge of special functions is required in modern engineering, physical science applications and computer modeling. These functions typically arise in such applications as communications systems, statistical probability distribution, electro-optics, nonlinear wave propagation, electromagnetic theory, potential theory, electric circuit theory, and quantum mechanics.
Potential topics include but are not limited to the following:
• Computer modeling of Special functions and polynomials
• Analytical properties and applications of Special functions.
• Inequalities for Special Functions
• Integration of products of Special Functions
• Properties of ordinary and general families of Special Polynomials
• Operational techniques involving Special Polynomials
• Classes of mixed Special Polynomials and their properties
• Other miscellaneous applications of Special Functions and Special Polynomials
Keywords
Hypergeometric functions and their extensions; Generalized functions and their extensions; Generalized inequalities and their extensions; Operational techniques; Mixed special polynomials; Applications; Computer modeling.
Published Papers
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Open Access
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Open Access
ARTICLE
Partial Bell Polynomials, Falling and Rising Factorials, Stirling Numbers, and Combinatorial Identities
Siqintuya Jin, Bai-Ni Guo, Feng Qi
Computer Modeling in Engineering & Sciences, Vol.132, No.3, pp. 781-799, 2022, DOI:10.32604/cmes.2022.019941
（This article belongs to this Special Issue:
Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)
Abstract In the paper, the authors collect, discuss, and find out several connections, equivalences, closed-form formulas, and
combinatorial identities concerning partial Bell polynomials, falling factorials, rising factorials, extended binomial
coefficients, and the Stirling numbers of the first and second kinds. These results are new, interesting, important,
useful, and applicable in combinatorial number theory.
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Some Identities of the Degenerate Poly-Cauchy and Unipoly Cauchy Polynomials of the Second Kind
Ghulam Muhiuddin, Waseem A. Khan, Deena Al-Kadi
Computer Modeling in Engineering & Sciences, Vol.132, No.3, pp. 763-779, 2022, DOI:10.32604/cmes.2022.017272
（This article belongs to this Special Issue:
Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)
Abstract In this paper, we introduce modied degenerate polyexponential Cauchy (or poly-Cauchy) polynomials and
numbers of the second kind and investigate some identities of these polynomials. We derive recurrence relations
and the relationship between special polynomials and numbers. Also, we introduce modied degenerate unipolyCauchy polynomials of the second kind and derive some fruitful properties of these polynomials. In addition,
positive associated beautiful zeros and graphical representations are displayed with the help of Mathematica.
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Asymptotic Approximations of Apostol-Tangent Polynomials in Terms of Hyperbolic Functions
Cristina B. Corcino, Wilson D. Castañeda Jr., Roberto B. Corcino
Computer Modeling in Engineering & Sciences, Vol.132, No.1, pp. 133-151, 2022, DOI:10.32604/cmes.2022.019965
（This article belongs to this Special Issue:
Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)
Abstract The tangent polynomials
Tn (z) are generalization of tangent numbers or the Euler zigzag numbers
Tn. In particular,
Tn (0) =
Tn. These polynomials are closely related to Bernoulli, Euler and Genocchi polynomials. One of the
extensions and analogues of special polynomials that attract the attention of several mathematicians is the Apostoltype polynomials. One of these Apostol-type polynomials is the Apostol-tangent polynomials
Tn(z, λ). When
λ = 1,
Tn (z, 1) =
Tn(z). The use of hyperbolic functions to derive asymptotic approximations of polynomials
together with saddle point method was applied to the Bernoulli and Euler polynomials by Lopez and…
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Open Access
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Image Encryption Algorithm Based on New Fractional Beta Chaotic Maps
Rabha W. Ibrahim, Hayder Natiq, Ahmed Alkhayyat, Alaa Kadhim Farhan, Nadia M. G. Al-Saidi, Dumitru Baleanu
Computer Modeling in Engineering & Sciences, Vol.132, No.1, pp. 119-131, 2022, DOI:10.32604/cmes.2022.018343
（This article belongs to this Special Issue:
Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)
Abstract In this study, a new algorithm of fractional beta chaotic maps is proposed to generate chaotic sequences for image
encryption. The proposed technique generates multi random sequences by shuffling the image pixel position.
This technique is used to blur the pixels connecting the input and encrypted images and to increase the attack
resistance. The proposed algorithm makes the encryption process sophisticated by using fractional chaotic maps,
which hold the properties of pseudo-randomness. The fractional beta sequences are utilized to alter the image pixels
to decryption attacks. The experimental results proved that the proposed image encryption algorithm successfully
encrypted and decrypted…
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Degenerate s-Extended Complete and Incomplete Lah-Bell Polynomials
Hye Kyung Kim, Dae Sik Lee
Computer Modeling in Engineering & Sciences, Vol.131, No.3, pp. 1479-1495, 2022, DOI:10.32604/cmes.2022.017616
（This article belongs to this Special Issue:
Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)
Abstract Degenerate versions of special polynomials and numbers applied to social problems, physics, and applied mathematics have been studied variously in recent years. Moreover, the (
s-)Lah numbers have many other interesting
applications in analysis and combinatorics. In this paper, we divide two parts. We first introduce new types of
both degenerate incomplete and complete
s-Bell polynomials respectively and investigate some properties of them
respectively. Second, we introduce the degenerate versions of complete and incomplete Lah-Bell polynomials as
multivariate forms for a new type of degenerate
s-extended Lah-Bell polynomials and numbers respectively. We
investigate relations between these polynomials and degenerate incomplete and…
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Properties of Certain Subclasses of Analytic Functions Involving q-Poisson Distribution
Bilal Khan, Zhi-Guo Liu, Nazar Khan, Aftab Hussain, Nasir Khan, Muhammad Tahir
Computer Modeling in Engineering & Sciences, Vol.131, No.3, pp. 1465-1477, 2022, DOI:10.32604/cmes.2022.016940
（This article belongs to this Special Issue:
Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)
Abstract By using the basic (or
q)-Calculus many subclasses of analytic and univalent functions have been generalized
and studied from different viewpoints and perspectives. In this paper, we aim to define certain new subclasses of
an analytic function. We then give necessary and sufficient conditions for each of the defined function classes.
We also study necessary and sufficient conditions for a function whose coefficients are probabilities of
q-Poisson
distribution. To validate our results, some known consequences are also given in the form of Remarks and
Corollaries.
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Modeling the Spread of Tuberculosis with Piecewise Differential Operators
Abdon Atangana, Ilknur Koca
Computer Modeling in Engineering & Sciences, Vol.131, No.2, pp. 787-814, 2022, DOI:10.32604/cmes.2022.019221
（This article belongs to this Special Issue:
Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)
Abstract Very recently, a new concept was introduced to capture crossover behaviors that exhibit changes in patterns. The
aim was to model real-world problems exhibiting crossover from one process to another, for example, randomness
to a power law. The concept was called piecewise calculus, as differential and integral operators are defined piece
wisely. These behaviors have been observed in the spread of several infectious diseases, for example, tuberculosis.
Therefore, in this paper, we aim at modeling the spread of tuberculosis using the concept of piecewise modeling.
Several cases are considered, conditions under which the unique system solution is obtained are presented…
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On Degenerate Array Type Polynomials
Lan Wu, Xue-Yan Chen, Muhammet Cihat Dağli, Feng Qi
Computer Modeling in Engineering & Sciences, Vol.131, No.1, pp. 295-305, 2022, DOI:10.32604/cmes.2022.018778
（This article belongs to this Special Issue:
Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)
Abstract In the paper, with the help of the Faá di Bruno formula and an identity of the Bell polynomials of the second
kind, the authors define degenerate λ-array type polynomials, establish two explicit formulas, and present several
recurrence relations of degenerate λ-array type polynomials and numbers.
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k-Order Fibonacci Polynomials on AES-Like Cryptology
Mustafa Asci, Suleyman Aydinyuz
Computer Modeling in Engineering & Sciences, Vol.131, No.1, pp. 277-293, 2022, DOI:10.32604/cmes.2022.017898
（This article belongs to this Special Issue:
Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)
Abstract The Advanced Encryption Standard (AES) is the most widely used symmetric cipher today. AES has an important
place in cryptology. Finite field, also known as Galois Fields, are cornerstones for understanding any cryptography.
This encryption method on AES is a method that uses polynomials on Galois fields. In this paper, we generalize
the AES-like cryptology on 2 × 2 matrices. We redefine the elements of k-order Fibonacci polynomials sequences
using a certain irreducible polynomial in our cryptology algorithm. So, this cryptology algorithm is called AES-like
cryptology on the k-order Fibonacci polynomial matrix.
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Complete Monotonicity of Functions Related to Trigamma and Tetragamma Functions
Mona Anis, Hanan Almuashi, Mansour Mahmoud
Computer Modeling in Engineering & Sciences, Vol.131, No.1, pp. 263-275, 2022, DOI:10.32604/cmes.2022.016927
（This article belongs to this Special Issue:
Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)
Abstract In this paper, we study the completely monotonic property of two functions involving the function △(x) =
and deduce the double inequality
, x > 0
which improve some recent results, where ψ(x) is the logarithmic derivative of the Gamma function. Also, we deduce the completely monotonic degree of a function involving ψ'(x).
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Note on a New Construction of Kantorovich Form q-Bernstein Operators Related to Shape Parameter λ
Qingbo Cai, Reşat Aslan
Computer Modeling in Engineering & Sciences, Vol.130, No.3, pp. 1479-1493, 2022, DOI:10.32604/cmes.2022.018338
（This article belongs to this Special Issue:
Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)
Abstract The main purpose of this paper is to introduce some approximation properties of a Kantorovich kind
q-Bernstein operators related to Bézier basis functions with shape parameter . Firstly, we compute some basic results such as moments and central moments, and derive the Korovkin type approximation theorem for these operators. Next, we estimate the order of convergence in terms of the usual modulus of continuity, for the functions belong to Lipschitz-type class and Peetre’s
K-functional, respectively. Lastly, with the aid of Maple software, we present the comparison of the convergence of these newly defined operators to the certain function with some…
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Lacunary Generating Functions of Hybrid Type Polynomials in Viewpoint of Symbolic Approach
Nusrat Raza, Umme Zainab and Serkan Araci
Computer Modeling in Engineering & Sciences, Vol.130, No.2, pp. 903-921, 2022, DOI:10.32604/cmes.2022.017669
（This article belongs to this Special Issue:
Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)
Abstract In this paper, we introduce mon-symbolic method to obtain the generating functions of the hybrid class of
Hermite-associated Laguerre and its associated polynomials. We obtain the series definitions of these hybrid
special polynomials. Also, we derive the double lacunary generating functions of the Hermite-Laguerre polynomials and the Hermite-Laguerre-Wright polynomials. Further, we find multiplicative and derivative operators
for the Hermite-Laguerre-Wright polynomials which helps to find the symbolic differential equation of the
Hermite-Laguerre-Wright polynomials. Some concluding remarks are also given.
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Some Formulas Involving Hypergeometric Functions in Four Variables
Hassen Aydi, Ashish Verma, Jihad Younis, Jung Rye Lee
Computer Modeling in Engineering & Sciences, Vol.130, No.2, pp. 887-902, 2022, DOI:10.32604/cmes.2022.016924
（This article belongs to this Special Issue:
Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)
Abstract Several (generalized) hypergeometric functions and a variety of their extensions have been presented and investigated in the literature by many authors. In the present paper, we investigate four new hypergeometric functions in four variables and then establish several recursion formulas for these new functions. Also, some interesting particular cases and consequences of our results are discussed.
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Open Access
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On ev and ve-Degree Based Topological Indices of Silicon Carbides
Jung Rye Lee, Aftab Hussain, Asfand Fahad, Ali Raza, Muhammad Imran Qureshi, Abid Mahboob, Choonkil Park
Computer Modeling in Engineering & Sciences, Vol.130, No.2, pp. 871-885, 2022, DOI:10.32604/cmes.2022.016836
（This article belongs to this Special Issue:
Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)
Abstract In quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) studies, computation of topological indices is a vital tool to predict biochemical and physio-chemical properties of chemical structures. Numerous topological indices have been inaugurated to describe different topological features. The ev and ve-degree are recently introduced novelties, having stronger prediction ability. In this article, we derive formulae of the ev-degree and ve-degree based topological indices for chemical structure of
Si2C3 −
I[
a,
b].
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Open Access
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Approximation by Szász Type Operators Involving Apostol-Genocchi Polynomials
Mine Menekşe Yılmaz
Computer Modeling in Engineering & Sciences, Vol.130, No.1, pp. 287-297, 2022, DOI:10.32604/cmes.2022.017385
（This article belongs to this Special Issue:
Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)
Abstract The goal of this paper is to give a form of the operator involving the generating function of Apostol-Genocchi polynomials of order
α. Applying the Korovkin theorem, we arrive at the convergence of the operator with the aid of moments and central moments. We determine the rate of convergence of the operator using several tools such as -functional, modulus of continuity, second modulus of continuity. We also give a type of Voronovskaya theorem for estimating error. Moreover, we investigate some results about convergence properties of the operator in a weighted space. Finally, we give numerical examples to support our theorems…
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Attractive Multistep Reproducing Kernel Approach for Solving Stiffness Differential Systems of Ordinary Differential Equations and Some Error Analysis
Radwan Abu-Gdairi, Shatha Hasan, Shrideh Al-Omari, Mohammad Al-Smadi, Shaher Momani
Computer Modeling in Engineering & Sciences, Vol.130, No.1, pp. 299-313, 2022, DOI:10.32604/cmes.2022.017010
（This article belongs to this Special Issue:
Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)
Abstract In this paper, an efficient multi-step scheme is presented based on reproducing kernel Hilbert space (RKHS) theory for solving ordinary stiff differential systems. The solution methodology depends on reproducing kernel functions to obtain analytic solutions in a uniform form for a rapidly convergent series in the posed Sobolev space. Using the Gram-Schmidt orthogonality process, complete orthogonal essential functions are obtained in a compact field to encompass Fourier series expansion with the help of kernel properties reproduction. Consequently, by applying the standard RKHS method to each subinterval, approximate solutions that converge uniformly to the exact solutions are obtained. For this purpose,…
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Some Results on Type 2 Degenerate Poly-Fubini Polynomials and Numbers
Ghulam Muhiuddin, Waseem A. Khan, Abdulghani Muhyi, Deena Al-Kadi
Computer Modeling in Engineering & Sciences, Vol.129, No.2, pp. 1051-1073, 2021, DOI:10.32604/cmes.2021.016546
（This article belongs to this Special Issue:
Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)
Abstract In this paper, we introduce type 2 degenerate poly-Fubini polynomials and derive several interesting characteristics
and properties. In addition, we define type 2 degenerate unipoly-Fubini polynomials and establish some certain
identities. Furthermore, we give some relationships between degenerate unipoly polynomials and special numbers
and polynomials. In the last section, certain beautiful zeros and graphical representations of type 2 degenerate
poly-Fubini polynomials are shown.
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Open Access
ARTICLE
Determinantal Expressions and Recursive Relations for the Bessel Zeta Function and for a Sequence Originating from a Series Expansion of the Power of Modified Bessel Function of the First Kind
Yan Hong, Bai-Ni Guo, Feng Qi
Computer Modeling in Engineering & Sciences, Vol.129, No.1, pp. 409-423, 2021, DOI:10.32604/cmes.2021.016431
（This article belongs to this Special Issue:
Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)
Abstract In the paper, by virtue of a general formula for any derivative of the ratio of two differentiable functions, with the
aid of a recursive property of the Hessenberg determinants, the authors establish determinantal expressions and
recursive relations for the Bessel zeta function and for a sequence originating from a series expansion of the power
of modified Bessel function of the first kind.
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Open Access
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Study of Degenerate Poly-Bernoulli Polynomials by λ-Umbral Calculus
Lee-Chae Jang, Dae San Kim, Hanyoung Kim, Taekyun Kim, Hyunseok Lee
Computer Modeling in Engineering & Sciences, Vol.129, No.1, pp. 393-408, 2021, DOI:10.32604/cmes.2021.016917
（This article belongs to this Special Issue:
Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)
Abstract Recently, degenerate poly-Bernoulli polynomials are defined in terms of degenerate polyexponential functions by
Kim-Kim-Kwon-Lee. The aim of this paper is to further examine some properties of the degenerate poly-Bernoulli
polynomials by using three formulas from the recently developed ‘λ-umbral calculus.’ In more detail, we represent
the degenerate poly-Bernoulli polynomials by Carlitz Bernoulli polynomials and degenerate Stirling numbers of
the first kind, by fully degenerate Bell polynomials and degenerate Stirling numbers of the first kind, and by higherorder degenerate Bernoulli polynomials and degenerate Stirling numbers of the second kind.
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Some Identities of the Higher-Order Type 2 Bernoulli Numbers and Polynomials of the Second Kind
Taekyun Kim, Dae San Kim, Dmitry V. Dolgy, Si-Hyeon Lee, Jongkyum Kwon
Computer Modeling in Engineering & Sciences, Vol.128, No.3, pp. 1121-1132, 2021, DOI:10.32604/cmes.2021.016532
（This article belongs to this Special Issue:
Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)
Abstract We introduce the higher-order type 2 Bernoulli numbers and polynomials of the second kind. In this paper, we
investigate some identities and properties for them in connection with central factorial numbers of the second
kind and the higher-order type 2 Bernoulli polynomials. We give some relations between the higher-order type 2
Bernoulli numbers of the second kind and their conjugates.
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