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Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling

Submission Deadline: 31 December 2021 (closed)

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.
  • ARTICLE

    k-Order Fibonacci Polynomials on AES-Like Cryptology

    Mustafa Asci, Suleyman Aydinyuz* CMES-Computer Modeling in Engineering & Sciences, Vol., , pp. 1-17, 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. More >

  • ARTICLE

    Degenerate s-Extended Complete and Incomplete Lah-Bell Polynomials

    Hye Kyung Kim1,*, Dae Sik Lee2 CMES-Computer Modeling in Engineering & Sciences, Vol., , pp. 1-17, 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… More >

  • ARTICLE

    Modeling the Spread of Tuberculosis with Piecewise Differential Operators

    Abdon Atangana1,2, Ilknur Koca3,* CMES-Computer Modeling in Engineering & Sciences, Vol., , pp. 1-28, 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… More >

  • ARTICLE

    On Degenerate Array Type Polynomials

    Lan Wu1, Xue-Yan Chen1, Muhammet Cihat Dağli2, Feng Qi3,4,* CMES-Computer Modeling in Engineering & Sciences, Vol., , pp. 1-11, 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. More >

  • ARTICLE

    Note on a New Construction of Kantorovich Form q-Bernstein Operators Related to Shape Parameter λ

    Qingbo Cai1, Reşat Aslan2,* CMES-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: )
    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… More >

  • ARTICLE

    Lacunary Generating Functions of Hybrid Type Polynomials in Viewpoint of Symbolic Approach

    Nusrat Raza1, Umme Zainab2 and Serkan Araci3,* CMES-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: )
    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. More >

  • ARTICLE

    Some Formulas Involving Hypergeometric Functions in Four Variables

    Hassen Aydi1,2,3, Ashish Verma4, Jihad Younis5, Jung Rye Lee6,* CMES-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: )
    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. More >

  • ARTICLE

    On ev and ve-Degree Based Topological Indices of Silicon Carbides

    Jung Rye Lee1, Aftab Hussain2, Asfand Fahad3, Ali Raza3, Muhammad Imran Qureshi3,*, Abid Mahboob4, Choonkil Park5 CMES-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: )
    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 Si2C3I[a,b]. More >

  • ARTICLE

    Approximation by Szász Type Operators Involving Apostol-Genocchi Polynomials

    Mine Menekşe Yılmaz* CMES-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: )
    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… More >

  • ARTICLE

    Attractive Multistep Reproducing Kernel Approach for Solving Stiffness Differential Systems of Ordinary Differential Equations and SomeError Analysis

    Radwan Abu-Gdairi1, Shatha Hasan2, Shrideh Al-Omari3,*, Mohammad Al-Smadi2,4, Shaher Momani4,5 CMES-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: )
    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,… More >

  • ARTICLE

    Some Results on Type 2 Degenerate Poly-Fubini Polynomials and Numbers

    Ghulam Muhiuddin1,*, Waseem A. Khan2, Abdulghani Muhyi3, Deena Al-Kadi4 CMES-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: )
    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. More >

  • 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 Hong1, Bai-Ni Guo2,*, Feng Qi3 CMES-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: )
    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. More >

  • ARTICLE

    Study of Degenerate Poly-Bernoulli Polynomials by λ-Umbral Calculus

    Lee-Chae Jang1, Dae San Kim2, Hanyoung Kim3, Taekyun Kim3,*, Hyunseok Lee3 CMES-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: )
    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. More >

  • ARTICLE

    Some Identities of the Higher-Order Type 2 Bernoulli Numbers and Polynomials of the Second Kind

    Taekyun Kim1,*, Dae San Kim2, Dmitry V. Dolgy3, Si-Hyeon Lee1, Jongkyum Kwon4,* CMES-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: )
    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. More >

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