# Topics in Special Functions and q-Special Functions: Theory, Methods, and Applications

Special functions, being natural generalizations of the elementary functions, have their origin in the solution of partial differential equations satisfying some set of conditions. Special functions can be defined in a variety of ways. Many special functions of a complex variable can be defined by means of either a series or an appropriate integral. Special functions like Bessel functions, Whittaker functions, Gauss hypergeometric function and the polynomials that go by the names of Jacobi, Legendre, Laguerre, Hermite, etc., have been continuously developed. Also, sequences of polynomials play a vital role in applied mathematics. Two important classes of polynomial sequences are the Sheffer and Appell sequences. The Appell and Sheffer polynomial sequences occur in different applications in many different branches of mathematics, theoretical physics, approximation theory, and other fields.

This special issue focuses on the applications of the special functions and polynomials to various areas of mathematics. Thorough knowledge of special functions is required in modern engineering and physical science applications. 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.

Edited by: Serkan Araci, H. M. Srivastava, Kottakkaran Sooppy Nisar

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1. ### On subclasses of analytic functions based on a quantum symmetric conformable differential operator with application

Quantum calculus (the calculus without limit) appeared for the first time in fluid mechanics, noncommutative geometry and combinatorics studies. Recently, it has been included into the field of geometric funct...

Authors: Rabha W. Ibrahim, Rafida M. Elobaid and Suzan J. Obaiys
Citation: Advances in Difference Equations 2020 2020:325
2. ### A new approach to interval-valued inequalities

The objective of this work is to advance and simplify the notion of Gronwallâ€™s inequality. By using an efficient partial order and concept of gH-differentiability on interval-valued functions, we investigate s...

Authors: Awais Younus, Muhammad Asif, Jehad Alzabut, Abdul Ghaffar and Kottakkaran Sooppy Nisar
Citation: Advances in Difference Equations 2020 2020:319
3. ### Approximation of functions in generalized Zygmund class by double Hausdorff matrix

Authors: H. K. Nigam, M. Mursaleen and Supriya Rani
Citation: Advances in Difference Equations 2020 2020:317
4. ### Numerical solution for the time-fractional Fokkerâ€“Planck equation via shifted Chebyshev polynomials of the fourth kind

This paper provides a numerical approach for solving the time-fractional Fokkerâ€“Planck equation (FFPE). The authors use the shifted Chebyshev collocation method and the finite difference method (FDM) to presen...

Authors: Haile Habenom and D. L. Suthar
Citation: Advances in Difference Equations 2020 2020:315
5. ### Modified Chebyshev collocation method for delayed predatorâ€“prey system

In this study, the approximate solutions of the predatorâ€“prey system with delay have been obtained by using the modified Chebyshev collocation method. The main technique is that this method transforms the orig...

Authors: J. Dengata and Shufang Ma
Citation: Advances in Difference Equations 2020 2020:313
6. ### Some results on degenerate Daehee and Bernoulli numbers and polynomials

In this paper, we study a degenerate version of the Daehee polynomials and numbers, namely the degenerate Daehee polynomials and numbers, which were actually called the degenerate Daehee polynomials and number...

Authors: Taekyun Kim, Dae San Kim, Han Young Kim and Jongkyum Kwon
Citation: Advances in Difference Equations 2020 2020:311
7. ### p-Adic integral on \(\mathbb{Z}_{p}\) associated with degenerate Bernoulli polynomials of the second kind

In this paper, by means of p-adic Volkenborn integrals we introduce and study two different degenerate versions of Bernoulli polynomials of the second kind, namely partially and fully degenerate Bernoulli polynom...

Authors: Lee-Chae Jang, Dae San Kim, Taekyun Kim and Hyunseok Lee
Citation: Advances in Difference Equations 2020 2020:278
8. ### New estimates considering the generalized proportional Hadamard fractional integral operators

In the article, we describe the GrÃ¼ss type inequality, provide some related inequalities by use of suitable fractional integral operators, address several variants by utilizing the generalized proportional Had...

Authors: Shuang-Shuang Zhou, Saima Rashid, Fahd Jarad, Humaira Kalsoom and Yu-Ming Chu
Citation: Advances in Difference Equations 2020 2020:275
9. ### Some fractional calculus findings associated with the incomplete I-functions

In this article, several interesting properties of the incomplete I-functions associated with the Marichevâ€“Saigoâ€“Maeda (MSM) fractional operators are studied and investigated. It is presented that the order of th...

Authors: Kamlesh Jangid, Sanjay Bhatter, Sapna Meena, Dumitru Baleanu, Maysaa Al Qurashi and Sunil Dutt Purohit
Citation: Advances in Difference Equations 2020 2020:265
10. ### A fractional derivative with two singular kernels and application to a heat conduction problem

In this article, we suggest a new notion of fractional derivative involving two singular kernels. Some properties related to this new operator are established and some examples are provided. We also present so...

Authors: Dumitru Baleanu, Mohamed Jleli, Sunil Kumar and Bessem Samet
Citation: Advances in Difference Equations 2020 2020:252
11. ### Further extension of Voigt function and its properties

In this paper, by using the confluent hypergeometric function of the first kind, we propose a further extension of the Voigt function and obtain its useful properties as (for example) explicit representation a...

Authors: Nabiullah Khan, Mohd Ghayasuddin, Waseem A. Khan, Thabet Abdeljawad and Kottakkaran Sooppy Nisar
Citation: Advances in Difference Equations 2020 2020:229
12. ### Monotonicity properties for a ratio of finite many gamma functions

In the paper, the authors consider a ratio of finite many gamma functions and find its monotonicity properties such as complete monotonicity, the Bernstein function property, and logarithmically complete monot...

Authors: Feng Qi and Dongkyu Lim
Citation: Advances in Difference Equations 2020 2020:193
13. ### Approximation by modified Kantorovichâ€“SzÃ¡sz type operators involving Charlier polynomials

In this paper, we give some direct approximation results by modified Kantorovichâ€“SzÃ¡sz type operators involving Charlier polynomials. Further, approximation results are also developed in polynomial weighted sp...

Authors: K. J. Ansari, M. Mursaleen, M. Shareef KP and M. Ghouse
Citation: Advances in Difference Equations 2020 2020:192
14. ### Degenerate polyexponential functions and type 2 degenerate poly-Bernoulli numbers and polynomials

The polyexponential functions were introduced by Hardy and rediscovered by Kim, as inverses to the polylogarithm functions. Recently, the type 2 poly-Bernoulli numbers and polynomials were defined by means of ...

Authors: Taekyun Kim, Dae San Kim, Jongkyum Kwon and Hyunseok Lee
Citation: Advances in Difference Equations 2020 2020:168
15. ### Existence of mild solutions for impulsive neutral Hilfer fractional evolution equations

In this paper, we investigate the existence of mild solutions for neutral Hilfer fractional evolution equations with noninstantaneous impulsive conditions in a Banach space. We obtain the existence results by ...

Authors: Pallavi Bedi, Anoop Kumar, Thabet Abdeljawad and Aziz Khan
Citation: Advances in Difference Equations 2020 2020:155
16. ### Fractional calculus operators with Appell function kernels applied to Srivastava polynomials and extended Mittag-Leffler function

This article aims to establish certain image formulas associated with the fractional calculus operators with Appell function in the kernel and Caputo-type fractional differential operators involving Srivastava...

Authors: Kottakkaran Sooppy Nisar, D. L. Suthar, R. Agarwal and S. D. Purohit
Citation: Advances in Difference Equations 2020 2020:148