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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. 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. The objective of this work is to advance and simplify the notion of Gronwall’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. This paper provides a numerical approach for solving the time-fractional Fokker–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
  4. 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
  5. In the article, we describe the Grüss 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
  6. In this article, several interesting properties of the incomplete I-functions associated with the Marichev–Saigo–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
  7. 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
  8. 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
  9. In this paper, we give some direct approximation results by modified Kantorovich–Szász 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
  10. 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
  11. 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
  12. 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