Skip to main content

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

  1. This paper deals with the derivation of some new dynamic Hilbert-type inequalities in time scale nabla calculus. In proving the results, the basic idea is to use some algebraic inequalities, Hölder’s inequalit...

    Authors: H. M. Rezk, Ghada AlNemer, H. A. Abd El-Hamid, Abdel-Haleem Abdel-Aty, Kottakkaran Sooppy Nisar and M. Zakarya

    Citation: Advances in Difference Equations 2020 2020:619

    Content type: Research

    Published on:

  2. Motivated by a recent study on certain families of the incomplete H-functions (Srivastava et al. in Russ. J. Math. Phys. 25(1):116–138, 2018), we aim to investigate and develop several interesting properties rela...

    Authors: K. Jangid, R. K. Parmar, R. Agarwal and Sunil D. Purohit

    Citation: Advances in Difference Equations 2020 2020:606

    Content type: Research

    Published on:

  3. We provide an effective finite element method to solve the Schrödinger eigenvalue problem with an inverse potential on a spherical domain. To overcome the difficulties caused by the singularities of coefficien...

    Authors: Yubing Sui, Donghao Zhang, Junying Cao and Jun Zhang

    Citation: Advances in Difference Equations 2020 2020:582

    Content type: Research

    Published on:

  4. In the article, we introduce the generalized exponentially μ-preinvex function, derive a new q-integral identity for second order q-differentiable function, and establish several new q-trapezoidal type integral i...

    Authors: Muhammad Uzair Awan, Sadia Talib, Artion Kashuri, Muhammad Aslam Noor, Khalida Inayat Noor and Yu-Ming Chu

    Citation: Advances in Difference Equations 2020 2020:575

    Content type: Research

    Published on:

  5. In this paper, we introduce one discrete random variable, namely the negative λ-binomial random variable. We deduce the expectation of the negative λ-binomial random variable. We also get the variance and explici...

    Authors: Yuankui Ma and Taekyun Kim

    Citation: Advances in Difference Equations 2020 2020:569

    Content type: Research

    Published on:

  6. Dedekind sums occur in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. In 1892, Dedekind showed a reciprocity relation for the Dedekind sum...

    Authors: Taekyun Kim, Dae San Kim, Hyunseok Lee and Lee-Chae Jang

    Citation: Advances in Difference Equations 2020 2020:563

    Content type: Research

    Published on:

  7. In this paper, we assess an integral containing incomplete H-functions and utilize it to build up an expansion formula for the incomplete H-functions including the Bessel function. Next, we evaluate an integral c...

    Authors: Sapna Meena, Sanjay Bhatter, Kamlesh Jangid and Sunil Dutt Purohit

    Citation: Advances in Difference Equations 2020 2020:562

    Content type: Research

    Published on:

  8. In this paper, we present two new unified integral formulas involving a generalized V-function. Some interesting special cases of the main results are also considered in the form of corollaries. Due to the gen...

    Authors: S. Chandak, S. K. Q. Al-Omari and D. L. Suthar

    Citation: Advances in Difference Equations 2020 2020:560

    Content type: Research

    Published on:

  9. A spline is a piecewise defined special function that is usually comprised of polynomials of a certain degree. These polynomials are supposed to generate a smooth curve by connecting at given data points. In t...

    Authors: Muhammad Kashif Iqbal, Muhammad Abbas, Tahir Nazir and Nouman Ali

    Citation: Advances in Difference Equations 2020 2020:558

    Content type: Research

    Published on:

  10. A Bézier model with shape parameters is one of the momentous research topics in geometric modeling and computer-aided geometric design. In this study, a new recursive formula in explicit expression is construc...

    Authors: Sidra Maqsood, Muhammad Abbas, Kenjiro T. Miura, Abdul Majeed and Azhar Iqbal

    Citation: Advances in Difference Equations 2020 2020:550

    Content type: Research

    Published on:

  11. The article considers several polynomials induced by admissible lower triangular matrices and studies their subordination properties. The concept generalizes the notion of stable functions in the unit disk. Se...

    Authors: Saiful R. Mondal, Kottakkaran Sooppy Nisar and Thabet Abdeljawad

    Citation: Advances in Difference Equations 2020 2020:538

    Content type: Research

    Published on:

  12. Recently, the nth Lah–Bell number was defined as the number of ways a set of n elements can be partitioned into nonempty linearly ordered subsets for any nonnegative integer n. Further, as natural extensions of t...

    Authors: Yuankui Ma, Dae San Kim, Taekyun Kim, Hanyoung Kim and Hyunseok Lee

    Citation: Advances in Difference Equations 2020 2020:510

    Content type: Research

    Published on:

  13. Recently, special functions of fractional order calculus have had many applications in various areas of mathematical analysis, physics, probability theory, optimization theory, graph theory, control systems, e...

    Authors: M. Zayed, M. Hidan, M. Abdalla and M. Abul-Ez

    Citation: Advances in Difference Equations 2020 2020:506

    Content type: Research

    Published on:

  14. In this article, we give a definition and discuss several properties of the δ-β-Gabor integral operator in a class of locally integrable Boehmians. We derive delta sequences, convolution products and establish a ...

    Authors: Shrideh Khalaf Al-Omari, Dumitru Baleanu and Kottakkaran Sooppy Nisar

    Citation: Advances in Difference Equations 2020 2020:500

    Content type: Research

    Published on:

  15. In the present work, a numerical technique for solving a general form of nonlinear fractional order integro-differential equations (GNFIDEs) with linear functional arguments using Chebyshev series is presented...

    Authors: Khalid K. Ali, Mohamed A. Abd El Salam, Emad M. H. Mohamed, Bessem Samet, Sunil Kumar and M. S. Osman

    Citation: Advances in Difference Equations 2020 2020:494

    Content type: Research

    Published on:

  16. The aim of the paper is to derive certain formulas involving integral transforms and a family of generalized Wright functions, expressed in terms of the generalized Wright hypergeometric function and in terms ...

    Authors: Nabiullah Khan, Talha Usman, Mohd Aman, Shrideh Al-Omari and Serkan Araci

    Citation: Advances in Difference Equations 2020 2020:491

    Content type: Research

    Published on:

  17. We investigate non-Gaussian noise second-order filtering and fixed-order smoothing problems for non-Gaussian stochastic discrete systems with packet dropouts. We present a novel Kalman-like nonlinear non-Gauss...

    Authors: Huihong Zhao, Zhifang Li, Bin Li and Tongxing Li

    Citation: Advances in Difference Equations 2020 2020:474

    Content type: Research

    Published on:

  18. Recently, Hoffman (Commun. Number Theory Phys. 13:529–567, 2019), Kaneko and Tsumura (Tsukuba J. Math. (in press), 2020) introduced and systematically studied two variants of multiple zeta values of level two, i....

    Authors: Junjie Quan

    Citation: Advances in Difference Equations 2020 2020:450

    Content type: Research

    Published on:

  19. In this article, we establish a new generalized q-integral identity involving a q-differentiable function. Using this new auxiliary result, we obtain some new associated quantum bounds essentially using the class...

    Authors: Muhammad Uzair Awan, Sadia Talib, Artion Kashuri, Muhammad Aslam Noor and Yu-Ming Chu

    Citation: Advances in Difference Equations 2020 2020:424

    Content type: Research

    Published on:

  20. Quasi Bézier curves (or QB-curves, for short) possess the excellent geometric features of classical Bézier curves and also have good shape adjustability. In this paper, an algorithm for a multi-degree reductio...

    Authors: Xianzhi Hu, Gang Hu, Muhammad Abbas and Md Yushalify Misro

    Citation: Advances in Difference Equations 2020 2020:413

    Content type: Research

    Published on:

  21. Recently, some mathematicians have been studying a lot of degenerate versions of special polynomials and numbers in some arithmetic and combinatorial aspects. Our research is also interested in this field. In ...

    Authors: Hye Kyung Kim and Lee-Chae Jang

    Citation: Advances in Difference Equations 2020 2020:392

    Content type: Research

    Published on:

  22. In this paper, we establish the generalized Riemann–Liouville (RL) fractional integrals in the sense of another increasing, positive, monotone, and measurable function Ψ. We determine certain new double-weighted ...

    Authors: Gauhar Rahman, Kottakkaran Sooppy Nisar, Behzad Ghanbari and Thabet Abdeljawad

    Citation: Advances in Difference Equations 2020 2020:368

    Content type: Research

    Published on:

  23. This article aims to introduce a hybrid family of 2-variable Boas–Buck-general polynomials by taking Boas–Buck polynomials as a base with the 2-variable general polynomials. These polynomials are framed within...

    Authors: Ghazala Yasmin, Hibah Islahi and Abdulghani Muhyi

    Citation: Advances in Difference Equations 2020 2020:362

    Content type: Research

    Published on:

  24. Fourier expansions of higher-order Apostol–Genocchi and Apostol–Bernoulli polynomials are obtained using Laurent series and residues. The Fourier expansion of higher-order Apostol–Euler polynomials is obtained...

    Authors: Cristina B. Corcino and Roberto B. Corcino

    Citation: Advances in Difference Equations 2020 2020:346

    Content type: Research

    Published on:

  25. In this paper, we present a new bound for the Jensen gap with the help of a Green function. Using the bound, we deduce a converse of the Hölder inequality as well. Finally, we present some applications of the ...

    Authors: Shahid Khan, Muhammad Adil Khan, Saad Ihsan Butt and Yu-Ming Chu

    Citation: Advances in Difference Equations 2020 2020:333

    Content type: Research

    Published on:

  26. 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

    Content type: Research

    Published on:

  27. 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

    Content type: Research

    Published on:

  28. 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

    Content type: Research

    Published on:

  29. 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

    Content type: Research

    Published on:

  30. 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

    Content type: Research

    Published on:

  31. 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

    Content type: Research

    Published on:

  32. 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

    Content type: Research

    Published on:

  33. 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

    Content type: Research

    Published on:

  34. 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

    Content type: Research

    Published on:

  35. 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

    Content type: Research

    Published on:

  36. 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

    Content type: Research

    Published on:

  37. 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

    Content type: Research

    Published on: