# 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. ### Five semi analytical and numerical simulations for the fractional nonlinear space-time telegraph equation

The accuracy of analytical obtained solutions of the fractional nonlinear space–time telegraph equation that has been constructed in (Hamed and Khater in J. Math., 2020) is checked through five recent semi-analyt...

Authors: Mostafa M. A. Khater, Choonkil Park, Jung Rye Lee, Mohamed S. Mohamed and Raghda A. M. Attia

Citation: Advances in Difference Equations 2021 2021:227

Content type: Research

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2. ### On the ω-multiple Charlier polynomials

Authors: Mehmet Ali Özarslan and Gizem Baran

Citation: Advances in Difference Equations 2021 2021:119

Content type: Research

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3. ### Upper semicontinuity of attractors for nonclassical diffusion equations with arbitrary polynomial growth

In this paper, we mainly investigate upper semicontinuity and regularity of attractors for nonclassical diffusion equations with perturbed parameters ν and the nonlinear term f satisfying the polynomial growth of...

Authors: Yongqin Xie, Jun Li and Kaixuan Zhu

Citation: Advances in Difference Equations 2021 2021:75

Content type: Research

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4. ### Some extensions for the several combinatorial identities

In this paper, we give some extensions for Mortenson’s identities in series with the Bell polynomial using the partial fraction decomposition. As applications, we obtain some combinatorial identities involving...

Authors: Gao-Wen Xi and Qiu-Ming Luo

Citation: Advances in Difference Equations 2021 2021:38

Content type: Research

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5. ### On the weighted fractional integral inequalities for Chebyshev functionals

The goal of this present paper is to study some new inequalities for a class of differentiable functions connected with Chebyshev’s functionals by utilizing a fractional generalized weighted fractional integra...

Authors: Gauhar Rahman, Kottakkaran Sooppy Nisar, Sami Ullah Khan, Dumitru Baleanu and V. Vijayakumar

Citation: Advances in Difference Equations 2021 2021:18

Content type: Research

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6. ### Approximation of functions by a class of Durrmeyer–Stancu type operators which includes Euler’s beta function

Authors: Abdullah Alotaibi, Faruk Özger, S. A. Mohiuddine and Mohammed A. Alghamdi

Citation: Advances in Difference Equations 2021 2021:13

Content type: Research

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7. ### The q-Sumudu transform and its certain properties in a generalized q-calculus theory

In this paper we consider a generalization to the q-calculus theory in the space of q-integrable functions. We introduce q-delta sequences and develop q-convolution products to derive certain q-convolution theore...

Authors: Shrideh Khalaf Al-Omari

Citation: Advances in Difference Equations 2021 2021:10

Content type: Research

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8. ### Novel finite point approach for solving time-fractional convection-dominated diffusion equations

In this paper, a stabilized numerical method with high accuracy is proposed to solve time-fractional singularly perturbed convection-diffusion equation with variable coefficients. The tailored finite point met...

Authors: Xiaomin Liu, Muhammad Abbas, Honghong Yang, Xinqiang Qin and Tahir Nazir

Citation: Advances in Difference Equations 2021 2021:4

Content type: Research

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9. ### Some inequalities for Csiszár divergence via theory of time scales

In this paper, we present some inequalities for Csiszár f-divergence between two probability measures on time scale. These results extend some known results in the literature and offer new results in h-discrete c...

Authors: Iqrar Ansari, Khuram Ali Khan, Ammara Nosheen, Ðilda Pečarić and Josip Pečarić

Citation: Advances in Difference Equations 2020 2020:698

Content type: Research

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10. ### A new generalization of Mittag-Leffler function via q-calculus

The present paper deals with a new different generalization of the Mittag-Leffler function through q-calculus. We then investigate its remarkable properties like convergence, recurrence relation, integral represe...

Citation: Advances in Difference Equations 2020 2020:695

Content type: Research

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11. ### Value distribution of meromorphic functions concerning rational functions and differences

Let c be a nonzero constant and n a positive integer, let f be a transcendental meromorphic function of finite order, and let R be a nonconstant rational function. Under some conditions, we study the relationship...

Authors: Mingliang Fang, Degui Yang and Dan Liu

Citation: Advances in Difference Equations 2020 2020:692

Content type: Research

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12. ### Some Ramanujan-type circular summation formulas

In this paper, we give two Ramanujan-type circular summation formulas by applying the way of elliptic functions and the properties of theta functions. As applications, we obtain the corresponding imaginary tra...

Authors: Ji-Ke Ge and Qiu-Ming Luo

Citation: Advances in Difference Equations 2020 2020:690

Content type: Research

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13. ### An extension of beta function, its statistical distribution, and associated fractional operator

Recently, various forms of extended beta function have been proposed and presented by many researchers. The principal goal of this paper is to present another expansion of beta function using Appell series and...

Authors: Ankita Chandola, Rupakshi Mishra Pandey, Ritu Agarwal and Sunil Dutt Purohit

Citation: Advances in Difference Equations 2020 2020:684

Content type: Research

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14. ### Approximation by bivariate generalized Bernstein–Schurer operators and associated GBS operators

We construct the bivariate form of Bernstein–Schurer operators based on parameter α. We establish the Voronovskaja-type theorem and give an estimate of the order of approximation with the help of Peetre’s K-funct...

Authors: S. A. Mohiuddine

Citation: Advances in Difference Equations 2020 2020:676

Content type: Research

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15. ### A note on generalized q-difference equations for general Al-Salam–Carlitz polynomials

In this paper, we deduce the generalized q-difference equations for general Al-Salam–Carlitz polynomials and generalize Arjika’s recent results (Arjika in J. Differ. Equ. Appl. 26:987–999, 2020). In addition, we ...

Authors: Jian Cao, Binbin Xu and Sama Arjika

Citation: Advances in Difference Equations 2020 2020:668

Content type: Research

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16. ### A necessary and sufficient condition for sequences to be minimal completely monotonic

In this article, we present a necessary and sufficient condition under which sequences are minimal completely monotonic.

Authors: Xi-Feng Wang, Mourad E. H. Ismail, Necdet Batir and S. Guo

Citation: Advances in Difference Equations 2020 2020:665

Content type: Research

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17. ### Sequence spaces derived by the triple band generalized Fibonacci difference operator

Authors: Taja Yaying, Bipan Hazarika, S. A. Mohiuddine, M. Mursaleen and Khursheed J. Ansari

Citation: Advances in Difference Equations 2020 2020:639

Content type: Research

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18. ### Hilbert-type inequalities for time scale nabla calculus

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

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19. ### Fractional calculus and integral transforms of the product of a general class of polynomial and incomplete Fox–Wright functions

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

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20. ### On basic Horn hypergeometric functions $$\mathbf{H}_{3}$$ and $$\mathbf{H}_{4}$$

Authors: Ayman Shehata

Citation: Advances in Difference Equations 2020 2020:595

Content type: Research

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21. ### Generalizations of Hermite–Hadamard like inequalities involving $$\chi _{{\kappa }}$$-Hilfer fractional integrals

Citation: Advances in Difference Equations 2020 2020:594

Content type: Research

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22. ### An efficient finite element method and error analysis for eigenvalue problem of Schrödinger equation with an inverse square potential on spherical domain

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

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23. ### A new q-integral identity and estimation of its bounds involving generalized exponentially μ-preinvex functions

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

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24. ### A note on negative λ-binomial distribution

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

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25. ### Identities on poly-Dedekind sums

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

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26. ### Some expansion formulas for incomplete H- and H̅-functions involving Bessel functions

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

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27. ### Unified integral associated with the generalized V-function

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

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28. ### Application of new quintic polynomial B-spline approximation for numerical investigation of Kuramoto–Sivashinsky equation

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...

Citation: Advances in Difference Equations 2020 2020:558

Content type: Research

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29. ### Geometric modeling and applications of generalized blended trigonometric Bézier curves with shape parameters

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

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30. ### Some subordination involving polynomials induced by lower triangular matrices

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

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31. ### Numerical solution of certain Cauchy singular integral equations using a collocation scheme

The present study is devoted to developing a computational collocation technique for solving the Cauchy singular integral equation of the second kind (CSIE-2). Although, several studies have investigated the n...

Authors: Ali Seifi

Citation: Advances in Difference Equations 2020 2020:537

Content type: Research

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32. ### Some identities of Lah–Bell polynomials

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

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33. ### Fractional order of Legendre-type matrix polynomials

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

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34. ### δ-β-Gabor integral operators for a space of locally integrable generalized functions

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

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35. ### Generating functions for some families of the generalized Al-Salam–Carlitz q-polynomials

Authors: Hari Mohan Srivastava and Sama Arjika

Citation: Advances in Difference Equations 2020 2020:498

Content type: Research

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36. ### Numerical solution for generalized nonlinear fractional integro-differential equations with linear functional arguments using Chebyshev series

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

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37. ### Computation of certain integral formulas involving generalized Wright function

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

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38. ### A study on input noise second-order filtering and smoothing of linear stochastic discrete systems with packet dropouts

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

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39. ### Blending type approximation by τ-Baskakov-Durrmeyer type hybrid operators

Authors: S. A. Mohiuddine, Arun Kajla, M. Mursaleen and Mohammed A. Alghamdi

Citation: Advances in Difference Equations 2020 2020:467

Content type: Research

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40. ### New generating functions of I-function satisfying Truesdell’s $$F_{q}$$-equation

Authors: Altaf Ahmad Bhat, Asifa Tassaddiq, D. K. Jain and Humera Naaz

Citation: Advances in Difference Equations 2020 2020:464

Content type: Research

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41. ### Alternating double t-values and T-values

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

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42. ### Extension of generalized Fox’s H-function operator to certain set of generalized integrable functions

Authors: Shrideh Khalaf Al-Omari

Citation: Advances in Difference Equations 2020 2020:448

Content type: Research

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43. ### Iq-Calculus and Iq-Hermite–Hadamard inequalities for interval-valued functions

In this paper, we introduce the Iq-derivative and Iq-integral for interval-valued functions and give their basic properties. As a promotion of q-Hermite–Hadamard inequalities, we also give the Iq-Hermite-Hadamard...

Authors: Tianyi Lou, Guoju Ye, Dafang Zhao and Wei Liu

Citation: Advances in Difference Equations 2020 2020:446

Content type: Research

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44. ### Estimates of quantum bounds pertaining to new q-integral identity with applications

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...

Citation: Advances in Difference Equations 2020 2020:424

Content type: Research

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45. ### Approximate multi-degree reduction of Q-Bézier curves via generalized Bernstein polynomial functions

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

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46. ### A note on degenerate poly-Genocchi numbers and polynomials

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

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47. ### On generalized fractional integral inequalities for the monotone weighted Chebyshev functionals

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 ...

Citation: Advances in Difference Equations 2020 2020:368

Content type: Research

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48. ### Certain results on a hybrid class of the Boas–Buck polynomials

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

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49. ### Fourier expansions for higher-order Apostol–Genocchi, Apostol–Bernoulli and Apostol–Euler polynomials

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

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50. ### A new bound for the Jensen gap pertaining twice differentiable functions with applications

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 ...