# 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. ### 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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2. ### 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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3. ### 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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4. ### 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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5. ### 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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6. ### 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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7. ### 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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8. ### 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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9. ### 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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10. ### 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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11. ### 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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12. ### 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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13. ### 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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14. ### 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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15. ### 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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16. ### 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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17. ### 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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18. ### 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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19. ### 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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20. ### 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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21. ### 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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22. ### 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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23. ### 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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24. ### 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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25. ### 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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26. ### 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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27. ### 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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28. ### 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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29. ### 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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30. ### 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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31. ### 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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32. ### 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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33. ### 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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34. ### 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 ...

Citation: Advances in Difference Equations 2020 2020:333

Content type: Research

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35. ### 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

Content type: Research

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36. ### A new approach to interval-valued inequalities

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

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37. ### 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

Content type: Research

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38. ### Numerical solution for the time-fractional Fokker–Planck equation via shifted Chebyshev polynomials of the fourth kind

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

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39. ### Modified Chebyshev collocation method for delayed predator–prey system

In this study, the approximate solutions of the predator–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

Content type: Research

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40. ### 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

Content type: Research

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41. ### 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

Content type: Research

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42. ### New estimates considering the generalized proportional Hadamard fractional integral operators

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

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43. ### Some fractional calculus findings associated with the incomplete I-functions

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

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44. ### 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

Content type: Research

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45. ### 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

Content type: Research

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46. ### 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

Content type: Research

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47. ### Approximation by modified Kantorovich–Szász type operators involving Charlier polynomials

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

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48. ### 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

Content type: Research

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49. ### 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

Content type: Research

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50. ### 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

Content type: Research

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