Some fractional-calculus results involving the generalized Lommel–Wright and related functions

Prieto, Ana Teresa; Romero, Susana Salinas de; Srivástava, H. M.

doi:10.1016/j.aml.2006.02.018

Cited by 10 publications

(7 citation statements)

References 15 publications

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“…Let us start with proving (24). Using the first relation (27), the function z −σ J 2,q ν,σ (z) can be represented as follows…”

Section: Theorem 1 Let Z ∈ C\(−∞ 0] Be a Complex Variable And Letmentioning

confidence: 99%

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A Survey on Bessel Type Functions as Multi-Index Mittag-Leffler Functions: Differential and Integral Relations

Paneva-Konovska¹

2019

Int. J. of Appl. Math.

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As recently observed by Bazhlekova and Dimovski [1], the nth derivative of the 2-parametric Mittag-Leffler function gives a 3-parametric Mittag-Leffler function, known as the Prabhakar function. Following the analogy, the n-th derivatives of the Bessel type functions are obtained, and it turns out that usually they are expressed in terms of the Bessel type functions with the same or more number of indices, up to a matching power function. Further, some special cases of the fractional order Riemann-Liouville derivatives and integrals of the Bessel type functions are calculated and interesting relations are proved.

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“…Let us start with proving (24). Using the first relation (27), the function z −σ J 2,q ν,σ (z) can be represented as follows…”

Section: Theorem 1 Let Z ∈ C\(−∞ 0] Be a Complex Variable And Letmentioning

confidence: 99%

“…Putting q = 1 in the equalities (24), (25) and (26), they give the corresponding results for the functions (5) and (6).…”

Section: Remarkmentioning

confidence: 99%

A Survey on Bessel Type Functions as Multi-Index Mittag-Leffler Functions: Differential and Integral Relations

Paneva-Konovska¹

2019

Int. J. of Appl. Math.

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“…[29] and [31]). Moreover, a rather systematic analysis (including interconnections) of many of the results involving (homogeneous or nonhomogeneous) linear differential equations associated with (for example) the Gauss hypergeometric equation (4.1) can be found in the works of Nishimoto et al ([37] and [38]) and the recent contribution on this subject by Wang et al [63] (see also some other recent applications considered by Lin et al [19] and Prieto et al [42]). …”

Section: Fractional Differintegral Operators Based Upon the Cauchy-gomentioning

confidence: 99%

An Elementary and Introductory Approach to Fractional Calculus and Its Applications

Srivástava¹

2017

Contributions

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A b s t r a c t: The subject of fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. It does indeed provide several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables. The main object of this paper * is to present a brief elementary and introductory approach to the theory of fractional calculus and its applications especially in developing solutions of certain interesting families of ordinary and partial fractional differintegral equations. Relevant connections of some of the results presented in this lecture with those obtained in many other earlier works on this subject will also be indicated.

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“…The last important issue that we should like to emphasize here is that all of the generalizations or modifications of BFs are, in fact, special cases of the Fox-Wright functions. Generally, the Fox-Wright functions can be represented as [62][63][64]…”

Section: Introductionmentioning

confidence: 99%

“…where > 0; , ∈ C; n ∈ N. It is interesting to note that J (z) = J 1,1 ,0 (z). • Generalized BF J (z) that is defined by 63,64…”

Section: Introductionmentioning

confidence: 99%

Generalized Bessel functions: Theory and their applications

Khosravian-Arab

Dehghan

Eslahchi

2017

Math Methods in App Sciences

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This paper presents 2 new classes of the Bessel functions on a compact domain[0, T] as generalized-tempered Bessel functions of the first-and second-kind which are denoted by GTBFs-1 and GTBFs-2. Two special cases corresponding to the GTBFs-1 and GTBFs-2 are considered. We first prove that these functions are as the solutions of 2 linear differential operators and then show that these operators are self-adjoint on suitable domains. Some interesting properties of these sets of functions such as orthogonality, completeness, fractional derivatives and integrals, recursive relations, asymptotic formulas, and so on are proved in detail. Finally, these functions are performed to approximate some functions and also to solve 3 practical differential equations of fractionalorders. KEYWORDS Bagley-Torvik equation, Bessel functions, fractional derivatives and integrals, Gaussian quadrature rules, relaxation-oscilation equation, self-adjoint operator, spectral Galerkin method, tempered fractional derivatives and integrals Math Meth Appl Sci. 2017;40:6389-6410.wileyonlinelibrary.com/journal/mma Esmaeili and Shamsi 30 were the first authors who introduced a new class of the Jacobi polynomials to solve some fractional differential equations. It is worthwhile noting that the new set, generally due to the nonpolynomial nature, is in fact, a set of orthogonal basis functions. These basis functions are also performed to solve some fractional optimal control problems and calculus of variations. [31][32][33] Later, Zayernouri and Karniadakis 34 introduced 2 new classes of Jacobi functions, which are orthogonal with respect to some suitable weight functions, as the eigenfunctions of some fractional Sturm-Liouville eigenvalue problems on [−1, 1]. They also used these basis functions to solve various fractional differential equations. (See for instance Zayernouri and Karniadakis 35-37 and other papers of the same authors. Also see Chen et al 38 and references therein.) After them, Khosravian-Arab et al 39 introduced 2 new classes of generalized Laguerre functions, which are also orthogonal with respect to some weight functions on [0, ∞).As we are aware, one of the most interesting and very practical functions which arises frequently in mathematical physics is the Bessel function(s) (BF(s)). From the historical point of view, the BFs arose in Daniel Bernoulli's investigation of the oscillations of a hanging chain, Euler's theory of the vibration of a circular membrane, and Bessel's studies of planetary motion. Recently, several applications of BFs have been discovered in physics and engineering such as propagation of waves, elasticity, fluid motion, and in many problems of potential theory and diffusion involving cylindrical symmetry. [40][41][42] Although, the BFs are usually known as the solutions of a second-order differential equation, so-called as Bessel's differential equation, which arises by the use of the separation of the variables for various problems in mathematical physics, but, in fact, there is a very close connection betw...

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Some fractional-calculus results involving the generalized Lommel–Wright and related functions

Cited by 10 publications

References 15 publications

A Survey on Bessel Type Functions as Multi-Index Mittag-Leffler Functions: Differential and Integral Relations

A Survey on Bessel Type Functions as Multi-Index Mittag-Leffler Functions: Differential and Integral Relations

An Elementary and Introductory Approach to Fractional Calculus and Its Applications

Generalized Bessel functions: Theory and their applications

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