A certain ( p , q ) $(p,q)$ -derivative operator and associated divided differences

Aracı, Serkan; Duran, Ugur; Açıkgöz, Mehmet; Srivástava, H. M.

doi:10.1186/s13660-016-1240-8

Cited by 54 publications

(39 citation statements)

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“…Theorem 2. 10 The initial value problem (2.12) of the impulsive (p, q)-difference equation of type II can be stated as an integral equation of the form…”

Section: Theorem 29mentioning

confidence: 99%

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Impulsive quantum $(p,q)$-difference equations

et al. 2020

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In this paper we study quantum (p, q)-difference equations with impulse and initial or boundary conditions. We consider first order impulsive (p, q)-difference boundary value problems and second order impulsive (p, q)-difference initial value problems. Existence and uniqueness results are proved via Banach's fixed point theorem.for first-order (p, q)-difference equations, we refer the reader to [1]. In addition, in [2], the authors defined the second-order (p, q)-difference byThen we see that if f (t) is defined on [0, T] then the function( 1.3)If f (t) = t α , α > 0, then we have the formula t 0

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“…Theorem 2. 10 The initial value problem (2.12) of the impulsive (p, q)-difference equation of type II can be stated as an integral equation of the form…”

Section: Theorem 29mentioning

confidence: 99%

“…The (p, q)-difference of a function f on [0, ∞) is defined by The (p, q)-calculus was introduced in [3]. For some recent results, see [4][5][6][7][8][9][10] and references cited therein. For p = 1, the (p, q)-calculus is reduced to the classical q-calculus initiated by Jackson [11,12].…”

Section: Introductionmentioning

confidence: 99%

Impulsive quantum $(p,q)$-difference equations

et al. 2020

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“…Many researchers have studied the Bernoulli numbers and polynomials, Euler numbers and polynomials, Genocchi numbers and polynomials, tangent numbers and polynomials, zeta function, and Hurwitz zeta function. Recently, some generalizations of the Bernoulli numbers and polynomials, Euler numbers and polynomials, Genocchi numbers and polynomials, tangent numbers and polynomials, zeta function, and Hurwitz zeta function were introduced (see [1][2][3][4][5][6][7][8][9][10][11]). Luo and Zhou [6] introduced the l-function and q-L-function.…”

Section: Introductionmentioning

confidence: 99%

“…In [10], Simsek defined the twisted (h, q)-Bernoulli numbers and polynomials of the twisted (h, q)-zeta function and L-function. Many (p, q)-extensions of some special numbers, polynomials, and functions have been studied (see [1][2][3][4][5]). In this paper, we introduce the multiple twisted (p, q)-L-function in the complex field and Carlitz-type higher order twisted (p, q)-Euler numbers and polynomials.…”

Section: Introductionmentioning

confidence: 99%

Some Properties for Multiple Twisted (p, q)-L-Function and Carlitz’s Type Higher-Order Twisted (p, q)-Euler Polynomials

Hwang

Ryoo

2019

Mathematics

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The main goal of this paper is to study some interesting identities for the multiple twisted (p, q)-L-function in a complex field. First, we construct new generating functions of the new Carlitz-type higher order twisted (p, q)-Euler numbers and polynomials. By applying the Mellin transformation to these generating functions, we obtain integral representations of the multiple twisted (p, q)-Euler zeta function and multiple twisted (p, q)-L-function, which interpolate the Carlitz-type higher order twisted (p, q)-Euler numbers and Carlitz-type higher order twisted (p, q)-Euler polynomials at non-positive integers, respectively. Second, we get some explicit formulas and properties, which are related to Carlitz-type higher order twisted (p, q)-Euler numbers and polynomials. Third, we give some new symmetric identities for the multiple twisted (p, q)-L-function. Furthermore, we also obtain symmetric identities for Carlitz-type higher order twisted (p, q)-Euler numbers and polynomials by using the symmetric property for the multiple twisted (p, q)-L-function.

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“…Many papers pertaining to approximation theory and special functions have been presented recently (cf. [11][12][13][14][15][16][17][18][19][20][21]). The first -variant and ( , )-variant of Bernstein-Durrmeyer operators were given in [22] and [10], respectively.…”

Section: Introductionmentioning

confidence: 99%