• East Asian mathematical journal
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• 제34권1호
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• pp.29-37
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• 2018

We introduce formal symbols of product type, of zero class, and of minimum type and show that the formal power series representations for $e^p$ and $e^q$ are formal symbols of product type giving the same pseudodifferential operator, where p and q are formal symbols of minimum type and p - q is of zero class.

• 대한수학회논문집
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• 제38권3호
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• pp.755-772
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• 2023

Our aim is to establish certain image formulas of the (p, q)-extended modified Bessel function of the second kind M_ν,p,q(z) by employing the Marichev-Saigo-Maeda fractional calculus (integral and differential) operators including their composition formulas and using certain integral transforms involving (p, q)-extended modified Bessel function of the second kind M_ν,p,q(z). Corresponding assertions for the Saigo's, Riemann-Liouville (R-L) and Erdélyi-Kober (E-K) fractional integral and differential operators are deduced. All the results are represented in terms of the Hadamard product of the (p, q)-extended modified Bessel function of the second kind M_ν,p,q(z) and Fox-Wright function _rΨ_s(z).

• Journal of applied mathematics & informatics
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• 제36권3_4호
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• pp.271-283
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• 2018

In this paper, we introduce various first order (p, q)-difference equations. We investigate solutions to equations which are linear (p, q)-difference equations and nonlinear (p, q)-difference equations. We also find some properties of (p, q)-calculus, exponential functions, and inverse function.

• Journal of applied mathematics & informatics
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• 제36권5_6호
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• pp.505-519
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• 2018

In this paper we define a (p, q)-Laplace transform. By using this definition, we obtain many properties including the linearity, scaling, translation, transform of derivatives, derivative of transforms, transform of integrals and so on. Finally, we solve the differential equation using the (p, q)-Laplace transform.

• Nonlinear Functional Analysis and Applications
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• 제28권4호
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• pp.1069-1086
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• 2023

The natural transform is represented by two (p, q)-analogues, and their comparative characteristics are established. To resolve some (p, q)-difference and functional equations, applications are carried out.

• Kyungpook Mathematical Journal
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• 제61권1호
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• pp.75-98
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• 2021

In this paper, we introduce new classes of post-quantum or (p, q)-starlike and convex functions with respect to symmetric points associated with a cardiod-shaped domain. We obtain (p, q)-Fekete-Szegö inequalities for functions in these classes. We also obtain estimates of initial (p, q)-logarithmic coefficients. In addition, we get q-Bieberbachde-Branges type inequalities for the special case of our classes when p = 1. Moreover, we also discuss some special cases of the obtained results.

• Kyungpook Mathematical Journal
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• 제63권3호
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• pp.463-484
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• 2023

An examination is conducted on the multinomial coefficients derived from generalized quantum deformed algebras, and on their recurrence relations. The 𝓡(p, q)-deformed multinomial probability distribution and the negative 𝓡(p, q)-deformed multinomial probability distribution are constructed, and the recurrence relations are determined. From our general result, we deduce particular cases that correspond to quantum algebras considered in the literature.

• 대한수학회보
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• 제55권2호
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• pp.599-610
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• 2018

We aim to present some formulas for Saigo hypergeometric fractional integral and differential operators involving (p, q)-extended Bessel function $J_{{\nu},p,q}(z)$, which are expressed in terms of Hadamard product of the (p, q)-extended Gauss hypergeometric function and the Fox-Wright function $_p{\Psi}_q(z)$. A number of interesting special cases of our main results are also considered. Further, it is emphasized that the results presented here, which are seemingly complicated series, can reveal their involved properties via those of the two known functions in their respective Hadamard product.

• Korean Journal of Mathematics
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• 제29권2호
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• pp.253-266
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• 2021

This work is devoted to the study of a q-harmonic analysis related to the q-analog of the Bessel-Wright integral transform [6]. We establish some important properties of this transform and we focalise our attention in studying the associated transmutation operator.

• Journal of applied mathematics & informatics
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• 제36권3_4호
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• pp.285-299
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• 2018

In this article, we present the Stancu generalization of (p, q)-$Sz{\acute{a}}sz$-Mirakyan Kantorovich type linear positive operators. Using Korovkin's result, approximation properties are investigated. First, we evaluate moments and direct results. By choosing p and q, the convergence rate have been estimated for better approximation. For the particular case ${\alpha}=0$, ${\beta}=0$ we obtain results for (p, q)-$Sz{\acute{a}}sz$-Mirakyan Kantorovich type operators.