• East Asian mathematical journal
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• v.34 no.1
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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.

• East Asian mathematical journal
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• v.32 no.1
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• pp.35-41
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• 2016

We introduce formal symbols of product type and of minimum type and show that the formal power series representation for $e^p$ is a formal symbol of product type, where p is a formal symbol of minimum type.

• Nonlinear Functional Analysis and Applications
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• v.26 no.1
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• pp.71-81
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• 2021

The main object of this present paper is to study some majorization problems for certain classes of analytic functions defined by means of q-calculus operator associated with exponential function.

• Kyungpook Mathematical Journal
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• v.53 no.3
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• pp.349-369
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• 2013

The present paper envisages the $q$-analogue of the exponential operators, determined by G. Dattoli and his collaborators for translation and diffusive operators which were utilized to establish analytical solutions of difference and integral equations. The generalization of their technique is expected to cover wide range of such utilization.

• Journal of the Chungcheong Mathematical Society
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• v.29 no.1
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• pp.73-82
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• 2016

The utility of exponential generating functions is that they are relevant for combinatorial problems involving sets and subsets. Sequences of polynomials play a fundamental role in applied mathematics, such sequences can be described using the exponential generating functions. The actuarial polynomials ${\alpha}^{({\beta})}_n(x)$, n = 0, 1, 2, ${\cdots}$, which was suggested by Toscano, have the following exponential generating function: $${\limits\sum^{\infty}_{n=0}}{\frac{{\alpha}^{({\beta})}_n(x)}{n!}}t^n={\exp}({\beta}t+x(1-e^t))$$. A linear functional on polynomial space can be identified with a formal power series. The set of formal power series is usually given the structure of an algebra under formal addition and multiplication. This algebra structure, the additive part of which agree with the vector space structure on the space of linear functionals, which is transferred from the space of the linear functionals. The algebra so obtained is called the umbral algebra, and the umbral calculus is the study of this algebra. In this paper, we investigate some umbral representations in the actuarial polynomials.

• Journal of applied mathematics & informatics
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• v.36 no.3_4
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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.

• Bulletin of the Korean Mathematical Society
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• v.56 no.4
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• pp.977-992
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• 2019

Mittag-Leffler stability of nonlinear fractional nabla difference systems is defined and the Lyapunov direct method is employed to provide sufficient conditions for Mittag-Leffler stability of, and in some cases the stability of, the zero solution of a system nonlinear fractional nabla difference equations. For this purpose, we obtain several properties of the exponential and one parameter Mittag-Leffler functions of fractional nabla calculus. Two examples are provided to illustrate the applicability of established results.

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.257-273
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• 2010

One can construct a theory of probability of fractional order in which the exponential function is replaced by the Mittag-Leffler function. In this framework, it seems of interest to generalize some useful classical mathematical tools, so that they are more suitable in fractional calculus. After a short background on fractional calculus based on modified Riemann Liouville derivative, one summarizes some definitions on probability density of fractional order (for the motive), and then one introduces successively fractional Euler's integrals (first and second kind) and fractional Hermite polynomials. Some properties of the Gaussian density of fractional order are exhibited. The fractional probability so introduced exhibits some relations with quantum probability.

• Journal of the Semiconductor & Display Technology
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• v.23 no.2
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• pp.60-64
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• 2024

The threshold voltage shift in thin-film transistors (TFTs) is modeled using stretched-exponential (SE) and stretched-hyperbola (SH) functions. These models are derived by introducing empirical parameters into reaction rate equations that describe defect generation or charge trapping caused by hydrogen diffusion in the dielectric or interface. Separately, the dielectric relaxation phenomena are also described by the same reaction rate equations based on defect diffusion. Dielectric relaxation was initially modeled using the SE model, and various models have been proposed using fractional calculus. In this study, the characteristics of the threshold voltage shift and the dielectric relaxation phenomena are compared and analyzed to explore the applicability of analytical models used in the field of dielectric relaxation, in addition to the conventional SE and SH models.

• School Mathematics
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• v.19 no.1
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• pp.95-116
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• 2017

Through the teaching experiments, we investigated a series of processes for obtaining the differential coefficient at x=0 of the exponential function $y=2^x$ based on the process of constructing the natural constant e and the understanding of it. and all of the students who participated in this study were students who had no experience of calculating the derivative of the exponential function. The purpose of this study was not to generalize the responses of students but to suggest implications for mathematical concept mapping related to calculus by analyzing various responses of students participating in experiments. It is expected that the accumulation of research data derived in this kind of research on the way of understanding and composition of learners will be an important basic data for presenting the learning model related to calculus.