scholarly journals A Study on Some New Results Arising from (p,q)-Calculus

2018 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz ◽  
Serkan Araci
Keyword(s):  
Exponential Function ◽  
Chain Rule ◽  
Binomial Coefficients ◽  
Final Part ◽  
Trigonometric Functions ◽  
Elementary Functions ◽  
Exponential Functions ◽  
Hyperbolic Functions ◽  
Quantum Calculus ◽  
A Chain

This paper includes some new investigations and results for post quantum calculus, denoted by (p,q)-calculus. A chain rule for (p,q)-derivative is developed. Also, a new (p,q)-analogue of the exponential function is introduced and some its properties including the addition property for (p,q)-exponential functions are investigated. Several useful results involving (p,q)-binomial coefficients and (p,q)-antiderivative are discovered. At the final part of this paper, (p,q)-analogue of some elementary functions including trigonometric functions and hyperbolic functions are considered and some properties and relations among them are analyzed extensively.

scholarly journals Research on Some New Results Arising from Multiple q-Calculus

2017 ◽  
Author(s):  
Uğur Duran ◽  
Mehmet Acikgoz ◽  
Serkan Araci
Keyword(s):  
Number Theory ◽  
Chain Rule ◽  
Trigonometric Functions ◽  
Elementary Functions ◽  
Hyperbolic Functions ◽  
Binomial Formula

In this paper, we develop the theory of the multiple q-analogue of the Heine’s binomial formula, chain rule and Leibnitz’s rule. We also derive many useful definitions and results involving multiple q-antiderivative and multiple q-Jackson’s integral. Finally, we list here multiple q-analogue of some elementary functions including trigonometric functions and hyperbolic functions. This may be a good consideration in developing the multiple q-calculus in combinatorics, number theory and other fields of mathematics.

scholarly journals Research on some new results arising from multiple q-calculus

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz ◽  
Serkan Araci
Keyword(s):  
Number Theory ◽  
Chain Rule ◽  
Trigonometric Functions ◽  
Elementary Functions ◽  
Hyperbolic Functions ◽  
Binomial Formula

In this paper, we develop the theory of the multiple q-analogue of the Heine?s binomial formula, chain rule and Leibniz?s rule. We also derive many useful definitions and results involving multiple q-antiderivative and multiple q-Jackson?s integral. Finally, we list here multiple q-analogue of some elementary functions including trigonometric functions and hyperbolic functions. This may be a good consideration in developing the multiple q-calculus in combinatorics, number theory and other fields of mathematics.

scholarly journals Relation of Some Known Functions in terms of Generalized Meijer G -Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Syed Ali Haider Shah ◽  
Shahid Mubeen ◽  
Gauhar Rahman ◽  
Jihad Younis
Keyword(s):  
Bessel Functions ◽  
Logarithmic Function ◽  
Trigonometric Functions ◽  
Modified Bessel Functions ◽  
Modified Bessel Function ◽  
Exponential Functions ◽  
Hyperbolic Functions ◽  
Binomial Expansion ◽  
Sine And Cosine Functions ◽  
Cosine Functions

The aim of this paper is to prove some identities in the form of generalized Meijer G -function. We prove the relation of some known functions such as exponential functions, sine and cosine functions, product of exponential and trigonometric functions, product of exponential and hyperbolic functions, binomial expansion, logarithmic function, and sine integral, with the generalized Meijer G -function. We also prove the product of modified Bessel function of first and second kind in the form of generalized Meijer G -function and solve an integral involving the product of modified Bessel functions.

scholarly journals The Wigner-Dunkl-Newton mechanics with time-reversal symmetry

2020 ◽  
Vol 66 (3 May-Jun) ◽  
pp. 308
Author(s):  
W. Sang Chung ◽  
H. Hassanabadi
Keyword(s):  
Time Reversal ◽  
Hamiltonian Formalism ◽  
Equation Of Motion ◽  
Time Reversal Symmetry ◽  
Elementary Functions ◽  
Exponential Functions ◽  
Hyperbolic Functions ◽  
Newton Equation

In this paper we use the Dunkl derivative with respect to time to construct theWigner-Dunkl-Newton mechanics with time-reversal symmetry. We deflne the WignerDunkl-Newton velocity and Wigner-Dunkl-Newton acceleration and construct the WignerDunkl-Newton equation of motion. We also discuss the Hamiltonian formalism in theWigner-Dunkl-Newton mechanics. We discuss some deformed elementary functions suchas the ”-deformed exponential functions, ”-deformed hyperbolic functions and ”-deformedtrigonometric functions. Using these we solve some problems in on dimensional WignerDunkl-Newton mechanics mechanics.

Defining trigonometric functions via complex sequences

2016 ◽  
Vol 100 (547) ◽  
pp. 9-23 ◽  
Author(s):  
Jan Gustavsson ◽  
Mikael P. Sundqvist
Keyword(s):  
Functional Equation ◽  
Simple Model ◽  
Exponential Function ◽  
Trigonometric Functions ◽  
Elementary Functions ◽  
Initial Value ◽  
Advantages And Disadvantages ◽  
Complex Sequences ◽  
Image Position ◽  
Compound Interest

In the literature we find several different ways of introducing elementary functions. For the exponential function, we mention the following ways of characterising the exponential function:(a) (b) , also for complex values of x;(c) x → exp (x) is the unique solution to the initial value problem [4](d) x → exp (x) is the inverse of (e)x → exp (x) is the unique continuous function satisfying thefunctional equation f (x + y) = f (x) f (y) and f(0) = 1 [6]; the corresponding definition is done for the logarithm in [7];(f) Define dr for rational r, and then use a continuity/density argument [8].All of them have their advantages and disadvantages. We like (a) and (c), mostly because they have natural interpretations, (a) in the setting of compound interest and (c) being a simple model of many processes in physics and other sciences, but also because they are related to methods and ideas that are (usually) introduced rather early to the students.

scholarly journals Parameter N Analysis on the Rational-Talbot Algorithm for Numerical Inversion of Laplace Transform

2021 ◽  
Vol 31 (2) ◽  
pp. 50-60
Author(s):  
Elisandra Freitas ◽  
George Ricardo Libardi Calixto ◽  
Juciara Alves Ferreira ◽  
Bárbara Denicol do Amaral Rodriguez ◽  
João Francisco Prolo Filho
Keyword(s):  
Laplace Transform ◽  
Free Parameter ◽  
Numerical Results ◽  
Numerical Inversion ◽  
Good Precision ◽  
Trigonometric Functions ◽  
Inversion Process ◽  
Elementary Functions ◽  
Exponential Functions ◽  
The Laplace Transform

This article investigates the numerical inversion of the Laplace Transform by the Rational-Talbot method and analyzes the influence on the variation of the free parameter N established by the technique when applied to certain functions. The set of elementary functions, for which the method is tested, has exponential and oscillatory characteristics. Based on the results obtained, it was concluded that the Rational-Talbot method is e cient for the inversion of decreasing exponential functions. At the same time, to perform the inversion process effectively for trigonometric forms, the algorithm requires a greater amount of terms in the sum. For higher values of N, the technique works well. In fact, this is observed in inverting the functions transform, that combine trigonometric and polynomial factors. The method numerical results have a good precision for the treatment of decreasing exponential functions when multiplied by trigonometric functions.

scholarly journals Application of the Efros Theorem to the Function Represented by the Inverse Laplace Transform of s−μ exp(−sν)

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 354
Author(s):  
Alexander Apelblat ◽  
Francesco Mainardi
Keyword(s):  
Laplace Transform ◽  
Operational Calculus ◽  
Inverse Laplace Transform ◽  
Elementary Functions ◽  
Hyperbolic Functions ◽  
Error Functions ◽  
Convolution Integrals ◽  
Special Case ◽  
Integral Identities ◽  
Finite Integrals

Using a special case of the Efros theorem which was derived by Wlodarski, and operational calculus, it was possible to derive many infinite integrals, finite integrals and integral identities for the function represented by the inverse Laplace transform. The integral identities are mainly in terms of convolution integrals with the Mittag–Leffler and Volterra functions. The integrands of determined integrals include elementary functions (power, exponential, logarithmic, trigonometric and hyperbolic functions) and the error functions, the Mittag–Leffler functions and the Volterra functions. Some properties of the inverse Laplace transform of s−μexp(−sν) with μ≥0 and 0<ν<1 are presented.

scholarly journals On Bleimann-Butzer-Hahn operators for exponential functions

2007 ◽  
Vol 75 (3) ◽  
pp. 409-415 ◽  
Author(s):  
Ulrich Abel ◽  
Mircea Ivan
Keyword(s):  
Exponential Type ◽  
Binomial Coefficients ◽  
Monotone Functions ◽  
Exponential Functions ◽  
Approximation Process

Some inequalities involving the binomial coefficients are obtained. They are used to determine the domain of convergence of the Bleimann, Butzer and Hahn approximation process for exponential type functions. An answer to Hermann's conjecture related to the Bleimann, Butzer and Hahn operators for monotone functions is given.

A chain rule for multivariate divided differences

2010 ◽  
Vol 50 (3) ◽  
pp. 577-586 ◽  
Author(s):  
Michael S. Floater
Keyword(s):  
Chain Rule ◽  
Divided Differences ◽  
A Chain
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