EXCEL-orientated procedure for calculating the values of special functions  with interval argument assigned on the hyperbolic form

Aim of the work is to propose the main terms of the EXCEL-orientated procedures for calculating the values of elementary and special functions with interval argument that is assigned on the hyperbolic form. The results of the work. The methods of presenting the interval values in the hyperbolic form and the rules of addition, subtraction, multiplication, and division of this values were considered. The procedures of calculating the function values, whose arguments can be degenerate or interval values were described. Namely, the direct and the reverse functions of the linear trigonometry, the direct and the reverse functions of the hyperbolic trigonometry, exponential function, arbitrary exponential function and power function, Gamma-function, incomplete Gamma-function, digamma-function, trigamma-function, tetragamma-function, pentagamma-function, Beta-function and its partial derivatives, integral exponential function, integral logarithm, dilogarithm, Frenel integrals, sine integral, cosine integral, hyperbolic sine integral, hyperbolic cosine integral. The basic terms of the EXCEL-orientated procedures for calculating the values of elementary and special functions with interval argument that is assigned on the hyperbolic form were proposed. The numerical examples were provided, that illustrate the application of the proposed methods.

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Extended k-Gamma and k-Beta Functions of Matrix Arguments

Mathematics ◽

10.3390/math8101715 ◽

2020 ◽

Vol 8 (10) ◽

pp. 1715

Author(s):

Ghazi S. Khammash ◽

Praveen Agarwal ◽

Junesang Choi

Keyword(s):

Gamma Function ◽

Hypergeometric Functions ◽

Special Functions ◽

Integral Formula ◽

Beta Function ◽

Integral Representations ◽

Matrix Argument ◽

Functional Relations

Various k-special functions such as k-gamma function, k-beta function and k-hypergeometric functions have been introduced and investigated. Recently, the k-gamma function of a matrix argument and k-beta function of matrix arguments have been presented and studied. In this paper, we aim to introduce an extended k-gamma function of a matrix argument and an extended k-beta function of matrix arguments and investigate some of their properties such as functional relations, inequality, integral formula, and integral representations. Also an application of the extended k-beta function of matrix arguments to statistics is considered.

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Modified Sumudu Transform and Its Properties

10.20944/preprints202012.0614.v1 ◽

2020 ◽

Author(s):

Ugur Duran

Keyword(s):

Laplace Transform ◽

Power Function ◽

Exponential Function ◽

Hyperbolic Sine ◽

Hyperbolic Cosine ◽

Sumudu Transform ◽

And Function ◽

Sumudu Transforms ◽

Dual Transform

Saif et al. (J. Math. Comput. Sci. 21 (2020) 127-135) considered modified Laplace transform and developed some of their certain properties and relations. Motivated by this work, in this paper, we define modified Sumudu transform and investigate many properties and relations including modified Sumudu transforms of the power function, sine, cosine, hyperbolic sine, hyperbolic cosine, exponential function, and function derivatives. Moreover, we attain two shifting properties and a scale preserving theorem for the modified Sumudu transform. We give modified inverse Sumudu transform and investigate some relations and examples. Furthermore, we show that the modified Sumudu transform is the theoretical dual transform to the modified Laplace transform.

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Degenerate Sumudu Transform and Its Properties

10.20944/preprints202012.0626.v1 ◽

2020 ◽

Author(s):

Ugur Duran

Keyword(s):

Laplace Transform ◽

Gamma Function ◽

Power Functions ◽

Hyperbolic Sine ◽

Hyperbolic Cosine ◽

Sumudu Transform ◽

And Function ◽

Sumudu Transforms ◽

Dual Transform ◽

Math Phys

Kim-Kim (Russ. J. Math. Phys. 2017, 24, 241-248) defined the degenerate Laplace transform and investigated some of their certain properties. Motivated by this study, in this paper, we introduce the degenerate Sumudu transform and establish some properties and relations. We derive degenerate Sumudu transforms of power functions, degenerate sine, degenerate cosine, degenerate hyperbolic sine, degenerate hyperbolic cosine, degenerate exponential function, and function derivatives. We also acquire a relationship between degenerate Sumudu transform and degenerate gamma function. Moreover, we investigate a scale preserving theorem for the degenerate Sumudu transform. Furthermore, we show that the degenerate Sumudu transform is the theoretical dual transform to the degenerate Laplace transform.

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On solutions of PDEs by using algebras

10.22541/au.161746556.68739421/v1 ◽

2021 ◽

Author(s):

ELIFALET LÓPEZ-GONZÁLEZ

Keyword(s):

Vector Field ◽

Exponential Function ◽

Laplace’S Equation ◽

Laplace's Equation ◽

Differentiable Functions ◽

Hyperbolic Sine ◽

Hyperbolic Cosine ◽

Dependent Variables ◽

Planar Vector ◽

Cosine Functions

The components of complex differentiable functions define solutions for the Laplace’s equation. In this paper we generalize this result; for each PDE of the form $Au_{xx}+Bu_{xy}+Cu_{yy}=0$ we give an affine planar vector field $\varphi$ and an associative and commutative 2D algebra with unit $\mathbb A$, with respect to which the components of all functions of the form $\mathcal L\circ\varphi$ define solutions for this PDE, where $\mathcal L$ is differentiable in the sense of Lorch with respect to $\mathbb A$. In the same way, for each PDE of the form $Au_{xx}+Bu_{xy}+Cu_{yy}+Du_x+Eu_y+Fu=0$, the components of the exponential function $e^{\varphi}$ defined with respect to $\mathbb A$, define solutions for this PDE. In the case of PDEs of the form $Au_{xx}+Bu_{xy}+Cu_{yy}+Fu=0$, sine, cosine, hyperbolic sine, and hyperbolic cosine functions can be used instead of the exponential function. Also, solutions for two dependent variables $3^{\text{th}}$ order PDEs and a $4^{\text{th}}$ order PDE are constructed.

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New fractional approaches for n-polynomial P-convexity with applications in special function theory

Advances in Difference Equations ◽

10.1186/s13662-020-03000-5 ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 1

Author(s):

Shu-Bo Chen ◽

Saima Rashid ◽

Muhammad Aslam Noor ◽

Zakia Hammouch ◽

Yu-Ming Chu

Keyword(s):

Fractional Calculus ◽

Convex Functions ◽

Special Functions ◽

Real Life ◽

Fractional Integrals ◽

Digamma Function ◽

Novel Approach ◽

Inequality Theory ◽

New Strategy ◽

Significant Mechanism

Abstract Inequality theory provides a significant mechanism for managing symmetrical aspects in real-life circumstances. The renowned distinguishing feature of integral inequalities and fractional calculus has a solid possibility to regulate continuous issues with high proficiency. This manuscript contributes to a captivating association of fractional calculus, special functions and convex functions. The authors develop a novel approach for investigating a new class of convex functions which is known as an n-polynomial $\mathcal{P}$ P -convex function. Meanwhile, considering two identities via generalized fractional integrals, provide several generalizations of the Hermite–Hadamard and Ostrowski type inequalities by employing the better approaches of Hölder and power-mean inequalities. By this new strategy, using the concept of n-polynomial $\mathcal{P}$ P -convexity we can evaluate several other classes of n-polynomial harmonically convex, n-polynomial convex, classical harmonically convex and classical convex functions as particular cases. In order to investigate the efficiency and supremacy of the suggested scheme regarding the fractional calculus, special functions and n-polynomial $\mathcal{P}$ P -convexity, we present two applications for the modified Bessel function and $\mathfrak{q}$ q -digamma function. Finally, these outcomes can evaluate the possible symmetric roles of the criterion that express the real phenomena of the problem.

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Some extensions of the Prabhu-Srivastava theorem involving the (p,q)-gamma function

Filomat ◽

10.2298/fil1714507l ◽

2017 ◽

Vol 31 (14) ◽

pp. 4507-4513

Author(s):

Ji-Cai Liu ◽

Jichun Liu

Keyword(s):

Gamma Function ◽

Digamma Function ◽

Derivatives Of

In this paper, we obtain some limit formulas for derivatives of (p,q)-gamma function and (p,q)- digamma function at their poles. These limit formulas extend the Prabhu-Srivastava theorem involving gamma function and digamma function.

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Integral transforms of the κ-Hilfer fractional derivative

10.22541/au.163830515.53068352/v1 ◽

2021 ◽

Author(s):

Felix Costa ◽

Junior Cesar Alves Soares ◽

Stefânia Jarosz

Keyword(s):

Fractional Derivative ◽

Gamma Function ◽

Fundamental Theorem ◽

Beta Function ◽

Integral Transforms ◽

Fractional Operators ◽

Riesz Fractional Derivative ◽

Hilfer Fractional Derivative ◽

Fundamental Theorem Of Calculus

In this paper, some important properties concerning the κ-Hilfer fractional derivative are discussed. Integral transforms for these operators are derived as particular cases of the Jafari transform. These integral transforms are used to derive a fractional version of the fundamental theorem of calculus. Keywords: Integral transforms, Jafari transform, κ-gamma function, κ-beta function, κ-Hilfer fractional derivative, κ-Riesz fractional derivative, κ-fractional operators.

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On Fully Degenerate Daehee Numbers and Polynomials of the Second Kind

Journal of Mathematics ◽

10.1155/2020/7893498 ◽

2020 ◽

Vol 2020 ◽

pp. 1-9

Author(s):

Sang Jo Yun ◽

Jin-Woo Park

Keyword(s):

Exponential Function ◽

Special Functions ◽

Eulerian Numbers

In a study, Carlitz introduced the degenerate exponential function and applied that function to Bernoulli and Eulerian numbers and degenerate special functions have been studied by many researchers. In this paper, we define the fully degenerate Daehee polynomials of the second kind which are different from other degenerate Daehee polynomials and derive some new and interesting identities and properties of those polynomials.

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