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17 pages, 430 KB

Open AccessArticle

Inhomogeneous Whittaker Equation with Initial and Boundary Conditions

by M. S. Abu Zaytoon, Hannah Al Ali and M. H. Hamdan

Mathematics 2025, 13(17), 2770; https://doi.org/10.3390/math13172770 - 28 Aug 2025

Viewed by 182

In this study, a semi-analytical solution to the inhomogeneous Whittaker equation is developed for both initial and boundary value problems. A new class of special integral functions

{}^{f}{Zi}_{κ, μ} (x)

, along with their derivatives, is introduced to [...] Read more.

In this study, a semi-analytical solution to the inhomogeneous Whittaker equation is developed for both initial and boundary value problems. A new class of special integral functions

{}^{f}{Zi}_{κ, μ} (x)

, along with their derivatives, is introduced to facilitate the construction of the solution. The analytical properties of

{}^{f}{Zi}_{κ, μ} (x)

are rigorously investigated, and explicit closed-form expressions for

{}^{f}{Zi}_{κ, μ} (x)

and its derivatives are derived in terms of Whittaker functions

M_{κ, μ} (z)

and

W_{κ, μ} (z)

, confluent hypergeometric functions, and other special functions including Bessel functions, modified Bessel functions, and the incomplete gamma functions, along with their respective derivatives. These expressions are obtained for specific parameter values using symbolic computation in Maple. The results contribute to the broader analytical framework for solving inhomogeneous linear differential equations with applications in engineering, mathematical physics, and biological modeling. Full article

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17 pages, 329 KB

Open AccessArticle

On Some Formulas for Single and Double Integral Transforms Related to the Group SO(2, 2)

by I. A. Shilin and Junesang Choi

Symmetry 2024, 16(9), 1102; https://doi.org/10.3390/sym16091102 - 23 Aug 2024

Viewed by 1148

Abstract

We present a novel proof, using group theory, for a Meijer transform formula. This proof reveals the formula as a specific case of a broader generalized result. The generalization is achieved through a linear operator that intertwines two representations of the connected component [...] Read more.

We present a novel proof, using group theory, for a Meijer transform formula. This proof reveals the formula as a specific case of a broader generalized result. The generalization is achieved through a linear operator that intertwines two representations of the connected component of the identity of the group

SO (2, 2)

. Using this same approach, we derive a formula for the sum of three double integral transforms, where the kernels are represented by Bessel functions. It is particularly noteworthy that the group

SO (2, 2)

is connected to symmetry in several significant ways, especially in mathematical physics and geometry. Full article

(This article belongs to the Special Issue Discussion of Properties and Applications of Integral Transform)

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14 pages, 5257 KB

Open AccessEditor’s ChoiceArticle

Simple Method of Light Field Calculation for Shaping of 3D Light Curves

by Svetlana N. Khonina, Alexey P. Porfirev, Sergey G. Volotovskiy, Andrey V. Ustinov and Sergey V. Karpeev

Photonics 2023, 10(8), 941; https://doi.org/10.3390/photonics10080941 - 17 Aug 2023

Cited by 5 | Viewed by 2013

Abstract

We propose a method for generating three-dimensional light fields with given intensity and phase distributions using purely phase transmission functions. The method is based on a generalization of the well-known approach to the design of diffractive optical elements that focus an incident laser [...] Read more.

We propose a method for generating three-dimensional light fields with given intensity and phase distributions using purely phase transmission functions. The method is based on a generalization of the well-known approach to the design of diffractive optical elements that focus an incident laser beam into an array of light spots in space. To calculate purely phase transmission functions, we use amplitude encoding, which made it possible to implement the designed elements using a single spatial light modulator. The generation of light beams in the form of rings, spirals, Lissajous figures, and multi-petal “rose” distributions uniformly elongated along the optical axis in the required segment is demonstrated. It is also possible to control the three-dimensional structure of the intensity and phase of the shaped light fields along the propagation axis. The experimentally generated intensity distributions are in good agreement with the numerically obtained results and show high potential for the application of the proposed method in laser manipulation with nano- and microparticles, as well as in laser material processing. Full article

(This article belongs to the Special Issue Light Focusing and Optical Vortices)

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30 pages, 457 KB

Open AccessArticle

Infinite Series and Logarithmic Integrals Associated to Differentiation with Respect to Parameters of the Whittaker W_κ,μ(x) Function II

by Alexander Apelblat and Juan Luis González-Santander

Axioms 2023, 12(4), 382; https://doi.org/10.3390/axioms12040382 - 16 Apr 2023

Viewed by 1658

Abstract

In the first part of this investigation, we considered the parameter differentiation of the Whittaker function

M_{κ, μ} (x)

. In this second part, first derivatives with respect to the parameters of the Whittaker function

W_{κ, μ} (x)

are [...] Read more.

In the first part of this investigation, we considered the parameter differentiation of the Whittaker function

M_{κ, μ} (x)

. In this second part, first derivatives with respect to the parameters of the Whittaker function

W_{κ, μ} (x)

are calculated. Using the confluent hypergeometric function, these derivatives can be expressed as infinite sums of quotients of the digamma and gamma functions. Furthermore, it is possible to obtain these parameter derivatives in terms of infinite integrals, with integrands containing elementary functions (products of algebraic, exponential, and logarithmic functions), from the integral representation of

W_{κ, μ} (x)

. These infinite sums and integrals can be expressed in closed form for particular values of the parameters. Finally, an integral representation of the integral Whittaker function

{wi}_{κ, μ} (x)

and its derivative with respect to

κ

, as well as some reduction formulas for the integral Whittaker functions

{Wi}_{κ, μ} (x)

and

{wi}_{κ, μ} (x)

, are calculated. Full article

(This article belongs to the Special Issue Dedicated to Professor Hari Mohan Srivastava on the Occasion of His 82nd Birthday)

29 pages, 433 KB

Open AccessArticle

Infinite Series and Logarithmic Integrals Associated to Differentiation with Respect to Parameters of the Whittaker M_κ_,_μ(x) Function I

by Alexander Apelblat and Juan Luis González-Santander

Axioms 2023, 12(4), 381; https://doi.org/10.3390/axioms12040381 - 16 Apr 2023

Viewed by 1741

Abstract

In this paper, first derivatives of the Whittaker function

M_{κ, μ} (x)

are calculated with respect to the parameters. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of quotients of the digamma and gamma functions. Moreover, [...] Read more.

In this paper, first derivatives of the Whittaker function

M_{κ, μ} (x)

are calculated with respect to the parameters. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of quotients of the digamma and gamma functions. Moreover, from the integral representation of

M_{κ, μ} (x)

it is possible to obtain these parameter derivatives in terms of finite and infinite integrals with integrands containing elementary functions (products of algebraic, exponential, and logarithmic functions). These infinite sums and integrals can be expressed in closed form for particular values of the parameters. For this purpose, we have obtained the parameter derivative of the incomplete gamma function in closed form. As an application, reduction formulas for parameter derivatives of the confluent hypergeometric function are derived, along with finite and infinite integrals containing products of algebraic, exponential, logarithmic, and Bessel functions. Finally, reduction formulas for the Whittaker functions

M_{κ, μ} (x)

and integral Whittaker functions

{Mi}_{κ, μ} (x)

and

{mi}_{κ, μ} (x)

are calculated. Full article

(This article belongs to the Special Issue Dedicated to Professor Hari Mohan Srivastava on the Occasion of His 82nd Birthday)

24 pages, 420 KB

Open AccessArticle

Unified Theory of Zeta-Functions Allied to Epstein Zeta-Functions and Associated with Maass Forms

by Nianliang Wang, Takako Kuzumaki and Shigeru Kanemitsu

Mathematics 2023, 11(4), 917; https://doi.org/10.3390/math11040917 - 11 Feb 2023

Viewed by 1873

Abstract

In this paper, we shall establish a hierarchy of functional equations (as a G-function hierarchy) by unifying zeta-functions that satisfy the Hecke functional equation and those corresponding to Maass forms in the framework of the ramified functional equation with (essentially) two gamma [...] Read more.

In this paper, we shall establish a hierarchy of functional equations (as a G-function hierarchy) by unifying zeta-functions that satisfy the Hecke functional equation and those corresponding to Maass forms in the framework of the ramified functional equation with (essentially) two gamma factors through the Fourier–Whittaker expansion. This unifies the theory of Epstein zeta-functions and zeta-functions associated to Maass forms and in a sense gives a method of construction of Maass forms. In the long term, this is a remote consequence of generalizing to an arithmetic progression through perturbed Dirichlet series. Full article

(This article belongs to the Special Issue Analytic Methods in Number Theory and Allied Fields)

32 pages, 495 KB

Open AccessArticle

A Unifying Principle in the Theory of Modular Relations

by Guodong Liu, Kalyan Chakraborty and Shigeru Kanemitsu

Mathematics 2023, 11(3), 535; https://doi.org/10.3390/math11030535 - 19 Jan 2023

Cited by 1 | Viewed by 1951

Abstract

The Voronoĭ summation formula is known to be equivalent to the functional equation for the square of the Riemann zeta function in case the function in question is the Mellin tranform of a suitable function. There are some other famous summation formulas which [...] Read more.

The Voronoĭ summation formula is known to be equivalent to the functional equation for the square of the Riemann zeta function in case the function in question is the Mellin tranform of a suitable function. There are some other famous summation formulas which are treated as independent of the modular relation. In this paper, we shall establish a far-reaching principle which furnishes the following. Given a zeta function

Z (s)

satisfying a suitable functional equation, one can generalize it to

Z_{f} (s)

in the form of an integral involving the Mellin transform

F (s)

of a certain suitable function

f (x)

and process it further as

{\tilde{Z}}_{f} (s)

. Under the condition that

F (s)

is expressed as an integral, and the order of two integrals is interchangeable, one can obtain a closed form for

{\tilde{Z}}_{f} (s)

. Ample examples are given: the Lipschitz summation formula, Koshlyakov’s generalized Dedekind zeta function and the Plana summation formula. In the final section, we shall elucidate Hamburger’s results in light of RHBM correspondence (i.e., through Fourier–Whittaker expansion). Full article

(This article belongs to the Special Issue Analytic Methods in Number Theory and Allied Fields)

34 pages, 577 KB

Open AccessArticle

The Integral Mittag-Leffler, Whittaker and Wright Functions

by Alexander Apelblat and Juan Luis González-Santander

Mathematics 2021, 9(24), 3255; https://doi.org/10.3390/math9243255 - 15 Dec 2021

Cited by 11 | Viewed by 2928

Abstract

Integral Mittag-Leffler, Whittaker and Wright functions with integrands similar to those which already exist in mathematical literature are introduced for the first time. For particular values of parameters, they can be presented in closed-form. In most reported cases, these new integral functions are [...] Read more.

Integral Mittag-Leffler, Whittaker and Wright functions with integrands similar to those which already exist in mathematical literature are introduced for the first time. For particular values of parameters, they can be presented in closed-form. In most reported cases, these new integral functions are expressed as generalized hypergeometric functions but also in terms of elementary and special functions. The behavior of some of the new integral functions is presented in graphical form. By using the MATHEMATICA program to obtain infinite sums that define the Mittag-Leffler, Whittaker, and Wright functions and also their corresponding integral functions, these functions and many new Laplace transforms of them are also reported in the Appendices for integral and fractional values of parameters. Full article

(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)

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