Editorial

4 pages, 178 KiB

Open AccessEditorial

Recent Advances in Special Functions and Their Applications

by Junesang Choi

Symmetry 2023, 15(12), 2159; https://doi.org/10.3390/sym15122159 - 05 Dec 2023

Viewed by 853

Due to their remarkable properties, a plethora of special functions have been crafted and harnessed across a diverse spectrum of fields spanning centuries [...] Full article

(This article belongs to the Special Issue Recent Advances in Special Functions and Their Applications)

Research

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17 pages, 340 KiB

Open AccessArticle

New Formulas Involving Fibonacci and Certain Orthogonal Polynomials

by Waleed Mohamed Abd-Elhameed, Hany M. Ahmed, Anna Napoli and Victor Kowalenko

Symmetry 2023, 15(3), 736; https://doi.org/10.3390/sym15030736 - 16 Mar 2023

Cited by 3 | Viewed by 1141

Abstract

In this paper, new formulas for the Fibonacci polynomials, including high-order derivatives and repeated integrals of them, are derived in terms of the polynomials themselves. The results are then used to solve connection problems between the Fibonacci and orthogonal polynomials. The inverse cases [...] Read more.

In this paper, new formulas for the Fibonacci polynomials, including high-order derivatives and repeated integrals of them, are derived in terms of the polynomials themselves. The results are then used to solve connection problems between the Fibonacci and orthogonal polynomials. The inverse cases are also studied. Finally, new results for the linear products of the Fibonacci and orthogonal polynomials are determined using the earlier result for the moments formula of Fibonacci polynomials. Full article

(This article belongs to the Special Issue Recent Advances in Special Functions and Their Applications)

13 pages, 286 KiB

Open AccessArticle

A Note on Certain General Transformation Formulas for the Appell and the Horn Functions

by Insuk Kim and Arjun K. Rathie

Symmetry 2023, 15(3), 696; https://doi.org/10.3390/sym15030696 - 10 Mar 2023

Cited by 2 | Viewed by 899

Abstract

In a number of problems in applied mathematics, physics (theoretical and mathematical), statistics, and other fields the hypergeometric functions of one and several variables naturally appear. Hypergeometric functions in one and several variables have several known applications today. The Appell’s four functions and [...] Read more.

In a number of problems in applied mathematics, physics (theoretical and mathematical), statistics, and other fields the hypergeometric functions of one and several variables naturally appear. Hypergeometric functions in one and several variables have several known applications today. The Appell’s four functions and the Horn’s functions have shown to be particularly useful in providing solutions to a variety of problems in both pure and applied mathematics. The Hubbell rectangular source and its generalization, non-relativistic theory, and the hydrogen dipole matrix elements are only a few examples of the numerous scientific and chemical domains where Appell functions are used. The Appell series is also used in quantum field theory, especially in the evaluation of Feynman integrals. Additionally, since 1985, computational sciences such as artificial intelligence (AI) and information processing (IP) have used the well-known Horn functions as a key idea. In literature, there have been published a significant number of results of double series in particular of Appell and Horn functions in a series of interesting and useful research publications. We find three general transformation formulas between Appell functions

F_{2}

and

F_{4}

and two general transformation formulas between Appell function

F_{2}

and Horn function

H_{4}

in the present study, which are mostly inspired by their work and naturally exhibit symmetry. By using the generalizations of the Kummer second theorem in the integral representation of the Appell function

F_{2}

, this is accomplished. As special cases of our main findings, both previously known and new results have been found. Full article

(This article belongs to the Special Issue Recent Advances in Special Functions and Their Applications)

11 pages, 266 KiB

Open AccessArticle

Some New Results for the Kampé de Fériet Function with an Application

by Insuk Kim, Richard B. Paris and Arjun K. Rathie

Symmetry 2022, 14(12), 2588; https://doi.org/10.3390/sym14122588 - 07 Dec 2022

Cited by 1 | Viewed by 950

Abstract

The generalized hypergeometric functions in one and several variables and their natural generalizations appear in many mathematical problems and their applications. The theory of generalized hypergeometric functions in several variables comes from the fact that the solutions of the partial differential equations appearing [...] Read more.

The generalized hypergeometric functions in one and several variables and their natural generalizations appear in many mathematical problems and their applications. The theory of generalized hypergeometric functions in several variables comes from the fact that the solutions of the partial differential equations appearing in a large number of applied problems of mathematical physics have been expressed in terms of such generalized hypergeometric functions. In particular, the Kampé de Fériet function (in two variables) has proved its practical utility in representing solutions to a wide range of problems in pure and applied mathematics, statistics, and mathematical physics. In this context, in a very recent paper, Progri successfully calculated the

_{2} F_{2}

generalized hypergeometric function for a particular set of parameters and expressed the result in terms of the difference between two Kampé de Fériet functions. Inspired by his work, in the present paper, we obtain three results for a terminating

_{3} F_{2}

series of arguments 1 and 2, together with a transformation formula of a

_{3} F_{2} (z)

generalized hypergeometric function in terms of the difference between two Kampé de Fériet functions. One application of this result is also provided. The paper concludes with six reduction formulas for the Kampé de Fériet function. Of note, symmetry occurs naturally in the generalized hypergeometric functions

_{p} F_{q}

and the Kampé de Fériet function involving two variables, which are the two most important functions discussed in this paper. Full article

(This article belongs to the Special Issue Recent Advances in Special Functions and Their Applications)

12 pages, 322 KiB

Open AccessArticle

Lagrange-Based Hypergeometric Bernoulli Polynomials

by Sahar Albosaily, Yamilet Quintana, Azhar Iqbal and Waseem A. Khan

Symmetry 2022, 14(6), 1125; https://doi.org/10.3390/sym14061125 - 30 May 2022

Cited by 4 | Viewed by 1234

Abstract

Special polynomials play an important role in several subjects of mathematics, engineering, and theoretical physics. Many problems arising in mathematics, engineering, and mathematical physics are framed in terms of differential equations. In this paper, we introduce the family of the Lagrange-based hypergeometric Bernoulli [...] Read more.

Special polynomials play an important role in several subjects of mathematics, engineering, and theoretical physics. Many problems arising in mathematics, engineering, and mathematical physics are framed in terms of differential equations. In this paper, we introduce the family of the Lagrange-based hypergeometric Bernoulli polynomials via the generating function method. We state some algebraic and differential properties for this family of extensions of the Lagrange-based Bernoulli polynomials, as well as a matrix-inversion formula involving these polynomials. Moreover, a generating relation involving the Stirling numbers of the second kind was derived. In fact, future investigations in this subject could be addressed for the potential applications of these polynomials in the aforementioned disciplines. Full article

(This article belongs to the Special Issue Recent Advances in Special Functions and Their Applications)

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12 pages, 292 KiB

Open AccessArticle

Certain Generalizations of Quadratic Transformations of Hypergeometric and Generalized Hypergeometric Functions

by Mohd Idris Qureshi, Junesang Choi and Tafaz Rahman Shah

Symmetry 2022, 14(5), 1073; https://doi.org/10.3390/sym14051073 - 23 May 2022

Cited by 3 | Viewed by 1328

Abstract

There have been numerous investigations on the hypergeometric series

_{2} F_{1}

and the generalized hypergeometric series

_{p} F_{q}

such as differential equations, integral representations, analytic continuations, asymptotic expansions, reduction cases, extensions of one and several variables, continued fractions, Riemann’s equation, group [...] Read more.

There have been numerous investigations on the hypergeometric series

_{2} F_{1}

and the generalized hypergeometric series

_{p} F_{q}

such as differential equations, integral representations, analytic continuations, asymptotic expansions, reduction cases, extensions of one and several variables, continued fractions, Riemann’s equation, group of the hypergeometric equation, summation, and transformation formulae. Among the various approaches to these functions, the transformation formulae for the hypergeometric series

_{2} F_{1}

and the generalized hypergeometric series

_{p} F_{q}

are significant, both in terms of applications and theory. The purpose of this paper is to establish a number of transformation formulae for

_{p} F_{q}

, whose particular cases would include Gauss’s and Kummer’s quadratic transformation formulae for

_{2} F_{1}

, as well as their two extensions for

_{3} F_{2}

, by making advantageous use of a recently introduced sequence and some techniques commonly used in dealing with

_{p} F_{q}

theory. The

_{p} F_{q}

function, which is the most significant function investigated in this study, exhibits natural symmetry. Full article

(This article belongs to the Special Issue Recent Advances in Special Functions and Their Applications)

14 pages, 277 KiB

Open AccessArticle

Another Method for Proving Certain Reduction Formulas for the Humbert Function ψ₂ Due to Brychkov et al. with an Application

by Asmaa O. Mohammed, Adem Kilicman, Mohamed M. Awad, Arjun K. Rathie and Medhat A. Rakha

Symmetry 2022, 14(5), 868; https://doi.org/10.3390/sym14050868 - 23 Apr 2022

Cited by 1 | Viewed by 1591

Abstract

Recently, Brychkov et al. established several new and interesting reduction formulas for the Humbert functions (the confluent hypergeometric functions of two variables). The primary objective of this study was to provide an alternative and simple approach for proving four reduction formulas for the [...] Read more.

Recently, Brychkov et al. established several new and interesting reduction formulas for the Humbert functions (the confluent hypergeometric functions of two variables). The primary objective of this study was to provide an alternative and simple approach for proving four reduction formulas for the Humbert function

ψ_{2}

. We construct intriguing series comprising the product of two confluent hypergeometric functions as an application. Numerous intriguing new and previously known outcomes are also achieved as specific instances of our primary discoveries. It is well-known that the hypergeometric functions in one and two variables and their confluent forms occur naturally in a wide variety of problems in applied mathematics, statistics, operations research, physics (theoretical and mathematical) and engineering mathematics, so the results established in this paper may be potentially useful in the above fields. Symmetry arises spontaneously in the abovementioned functions. Full article

(This article belongs to the Special Issue Recent Advances in Special Functions and Their Applications)

13 pages, 381 KiB

Open AccessArticle

Certain Integral Formulae Associated with the Product of Generalized Hypergeometric Series and Several Elementary Functions Derived from Formulas for the Beta Function

by Junesang Choi, Shantha Kumari Kurumujji, Adem Kilicman and Arjun Kumar Rathie

Symmetry 2022, 14(2), 389; https://doi.org/10.3390/sym14020389 - 15 Feb 2022

Cited by 1 | Viewed by 1717

Abstract

The literature has an astonishingly large number of integral formulae involving a range of special functions. In this paper, by using three Beta function formulae, we aim to establish three integral formulas whose integrands are products of the generalized hypergeometric series [...] Read more.

The literature has an astonishingly large number of integral formulae involving a range of special functions. In this paper, by using three Beta function formulae, we aim to establish three integral formulas whose integrands are products of the generalized hypergeometric series

_{p + 1} F_{p}

and the integrands of the three Beta function formulae. Among the many particular instances for our formulae, several are stated clearly. Moreover, an intriguing inequality that emerges throughout the proving procedure is shown. It is worth noting that the three integral formulae shown here may be expanded further by using a variety of more generalized special functions than

_{p + 1} F_{p}

. Symmetry occurs naturally in the Beta and

_{p + 1} F_{p}

functions, which are two of the most important functions discussed in this study. Full article

(This article belongs to the Special Issue Recent Advances in Special Functions and Their Applications)

11 pages, 264 KiB

Open AccessArticle

Some Explicit Expressions for Twisted Catalan-Daehee Numbers

by Dongkyu Lim

Symmetry 2022, 14(2), 189; https://doi.org/10.3390/sym14020189 - 19 Jan 2022

Cited by 3 | Viewed by 950

Abstract

In this paper, the author considers the twisted Catalan numbers and the twisted Catalan-Daehee numbers, which are arisen from p-adic fermionic integrals and p-adic invariant integrals, respectively. We give some explicit identities and properties for those twisted numbers and polynomials by [...] Read more.

In this paper, the author considers the twisted Catalan numbers and the twisted Catalan-Daehee numbers, which are arisen from p-adic fermionic integrals and p-adic invariant integrals, respectively. We give some explicit identities and properties for those twisted numbers and polynomials by using p-adic integrals or generating functions. Full article

(This article belongs to the Special Issue Recent Advances in Special Functions and Their Applications)

11 pages, 276 KiB

Open AccessArticle

A Note on the Summation of the Incomplete Gamma Function

by Robert Reynolds and Allan Stauffer

Symmetry 2021, 13(12), 2369; https://doi.org/10.3390/sym13122369 - 09 Dec 2021

Cited by 7 | Viewed by 2048

Abstract

We examine the improved infinite sum of the incomplete gamma function for large values of the parameters involved. We also evaluate the infinite sum and equivalent Hurwitz-Lerch zeta function at special values and produce a table of results for easy reading. Almost all [...] Read more.

We examine the improved infinite sum of the incomplete gamma function for large values of the parameters involved. We also evaluate the infinite sum and equivalent Hurwitz-Lerch zeta function at special values and produce a table of results for easy reading. Almost all Hurwitz-Lerch zeta functions have an asymmetrical zero distribution. Full article

(This article belongs to the Special Issue Recent Advances in Special Functions and Their Applications)

9 pages, 257 KiB

Open AccessArticle

Maximal Type Elements for Families

by Donal O’Regan

Symmetry 2021, 13(12), 2269; https://doi.org/10.3390/sym13122269 - 29 Nov 2021

Cited by 3 | Viewed by 966

Abstract

In this paper, we present a variety of existence theorems for maximal type elements in a general setting. We consider multivalued maps with continuous selections and multivalued maps which are admissible with respect to Gorniewicz and our existence theory is based on the [...] Read more.

In this paper, we present a variety of existence theorems for maximal type elements in a general setting. We consider multivalued maps with continuous selections and multivalued maps which are admissible with respect to Gorniewicz and our existence theory is based on the author’s old and new coincidence theory. Particularly, for the second section we present presents a collectively coincidence coercive type result for different classes of maps. In the third section we consider considers majorized maps and presents a variety of new maximal element type results. Coincidence theory is motivated from real-world physical models where symmetry and asymmetry play a major role. Full article

(This article belongs to the Special Issue Recent Advances in Special Functions and Their Applications)

13 pages, 274 KiB

Open AccessArticle

Finite Sums Involving Reciprocals of the Binomial and Central Binomial Coefficients and Harmonic Numbers

by Necdet Batir and Anthony Sofo

Symmetry 2021, 13(11), 2002; https://doi.org/10.3390/sym13112002 - 22 Oct 2021

Cited by 3 | Viewed by 1585

Abstract

We prove some finite sum identities involving reciprocals of the binomial and central binomial coefficients, as well as harmonic, Fibonacci and Lucas numbers, some of which recover previously known results, while the others are new. Full article

(This article belongs to the Special Issue Recent Advances in Special Functions and Their Applications)

13 pages, 1293 KiB

Open AccessArticle

Some Symmetric Properties and Location Conjecture of Approximate Roots for (p,q)-Cosine Euler Polynomials

by Cheon Seoung Ryoo and Jung Yoog Kang

Symmetry 2021, 13(8), 1520; https://doi.org/10.3390/sym13081520 - 18 Aug 2021

Cited by 3 | Viewed by 1211

Abstract

In this paper, we introduce

(p, q)

-cosine Euler polynomials. From these polynomials, we find several properties and identities. Moreover, we find the circle equations of approximate roots for

(p, q)

-cosine Euler polynomials by using a computer. Full article

(This article belongs to the Special Issue Recent Advances in Special Functions and Their Applications)

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22 pages, 5257 KiB

Open AccessArticle

q-Generalized Tangent Based Hybrid Polynomials

by Ghazala Yasmin, Hibah Islahi and Junesang Choi

Symmetry 2021, 13(5), 791; https://doi.org/10.3390/sym13050791 - 03 May 2021

Cited by 3 | Viewed by 1282

Abstract

In this paper, we incorporate two known polynomials to introduce so-called 2-variable q-generalized tangent based Apostol type Frobenius–Euler polynomials. Next we present a number of properties and formulas for these polynomials such as explicit expressions, series representations, summation formulas, addition formula, q [...] Read more.

In this paper, we incorporate two known polynomials to introduce so-called 2-variable q-generalized tangent based Apostol type Frobenius–Euler polynomials. Next we present a number of properties and formulas for these polynomials such as explicit expressions, series representations, summation formulas, addition formula, q-derivative and q-integral formulas, together with numerous particular cases of the new polynomials and their associated formulas demonstrated in two tables. Further, by using computer-aided programs (for example, Mathematica or Matlab), we draw graphs of some particular cases of the new polynomials, mainly, in order to observe in several angles how zeros of these polynomials are distributed and located. Lastly we provide numerous observations and questions which naturally arise amid the present investigation. Full article

(This article belongs to the Special Issue Recent Advances in Special Functions and Their Applications)

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