Symmetry

Research

12 pages, 283 KiB

Open AccessArticle

On Generating Functions for Parametrically Generalized Polynomials Involving Combinatorial, Bernoulli and Euler Polynomials and Numbers

by Abdelmejid Bayad and Yilmaz Simsek

Symmetry 2022, 14(4), 654; https://doi.org/10.3390/sym14040654 - 23 Mar 2022

Cited by 3 | Viewed by 1324

Abstract

The aim of this paper is to give generating functions for parametrically generalized polynomials that are related to the combinatorial numbers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the cosine-Bernoulli polynomials, the sine-Bernoulli polynomials, the cosine-Euler polynomials, and the sine-Euler [...] Read more.

The aim of this paper is to give generating functions for parametrically generalized polynomials that are related to the combinatorial numbers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the cosine-Bernoulli polynomials, the sine-Bernoulli polynomials, the cosine-Euler polynomials, and the sine-Euler polynomials. We investigate some properties of these generating functions. By applying Euler’s formula to these generating functions, we derive many new and interesting formulas and relations related to these special polynomials and numbers mentioned as above. Some special cases of the results obtained in this article are examined. With this special case, detailed comments and comparisons with previously available results are also provided. Furthermore, we come up with open questions about interpolation functions for these polynomials. The main results of this paper highlight the existing symmetry between numbers and polynomials in a more general framework. These include Bernouilli, Euler, and Catalan polynomials. Full article

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

19 pages, 440 KiB

Open AccessArticle

On Unconditionally Stable New Modified Fractional Group Iterative Scheme for the Solution of 2D Time-Fractional Telegraph Model

by Ajmal Ali, Thabet Abdeljawad, Azhar Iqbal, Tayyaba Akram and Muhammad Abbas

Symmetry 2021, 13(11), 2078; https://doi.org/10.3390/sym13112078 - 3 Nov 2021

Cited by 7 | Viewed by 1352

Abstract

In this study, a new modified group iterative scheme for solving the two-dimensional (2D) fractional hyperbolic telegraph differential equation with Dirichlet boundary conditions is obtained from the 2h-spaced standard and rotated Crank–Nicolson FD approximations. The findings of new four-point modified explicit group [...] Read more.

In this study, a new modified group iterative scheme for solving the two-dimensional (2D) fractional hyperbolic telegraph differential equation with Dirichlet boundary conditions is obtained from the 2h-spaced standard and rotated Crank–Nicolson FD approximations. The findings of new four-point modified explicit group relaxation method demonstrates the rapid rate of convergence of proposed method as compared to the existing schemes. Numerical tests are performed to test the capability of the group iterative scheme in comparison with the point iterative scheme counterparts. The stability of the derived modified group method is proven by the matrix norm algorithm. The obtained results are tabulated and concluded that exact solutions are exactly symmetric with approximate solutions. Full article

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

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6 pages, 260 KiB

Open AccessFeature PaperArticle

A Note on a Triple Integral

by Robert Reynolds and Allan Stauffer

Symmetry 2021, 13(11), 2056; https://doi.org/10.3390/sym13112056 - 1 Nov 2021

Cited by 1 | Viewed by 2288

Abstract

A closed form expression for a triple integral not previously considered is derived, in terms of the Lerch function. Almost all Lerch functions have an asymmetrical zero-distribution. The kernel of the integral involves the product of the logarithmic, exponential, quotient radical, and polynomial [...] Read more.

A closed form expression for a triple integral not previously considered is derived, in terms of the Lerch function. Almost all Lerch functions have an asymmetrical zero-distribution. The kernel of the integral involves the product of the logarithmic, exponential, quotient radical, and polynomial functions. Special cases are derived in terms of fundamental constants; results are summarized in a table. All results in this work are new. Full article

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

9 pages, 272 KiB

Open AccessArticle

p-Adic q-Twisted Dedekind-Type Sums

by Abdelmejid Bayad and Yilmaz Simsek

Symmetry 2021, 13(9), 1756; https://doi.org/10.3390/sym13091756 - 20 Sep 2021

Cited by 2 | Viewed by 1751

Abstract

The main purpose of this paper is to define p-adic and q-Dedekind type sums. Using the Volkenborn integral and the Teichmüller character representations of the Bernoulli polynomials, we give reciprocity law of these sums. These sums and their reciprocity law generalized [...] Read more.

The main purpose of this paper is to define p-adic and q-Dedekind type sums. Using the Volkenborn integral and the Teichmüller character representations of the Bernoulli polynomials, we give reciprocity law of these sums. These sums and their reciprocity law generalized some of the classical p-adic Dedekind sums and their reciprocity law. It is to be noted that the Dedekind reciprocity laws, is a fine study of the existing symmetry relations between the finite sums, considered in our study, and their symmetries through permutations of initial parameters. Full article

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

14 pages, 345 KiB

Open AccessArticle

Some Symmetry Identities for Carlitz’s Type Degenerate Twisted (p,q)-Euler Polynomials Related to Alternating Twisted (p,q)-Sums

by Cheon-Seoung Ryoo

Symmetry 2021, 13(8), 1371; https://doi.org/10.3390/sym13081371 - 28 Jul 2021

Cited by 2 | Viewed by 1289

Abstract

In this paper, we define a new form of Carlitz’s type degenerate twisted

(p, q)

-Euler numbers and polynomials by generalizing the degenerate Euler numbers and polynomials, Carlitz’s type degenerate q-Euler numbers and polynomials. Some interesting identities, explicit formulas, [...] Read more.

In this paper, we define a new form of Carlitz’s type degenerate twisted

(p, q)

-Euler numbers and polynomials by generalizing the degenerate Euler numbers and polynomials, Carlitz’s type degenerate q-Euler numbers and polynomials. Some interesting identities, explicit formulas, symmetric properties, a connection with Carlitz’s type degenerate twisted

(p, q)

-Euler numbers and polynomials are obtained. Finally, we investigate the zeros of the Carlitz’s type degenerate twisted

(p, q)

-Euler polynomials by using computer. Full article

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

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10 pages, 250 KiB

Open AccessArticle

On New Formulas of Fibonacci and Lucas Numbers Involving Golden Ratio Associated with Atomic Structure in Chemistry

by Rifat Battaloglu and Yilmaz Simsek

Symmetry 2021, 13(8), 1334; https://doi.org/10.3390/sym13081334 - 23 Jul 2021

Cited by 8 | Viewed by 2042

Abstract

The main purpose of this paper is to give many new formulas involving the Fibonacci numbers, the golden ratio, the Lucas numbers, and other special numbers. By using generating functions for the special numbers with their functional equations method, we also give many [...] Read more.

The main purpose of this paper is to give many new formulas involving the Fibonacci numbers, the golden ratio, the Lucas numbers, and other special numbers. By using generating functions for the special numbers with their functional equations method, we also give many new relations among the Fibonacci numbers, the Lucas numbers, the golden ratio, the Stirling numbers, and other special numbers. Moreover, some applications of the Fibonacci numbers and the golden ratio in chemistry are given. Full article

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

21 pages, 321 KiB

Open AccessArticle

On the Multilinear Fractional Transforms

by Öznur Kulak

Symmetry 2021, 13(5), 740; https://doi.org/10.3390/sym13050740 - 22 Apr 2021

Cited by 2 | Viewed by 1472

Abstract

In this paper we first introduce multilinear fractional wavelet transform on

R^{n} \times R_{+}^{n}

using Schwartz functions, i.e., infinitely differentiable complex-valued functions, rapidly decreasing at infinity. We also give multilinear fractional Fourier transform and prove the Hausdorff–Young inequality and Paley-type [...] Read more.

In this paper we first introduce multilinear fractional wavelet transform on

R^{n} \times R_{+}^{n}

using Schwartz functions, i.e., infinitely differentiable complex-valued functions, rapidly decreasing at infinity. We also give multilinear fractional Fourier transform and prove the Hausdorff–Young inequality and Paley-type inequality. We then study boundedness of the multilinear fractional wavelet transform on Lebesgue spaces and Lorentz spaces. Full article

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

16 pages, 569 KiB

Open AccessArticle

New Method for Generating New Families of Distributions

by Lamya A. Baharith and Wedad H. Aljuhani

Symmetry 2021, 13(4), 726; https://doi.org/10.3390/sym13040726 - 20 Apr 2021

Cited by 5 | Viewed by 1905

Abstract

This article presents a new method for generating distributions. This method combines two techniques—the transformed—transformer and alpha power transformation approaches—allowing for tremendous flexibility in the resulting distributions. The new approach is applied to introduce the alpha power Weibull—exponential distribution. The density of this [...] Read more.

This article presents a new method for generating distributions. This method combines two techniques—the transformed—transformer and alpha power transformation approaches—allowing for tremendous flexibility in the resulting distributions. The new approach is applied to introduce the alpha power Weibull—exponential distribution. The density of this distribution can take asymmetric and near-symmetric shapes. Various asymmetric shapes, such as decreasing, increasing, L-shaped, near-symmetrical, and right-skewed shapes, are observed for the related failure rate function, making it more tractable for many modeling applications. Some significant mathematical features of the suggested distribution are determined. Estimates of the unknown parameters of the proposed distribution are obtained using the maximum likelihood method. Furthermore, some numerical studies were carried out, in order to evaluate the estimation performance. Three practical datasets are considered to analyze the usefulness and flexibility of the introduced distribution. The proposed alpha power Weibull–exponential distribution can outperform other well-known distributions, showing its great adaptability in the context of real data analysis. Full article

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

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13 pages, 272 KiB

Open AccessArticle

Symmetric Properties for Dirichlet-Type Multiple (p,q)-L-Function

by Kyung-Won Hwang, Ravi P. Agarwal and Cheon-Seoung Ryoo

Symmetry 2021, 13(1), 95; https://doi.org/10.3390/sym13010095 - 7 Jan 2021

Cited by 1 | Viewed by 1368

Abstract

The main aim of this article is to investigate some interesting symmetric identities for the Dirichlet-type multiple

(p, q)

-L function. We use this function to examine the symmetry of the generalized higher-order

(p, q)

-Euler [...] Read more.

The main aim of this article is to investigate some interesting symmetric identities for the Dirichlet-type multiple

(p, q)

-L function. We use this function to examine the symmetry of the generalized higher-order

(p, q)

-Euler polynomials related to

χ

. First, the generalized higher-order

(p, q)

-Euler numbers and polynomials related to

χ

are defined. We also give a few new symmetric properties for the Dirichlet-type multiple

(p, q)

-L-function and generalized higher-order

(p, q)

-Euler polynomials related to

χ

. Full article

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

8 pages, 224 KiB

Open AccessArticle

Representation of Integers as Sums of Fibonacci and Lucas Numbers

by Ho Park, Bumkyu Cho, Durkbin Cho, Yung Duk Cho and Joonsang Park

Symmetry 2020, 12(10), 1625; https://doi.org/10.3390/sym12101625 - 1 Oct 2020

Cited by 6 | Viewed by 2291

Abstract

Motivated by the Elementary Problem B-416 in the Fibonacci Quarterly, we show that, given any integers n and r with

n \geq 2

, every positive integer can be expressed as a sum of Fibonacci numbers whose indices are distinct integers not congruent [...] Read more.

Motivated by the Elementary Problem B-416 in the Fibonacci Quarterly, we show that, given any integers n and r with

n \geq 2

, every positive integer can be expressed as a sum of Fibonacci numbers whose indices are distinct integers not congruent to r modulo n. Similar expressions are also dealt with for the case of Lucas numbers. Symmetric and anti-symmetric properties of Fibonacci and Lucas numbers are used in the proofs. Full article

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

18 pages, 1883 KiB

Open AccessArticle

Data Analysis of Beach Sands’ Chemical Analysis Using Multivariate Statistical Methods and Heavy Metal Distribution Maps: The Case of Moonlight Beach Sands, Kemer, Antalya, Turkey

by Fusun Yalcin

Symmetry 2020, 12(9), 1538; https://doi.org/10.3390/sym12091538 - 17 Sep 2020

Cited by 11 | Viewed by 2595

Abstract

Multivariate statistical methods are widely used in several disciplines of fundamental sciences. In the present study, the data analysis of the chemical analysis of the sands of Moonlight Beach in the Kemer region was examined using multivariate statistical methods. This study consists of [...] Read more.

Multivariate statistical methods are widely used in several disciplines of fundamental sciences. In the present study, the data analysis of the chemical analysis of the sands of Moonlight Beach in the Kemer region was examined using multivariate statistical methods. This study consists of three parts. The multivariate statistical analysis tests were described in the first part, then the pollution indexes were studied in the second part. Finally, the distribution maps of the chemical analyses and pollution indexes were generated using the obtained data. The heavy metals were mostly observed in location K1, while they were sorted as follows based on their concentrations: Mg > Fe > Al > Ti > Sr > Mn > Cr > Ni > Zn > Zr > Cu > Rb. Also, strong positive correlations were found between Si, Fe, Al, K, Ti, P. According to the results of factor analysis, it was found that four factors explained 83.5% of the total variance. On the other hand, the coefficient of determination (R²) was calculated as 63.6% in the regression model. Each unit increase in the value of Ti leads to an increase of 0.022 units in the value of Si. Potential Ecological Risk Index analysis results (RI < 150) revealed that the study area had no risk. However, the locations around Moonlight Beach are under risk in terms of Enrichment Factor and Contamination Factor values. The index values of heavy metals in the anomaly maps and their densities were found to be successful; and higher densities were observed based on heavy metal anomalies. Full article

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

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18 pages, 897 KiB

Open AccessArticle

Certain Identities Associated with (p,q)-Binomial Coefficients and (p,q)-Stirling Polynomials of the Second Kind

by Talha Usman, Mohd Saif and Junesang Choi

Symmetry 2020, 12(9), 1436; https://doi.org/10.3390/sym12091436 - 31 Aug 2020

Cited by 16 | Viewed by 2344

Abstract

The q-Stirling numbers (polynomials) of the second kind have been investigated and applied in a variety of research subjects including, even, the q-analogue of Bernstein polynomials. The

(p, q)

-Stirling numbers (polynomials) of the second kind have been [...] Read more.

The q-Stirling numbers (polynomials) of the second kind have been investigated and applied in a variety of research subjects including, even, the q-analogue of Bernstein polynomials. The

(p, q)

-Stirling numbers (polynomials) of the second kind have been studied, particularly, in relation to combinatorics. In this paper, we aim to introduce new

(p, q)

-Stirling polynomials of the second kind which are shown to be fit for the

(p, q)

-analogue of Bernstein polynomials. We also present some interesting identities involving the

(p, q)

-binomial coefficients. We further discuss certain vanishing identities associated with the q-and

(p, q)

-Stirling polynomials of the second kind. Full article

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

13 pages, 460 KiB

Open AccessArticle

Fibonacci Graphs

by Aysun Yurttas Gunes, Sadik Delen, Musa Demirci, Ahmet Sinan Cevik and Ismail Naci Cangul

Symmetry 2020, 12(9), 1383; https://doi.org/10.3390/sym12091383 - 19 Aug 2020

Cited by 6 | Viewed by 3817

Abstract

Apart from its applications in Chemistry, Biology, Physics, Social Sciences, Anthropology, etc., there are close relations between graph theory and other areas of Mathematics. Fibonacci numbers are of utmost interest due to their relation with the golden ratio and also due to many [...] Read more.

Apart from its applications in Chemistry, Biology, Physics, Social Sciences, Anthropology, etc., there are close relations between graph theory and other areas of Mathematics. Fibonacci numbers are of utmost interest due to their relation with the golden ratio and also due to many applications in different areas from Biology, Architecture, Anatomy to Finance. In this paper, we define Fibonacci graphs as graphs having degree sequence consisting of n consecutive Fibonacci numbers and use the invariant

Ω

to obtain some more information on these graphs. We give the necessary and sufficient conditions for the realizability of a set D of n successive Fibonacci numbers for every n and also list all possible realizations called Fibonacci graphs for