Research

14 pages, 334 KiB

Open AccessArticle

Some Symmetric Identities for the Multiple (p, q)-Hurwitz-Euler eta Function

by Kyung-Won Hwang and Cheon Seoung Ryoo

Symmetry 2019, 11(5), 645; https://doi.org/10.3390/sym11050645 - 08 May 2019

Cited by 5 | Viewed by 2217

The main purpose of this paper is to find some interesting symmetric identities for the

(p, q)

-Hurwitz-Euler eta function in a complex field. Firstly, we define the multiple

(p, q)

-Hurwitz-Euler eta function by generalizing the [...] Read more.

The main purpose of this paper is to find some interesting symmetric identities for the

(p, q)

-Hurwitz-Euler eta function in a complex field. Firstly, we define the multiple

(p, q)

-Hurwitz-Euler eta function by generalizing the Carlitz’s form

(p, q)

-Euler numbers and polynomials. We find some formulas and properties involved in Carlitz’s form

(p, q)

-Euler numbers and polynomials with higher order. We find new symmetric identities for multiple

(p, q)

-Hurwitz-Euler eta functions. We also obtain symmetric identities for Carlitz’s form

(p, q)

-Euler numbers and polynomials with higher order by using symmetry about multiple

(p, q)

-Hurwitz-Euler eta functions. Finally, we study the distribution and symmetric properties of the zero of Carlitz’s form

(p, q)

-Euler numbers and polynomials with higher order. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

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20 pages, 314 KiB

Open AccessArticle

A Note on the Truncated-Exponential Based Apostol-Type Polynomials

by H. M. Srivastava, Serkan Araci, Waseem A. Khan and Mehmet Acikgöz

Symmetry 2019, 11(4), 538; https://doi.org/10.3390/sym11040538 - 15 Apr 2019

Cited by 18 | Viewed by 3227

Abstract

In this paper, we propose to investigate the truncated-exponential-based Apostol-type polynomials and derive their various properties. In particular, we establish the operational correspondence between this new family of polynomials and the familiar Apostol-type polynomials. We also obtain some implicit summation formulas and symmetric [...] Read more.

In this paper, we propose to investigate the truncated-exponential-based Apostol-type polynomials and derive their various properties. In particular, we establish the operational correspondence between this new family of polynomials and the familiar Apostol-type polynomials. We also obtain some implicit summation formulas and symmetric identities by using their generating functions. The results, which we have derived here, provide generalizations of the corresponding known formulas including identities involving generalized Hermite-Bernoulli polynomials. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

16 pages, 276 KiB

Open AccessArticle

A Certain Family of Integral Operators Associated with the Struve Functions

by Shahid Mahmood, H.M. Srivastava, Sarfraz Nawaz Malik, Mohsan Raza, Neelam Shahzadi and Saira Zainab

Symmetry 2019, 11(4), 463; https://doi.org/10.3390/sym11040463 - 02 Apr 2019

Cited by 3 | Viewed by 2271

Abstract

This article presents the study of Struve functions and certain integral operators associated with the Struve functions. It contains the investigation of certain geometric properties like the strong starlikeness and strong convexity of the Struve functions. It also includes the criteria of univalence [...] Read more.

This article presents the study of Struve functions and certain integral operators associated with the Struve functions. It contains the investigation of certain geometric properties like the strong starlikeness and strong convexity of the Struve functions. It also includes the criteria of univalence for a family of certain integral operators associated with the generalized Struve functions. The starlikeness and uniform convexity of the said integral operators are also part of this research. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

19 pages, 5680 KiB

Open AccessArticle

Exploiting the Symmetry of Integral Transforms for Featuring Anuran Calls

by Amalia Luque, Jesús Gómez-Bellido, Alejandro Carrasco and Julio Barbancho

Symmetry 2019, 11(3), 405; https://doi.org/10.3390/sym11030405 - 20 Mar 2019

Cited by 3 | Viewed by 2646

Abstract

The application of machine learning techniques to sound signals requires the previous characterization of said signals. In many cases, their description is made using cepstral coefficients that represent the sound spectra. In this paper, the performance in obtaining cepstral coefficients by two integral [...] Read more.

The application of machine learning techniques to sound signals requires the previous characterization of said signals. In many cases, their description is made using cepstral coefficients that represent the sound spectra. In this paper, the performance in obtaining cepstral coefficients by two integral transforms, Discrete Fourier Transform (DFT) and Discrete Cosine Transform (DCT), are compared in the context of processing anuran calls. Due to the symmetry of sound spectra, it is shown that DCT clearly outperforms DFT, and decreases the error representing the spectrum by more than 30%. Additionally, it is demonstrated that DCT-based cepstral coefficients are less correlated than their DFT-based counterparts, which leads to a significant advantage for DCT-based cepstral coefficients if these features are later used in classification algorithms. Since the DCT superiority is based on the symmetry of sound spectra and not on any intrinsic advantage of the algorithm, the conclusions of this research can definitely be extrapolated to include any sound signal. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

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18 pages, 841 KiB

Open AccessArticle

Fuzzy Volterra Integro-Differential Equations Using General Linear Method

by Zanariah Abdul Majid, Faranak Rabiei, Fatin Abd Hamid and Fudziah Ismail

Symmetry 2019, 11(3), 381; https://doi.org/10.3390/sym11030381 - 15 Mar 2019

Cited by 5 | Viewed by 2721

Abstract

In this paper, a fuzzy general linear method of order three for solving fuzzy Volterra integro-differential equations of second kind is proposed. The general linear method is operated using the both internal stages of Runge-Kutta method and multivalues of a multisteps method. The [...] Read more.

In this paper, a fuzzy general linear method of order three for solving fuzzy Volterra integro-differential equations of second kind is proposed. The general linear method is operated using the both internal stages of Runge-Kutta method and multivalues of a multisteps method. The derivation of general linear method is based on the theory of B-series and rooted trees. Here, the fuzzy general linear method using the approach of generalized Hukuhara differentiability and combination of composite Simpson’s rules together with Lagrange interpolation polynomial is constructed for numerical solution of fuzzy volterra integro-differential equations. To illustrate the performance of the method, the numerical results are compared with some existing numerical methods. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

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13 pages, 258 KiB

Open AccessArticle

Upper Bound of the Third Hankel Determinant for a Subclass of q-Starlike Functions

by Shahid Mahmood, Hari M. Srivastava, Nazar Khan, Qazi Zahoor Ahmad, Bilal Khan and Irfan Ali

Symmetry 2019, 11(3), 347; https://doi.org/10.3390/sym11030347 - 07 Mar 2019

Cited by 82 | Viewed by 2939

Abstract

The main purpose of this article is to find the upper bound of the third Hankel determinant for a family of q-starlike functions which are associated with the Ruscheweyh-type q-derivative operator. The work is motivated by several special cases and consequences [...] Read more.

The main purpose of this article is to find the upper bound of the third Hankel determinant for a family of q-starlike functions which are associated with the Ruscheweyh-type q-derivative operator. The work is motivated by several special cases and consequences of our main results, which are pointed out herein. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

14 pages, 300 KiB

Open AccessArticle

Fractional Telegraph Equation and Its Solution by Natural Transform Decomposition Method

by Hassan Eltayeb, Yahya T. Abdalla, Imed Bachar and Mohamed H. Khabir

Symmetry 2019, 11(3), 334; https://doi.org/10.3390/sym11030334 - 06 Mar 2019

Cited by 45 | Viewed by 3590

Abstract

In this work, the natural transform decomposition method (NTDM) is applied to solve the linear and nonlinear fractional telegraph equations. This method is a combined form of the natural transform and the Adomian decomposition methods. In addition, we prove the convergence of our [...] Read more.

In this work, the natural transform decomposition method (NTDM) is applied to solve the linear and nonlinear fractional telegraph equations. This method is a combined form of the natural transform and the Adomian decomposition methods. In addition, we prove the convergence of our method. Finally, three examples have been employed to illustrate the preciseness and effectiveness of the proposed method. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

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16 pages, 347 KiB

Open AccessArticle

Some Difference Equations for Srivastava’s λ-Generalized Hurwitz–Lerch Zeta Functions with Applications

by Asifa Tassaddiq

Symmetry 2019, 11(3), 311; https://doi.org/10.3390/sym11030311 - 01 Mar 2019

Cited by 6 | Viewed by 2047

Abstract

In this article, we establish some new difference equations for the family of λ-generalized Hurwitz–Lerch zeta functions. These difference equations proved worthwhile to study these newly defined functions in terms of simpler functions. Several authors investigated such functions and their analytic properties, but [...] Read more.

In this article, we establish some new difference equations for the family of λ-generalized Hurwitz–Lerch zeta functions. These difference equations proved worthwhile to study these newly defined functions in terms of simpler functions. Several authors investigated such functions and their analytic properties, but no work has been reported for an estimation of their values. We perform some numerical computations to evaluate these functions for different values of the involved parameters. It is shown that the direct evaluation of involved integrals is not possible for the large values of parameter

s

; nevertheless, using our new difference equations, we can evaluate these functions for the large values of

s

. It is worth mentioning that for the small values of this parameter, our results are 100% accurate with the directly computed results using their integral representation. Difference equations so obtained are also useful for the computation of some new integrals of products of λ-generalized Hurwitz–Lerch zeta functions and verified to be consistent with the existing results. A derivative property of Mellin transforms proved fundamental to present this investigation. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

14 pages, 273 KiB

Open AccessArticle

Some General Classes of q-Starlike Functions Associated with the Janowski Functions

by Hari M. Srivastava, Muhammad Tahir, Bilal Khan, Qazi Zahoor Ahmad and Nazar Khan

Symmetry 2019, 11(2), 292; https://doi.org/10.3390/sym11020292 - 23 Feb 2019

Cited by 99 | Viewed by 3739

Abstract

By making use of the concept of basic (or q-) calculus, various families of q-extensions of starlike functions of order

α

in the open unit disk

U

were introduced and studied from many different viewpoints and perspectives. In this paper, we [...] Read more.

By making use of the concept of basic (or q-) calculus, various families of q-extensions of starlike functions of order

α

in the open unit disk

U

were introduced and studied from many different viewpoints and perspectives. In this paper, we first investigate the relationship between various known classes of q-starlike functions that are associated with the Janowski functions. We then introduce and study a new subclass of q-starlike functions that involves the Janowski functions. We also derive several properties of such families of q-starlike functions with negative coefficients including (for example) distortion theorems. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

14 pages, 273 KiB

Open AccessArticle

Some Subclasses of Uniformly Univalent Functions with Respect to Symmetric Points

by Shahid Mahmood, Hari M. Srivastava and Sarfraz Nawaz Malik

Symmetry 2019, 11(2), 287; https://doi.org/10.3390/sym11020287 - 22 Feb 2019

Cited by 8 | Viewed by 2314

Abstract

This article presents the study of certain analytic functions defined by bounded radius rotations associated with conic domain. Many geometric properties like coefficient estimate, radii problems, arc length, integral representation, inclusion results and growth rate of coefficients of Taylor’s series representation are investigated. [...] Read more.

This article presents the study of certain analytic functions defined by bounded radius rotations associated with conic domain. Many geometric properties like coefficient estimate, radii problems, arc length, integral representation, inclusion results and growth rate of coefficients of Taylor’s series representation are investigated. By varying the parameters in results, several well-known results in literature are obtained as special cases. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

18 pages, 339 KiB

Open AccessArticle

Existence Theory for Nonlinear Third-Order Ordinary Differential Equations with Nonlocal Multi-Point and Multi-Strip Boundary Conditions

by Ahmed Alsaedi, Mona Alsulami, Hari M. Srivastava, Bashir Ahmad and Sotiris K. Ntouyas

Symmetry 2019, 11(2), 281; https://doi.org/10.3390/sym11020281 - 22 Feb 2019

Cited by 8 | Viewed by 2532

Abstract

We investigate the solvability and Ulam stability for a nonlocal nonlinear third-order integro-multi-point boundary value problem on an arbitrary domain. The nonlinearity in the third-order ordinary differential equation involves the unknown function together with its first- and second-order derivatives. Our main results rely [...] Read more.

We investigate the solvability and Ulam stability for a nonlocal nonlinear third-order integro-multi-point boundary value problem on an arbitrary domain. The nonlinearity in the third-order ordinary differential equation involves the unknown function together with its first- and second-order derivatives. Our main results rely on the modern tools of functional analysis and are well illustrated with the aid of examples. An analogue problem involving non-separated integro-multi-point boundary conditions is also discussed. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

8 pages, 218 KiB

Open AccessArticle

Geometric Properties of Certain Analytic Functions Associated with the Dziok-Srivastava Operator

by Cai-Mei Yan and Jin-Lin Liu

Symmetry 2019, 11(2), 259; https://doi.org/10.3390/sym11020259 - 19 Feb 2019

Cited by 2 | Viewed by 1533

Abstract

The objective of the present paper is to derive certain geometric properties of analytic functions associated with the Dziok–Srivastava operator. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

8 pages, 273 KiB

Open AccessArticle

Continuous Wavelet Transform of Schwartz Tempered Distributions in S′ ( R n )

by Jagdish Narayan Pandey, Jay Singh Maurya, Santosh Kumar Upadhyay and Hari Mohan Srivastava

Symmetry 2019, 11(2), 235; https://doi.org/10.3390/sym11020235 - 15 Feb 2019

Cited by 17 | Viewed by 2644

Abstract

In this paper, we define a continuous wavelet transform of a Schwartz tempered distribution

f \in S^{^{'}} (R^{n})

with wavelet kernel

ψ \in S (R^{n})

and derive the corresponding wavelet inversion formula interpreting convergence in the [...] Read more.

In this paper, we define a continuous wavelet transform of a Schwartz tempered distribution

f \in S^{^{'}} (R^{n})

with wavelet kernel

ψ \in S (R^{n})

and derive the corresponding wavelet inversion formula interpreting convergence in the weak topology of

S^{^{'}} (R^{n})

. It turns out that the wavelet transform of a constant distribution is zero and our wavelet inversion formula is not true for constant distribution, but it is true for a non-constant distribution which is not equal to the sum of a non-constant distribution with a non-zero constant distribution. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

11 pages, 268 KiB

Open AccessFeature PaperArticle

A Dunkl–Type Generalization of Szász–Kantorovich Operators via Post–Quantum Calculus

by Md. Nasiruzzaman, Aiman Mukheimer and M. Mursaleen

Symmetry 2019, 11(2), 232; https://doi.org/10.3390/sym11020232 - 15 Feb 2019

Cited by 13 | Viewed by 2131

Abstract

In this paper, we define the

(p, q)

-variant of Szász–Kantorovich operators via Dunkl-type generalization generated by an exponential function and study the Korovkin-type results. We also obtain the convergence of our operators in weighted space by the modulus of [...] Read more.

In this paper, we define the

(p, q)

-variant of Szász–Kantorovich operators via Dunkl-type generalization generated by an exponential function and study the Korovkin-type results. We also obtain the convergence of our operators in weighted space by the modulus of continuity, Lipschitz class, and Peetre’s K-functionals. The extra parameter p provides more flexibility in approximation and plays an important role in symmetrizing these newly-defined operators. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

17 pages, 315 KiB

Open AccessFeature PaperArticle

Starlike Functions Related to the Bell Numbers

by Nak Eun Cho, Sushil Kumar, Virendra Kumar, V. Ravichandran and H. M. Srivastava

Symmetry 2019, 11(2), 219; https://doi.org/10.3390/sym11020219 - 13 Feb 2019

Cited by 42 | Viewed by 2982

Abstract

The present paper aims to establish the first order differential subordination relations between functions with a positive real part and starlike functions related to the Bell numbers. In addition, several sharp radii estimates for functions in the class of starlike functions associated with [...] Read more.

The present paper aims to establish the first order differential subordination relations between functions with a positive real part and starlike functions related to the Bell numbers. In addition, several sharp radii estimates for functions in the class of starlike functions associated with the Bell numbers are determined. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

11 pages, 774 KiB

Open AccessArticle

On Meromorphic Functions Defined by a New Operator Containing the Mittag–Leffler Function

by Suhila Elhaddad and Maslina Darus

Symmetry 2019, 11(2), 210; https://doi.org/10.3390/sym11020210 - 12 Feb 2019

Cited by 16 | Viewed by 2897

Abstract

This study defines a new linear differential operator via the Hadamard product between a q-hypergeometric function and Mittag–Leffler function. The application of the linear differential operator generates a new subclass of meromorphic function. Additionally, the study explores various properties and features, such [...] Read more.

This study defines a new linear differential operator via the Hadamard product between a q-hypergeometric function and Mittag–Leffler function. The application of the linear differential operator generates a new subclass of meromorphic function. Additionally, the study explores various properties and features, such as convex properties, distortion, growth, coefficient inequality and radii of starlikeness. Finally, the work discusses closure theorems and extreme points. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

17 pages, 551 KiB

Open AccessArticle

Hermite-Type Collocation Methods to Solve Volterra Integral Equations with Highly Oscillatory Bessel Kernels

by Chunhua Fang, Guo He and Shuhuang Xiang

Symmetry 2019, 11(2), 168; https://doi.org/10.3390/sym11020168 - 01 Feb 2019

Cited by 12 | Viewed by 2754

Abstract

In this paper, we present two kinds of Hermite-type collocation methods for linear Volterra integral equations of the second kind with highly oscillatory Bessel kernels. One method is direct Hermite collocation method, which used direct two-points Hermite interpolation in the whole interval. The [...] Read more.

In this paper, we present two kinds of Hermite-type collocation methods for linear Volterra integral equations of the second kind with highly oscillatory Bessel kernels. One method is direct Hermite collocation method, which used direct two-points Hermite interpolation in the whole interval. The other one is piecewise Hermite collocation method, which used a two-points Hermite interpolation in each subinterval. These two methods can calculate the approximate value of function value and derivative value simultaneously. Both methods are constructed easily and implemented well by the fast computation of highly oscillatory integrals involving Bessel functions. Under some conditions, the asymptotic convergence order with respect to oscillatory factor of these two methods are established, which are higher than the existing results. Some numerical experiments are included to show efficiency of these two methods. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

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19 pages, 558 KiB

Open AccessArticle

Certain Results of q -Sheffer–Appell Polynomials

by Ghazala Yasmin, Abdulghani Muhyi and Serkan Araci

Symmetry 2019, 11(2), 159; https://doi.org/10.3390/sym11020159 - 01 Feb 2019

Cited by 12 | Viewed by 2888

Abstract

In this paper, the class of

q

-Sheffer–Appell polynomials is introduced. The generating function, series definition, determinant definition and some other identities of this class are established. Certain members of

q

-Sheffer–Appell polynomials are investigated and some properties of these members are derived. [...] Read more.

In this paper, the class of

q

-Sheffer–Appell polynomials is introduced. The generating function, series definition, determinant definition and some other identities of this class are established. Certain members of

q

-Sheffer–Appell polynomials are investigated and some properties of these members are derived. In addition, the class of 2D

q

-Sheffer–Appell polynomials is introduced. Further, the graphs of some members of

q

-Sheffer–Appell polynomials and 2D

q

-Sheffer–Appell polynomials are plotted for different values of indices by using Matlab. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

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9 pages, 224 KiB

Open AccessArticle

The Order of Strongly Starlikeness of the Generalized α-Convex Functions

by Yuan Yuan, Rekha Srivastava and Jin-Lin Liu

Symmetry 2019, 11(1), 76; https://doi.org/10.3390/sym11010076 - 11 Jan 2019

Cited by 1 | Viewed by 2298

Abstract

We consider the order of the strongly-starlikeness of the generalized

α

-convex functions. Some sufficient conditions for functions to be p-valently strongly-starlike are given. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

10 pages, 243 KiB

Open AccessArticle

On a Generalization of the Initial-Boundary Problem for the Vibrating String Equation

by Djumaklich Amanov, Gafurjan Ibragimov and Adem Kılıçman

Symmetry 2019, 11(1), 73; https://doi.org/10.3390/sym11010073 - 10 Jan 2019

Cited by 4 | Viewed by 2320

Abstract

In the present paper, we study a generalization of the initial-boundary problem for the inhomogeneous vibrating string equation. The initial conditions include the higher order derivatives of the unknown function. The problem is studied under homogeneous boundary conditions of the first kind. The [...] Read more.

In the present paper, we study a generalization of the initial-boundary problem for the inhomogeneous vibrating string equation. The initial conditions include the higher order derivatives of the unknown function. The problem is studied under homogeneous boundary conditions of the first kind. The uniqueness and existence of a regular solution of the problem are proved. To prove the main result we use the spectral decomposition method. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

13 pages, 262 KiB

Open AccessArticle

Geometric Properties of Normalized Mittag–Leffler Functions

by Saddaf Noreen, Mohsan Raza, Jin-Lin Liu and Muhammad Arif

Symmetry 2019, 11(1), 45; https://doi.org/10.3390/sym11010045 - 03 Jan 2019

Cited by 15 | Viewed by 2610

Abstract

The aim of this paper is to investigate certain properties such as convexity of order

μ

, close-to-convexity of order

(1 + μ)

/2 and starlikeness of normalized Mittag–Leffler function. We use some inequalities to prove our results. We also discuss the close-to-convexity [...] Read more.

The aim of this paper is to investigate certain properties such as convexity of order

μ

, close-to-convexity of order

(1 + μ)

/2 and starlikeness of normalized Mittag–Leffler function. We use some inequalities to prove our results. We also discuss the close-to-convexity of Mittag–Leffler functions with respect to certain starlike functions. Furthermore, we find the conditions for the above-mentioned function to belong to the Hardy space

H^{p}

. Some of our results improve the results in the literature. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

12 pages, 289 KiB

Open AccessArticle

Some Generating Functions for q-Polynomials

by Howard S. Cohl, Roberto S. Costas-Santos and Tanay V. Wakhare

Symmetry 2018, 10(12), 758; https://doi.org/10.3390/sym10120758 - 16 Dec 2018

Cited by 1 | Viewed by 2909

Abstract

Demonstrating the striking symmetry between calculus and q-calculus, we obtain q-analogues of the Bateman, Pasternack, Sylvester, and Cesàro polynomials. Using these, we also obtain q-analogues for some of their generating functions. Our q-generating functions are given in terms of [...] Read more.

Demonstrating the striking symmetry between calculus and q-calculus, we obtain q-analogues of the Bateman, Pasternack, Sylvester, and Cesàro polynomials. Using these, we also obtain q-analogues for some of their generating functions. Our q-generating functions are given in terms of the basic hypergeometric series

_{4} ϕ_{5}

,

_{5} ϕ_{5}

,

_{4} ϕ_{3}

,

_{3} ϕ_{2}

,

_{2} ϕ_{1}

, and q-Pochhammer symbols. Starting with our q-generating functions, we are also able to find some new classical generating functions for the Pasternack and Bateman polynomials. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

20 pages, 362 KiB

Open AccessArticle

A New Representation for Srivastava’s λ-Generalized Hurwitz-Lerch Zeta Functions

by Asifa Tassaddiq

Symmetry 2018, 10(12), 733; https://doi.org/10.3390/sym10120733 - 08 Dec 2018

Cited by 12 | Viewed by 1977

Abstract

Taking inspiration principally from some of the latest research, we develop a new series representation for the λ-generalized Hurwitz-Lerch zeta functions. This representation led to important new results. The Fourier transform played a foundational role in this work. The duality property of [...] Read more.

Taking inspiration principally from some of the latest research, we develop a new series representation for the λ-generalized Hurwitz-Lerch zeta functions. This representation led to important new results. The Fourier transform played a foundational role in this work. The duality property of the Fourier transform became significant for checking the consistency of the results. Some known data has been verified as special cases of the results obtained in this investigation. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

15 pages, 324 KiB

Open AccessArticle

On Some Statistical Approximation by (p,q)-Bleimann, Butzer and Hahn Operators

by Khursheed J. Ansari, Ishfaq Ahmad, M. Mursaleen and Iqtadar Hussain

Symmetry 2018, 10(12), 731; https://doi.org/10.3390/sym10120731 - 07 Dec 2018

Cited by 12 | Viewed by 2334

Abstract

In this article, we propose a different generalization of

(p, q)

-BBH operators and carry statistical approximation properties of the introduced operators towards a function which has to be approximated where

(p, q)

-integers contains symmetric property. [...] Read more.

In this article, we propose a different generalization of

(p, q)

-BBH operators and carry statistical approximation properties of the introduced operators towards a function which has to be approximated where

(p, q)

-integers contains symmetric property. We establish a Korovkin approximation theorem in the statistical sense and obtain the statistical rates of convergence. Furthermore, we also introduce a bivariate extension of proposed operators and carry many statistical approximation results. The extra parameter p plays an important role to symmetrize the q-BBH operators. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

11 pages, 706 KiB

Open AccessArticle

Optimizing a Password Hashing Function with Hardware-Accelerated Symmetric Encryption

by Rafael Álvarez, Alicia Andrade and Antonio Zamora

Symmetry 2018, 10(12), 705; https://doi.org/10.3390/sym10120705 - 03 Dec 2018

Cited by 5 | Viewed by 3519

Abstract

Password-based key derivation functions (PBKDFs) are commonly used to transform user passwords into keys for symmetric encryption, as well as for user authentication, password hashing, and preventing attacks based on custom hardware. We propose two optimized alternatives that enhance the performance of a [...] Read more.

Password-based key derivation functions (PBKDFs) are commonly used to transform user passwords into keys for symmetric encryption, as well as for user authentication, password hashing, and preventing attacks based on custom hardware. We propose two optimized alternatives that enhance the performance of a previously published PBKDF. This design is based on (1) employing a symmetric cipher, the Advanced Encryption Standard (AES), as a pseudo-random generator and (2) taking advantage of the support for the hardware acceleration for AES that is available on many common platforms in order to mitigate common attacks to password-based user authentication systems. We also analyze their security characteristics, establishing that they are equivalent to the security of the core primitive (AES), and we compare their performance with well-known PBKDF algorithms, such as Scrypt and Argon2, with favorable results. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

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10 pages, 1382 KiB

Open AccessFeature PaperArticle

Complex Spirals and Pseudo-Chebyshev Polynomials of Fractional Degree

by Paolo Emilio Ricci

Symmetry 2018, 10(12), 671; https://doi.org/10.3390/sym10120671 - 28 Nov 2018

Cited by 10 | Viewed by 4500

Abstract

The complex Bernoulli spiral is connected to Grandi curves and Chebyshev polynomials. In this framework, pseudo-Chebyshev polynomials are introduced, and some of their properties are borrowed to form classical trigonometric identities; in particular, a set of orthogonal pseudo-Chebyshev polynomials of half-integer degree is [...] Read more.

The complex Bernoulli spiral is connected to Grandi curves and Chebyshev polynomials. In this framework, pseudo-Chebyshev polynomials are introduced, and some of their properties are borrowed to form classical trigonometric identities; in particular, a set of orthogonal pseudo-Chebyshev polynomials of half-integer degree is derived. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

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20 pages, 358 KiB

Open AccessArticle

Generalized Liouville–Caputo Fractional Differential Equations and Inclusions with Nonlocal Generalized Fractional Integral and Multipoint Boundary Conditions

by Ahmed Alsaedi, Madeaha Alghanmi, Bashir Ahmad and Sotiris K. Ntouyas

Symmetry 2018, 10(12), 667; https://doi.org/10.3390/sym10120667 - 26 Nov 2018

Cited by 10 | Viewed by 2271

Abstract

We develop the existence criteria for solutions of Liouville–Caputo-type generalized fractional differential equations and inclusions equipped with nonlocal generalized fractional integral and multipoint boundary conditions. Modern techniques of functional analysis are employed to derive the main results. Examples illustrating the main results are [...] Read more.

We develop the existence criteria for solutions of Liouville–Caputo-type generalized fractional differential equations and inclusions equipped with nonlocal generalized fractional integral and multipoint boundary conditions. Modern techniques of functional analysis are employed to derive the main results. Examples illustrating the main results are also presented. It is imperative to mention that our results correspond to the ones for a symmetric second-order nonlocal multipoint integral boundary value problem under suitable conditions (see the last section). Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

8 pages, 243 KiB

Open AccessArticle

Convolution and Partial Sums of Certain Multivalent Analytic Functions Involving Srivastava–Tomovski Generalization of the Mittag–Leffler Function

by Yi-Hui Xu and Jin-Lin Liu

Symmetry 2018, 10(11), 597; https://doi.org/10.3390/sym10110597 - 05 Nov 2018

Cited by 1 | Viewed by 2319

Abstract

We derive several properties such as convolution and partial sums of multivalent analytic functions associated with an operator involving Srivastava–Tomovski generalization of the Mittag–Leffler function. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

15 pages, 630 KiB

Open AccessArticle

Modified Kudryashov Method to Solve Generalized Kuramoto-Sivashinsky Equation

by Adem Kilicman and Rathinavel Silambarasan

Symmetry 2018, 10(10), 527; https://doi.org/10.3390/sym10100527 - 21 Oct 2018

Cited by 26 | Viewed by 3751

Abstract

The generalized Kuramoto–Sivashinsky equation is investigated using the modified Kudryashov method for the new exact solutions. The modified Kudryashov method converts the given nonlinear partial differential equation to algebraic equations, as a result of various steps, which upon solving the so-obtained equation systems [...] Read more.

The generalized Kuramoto–Sivashinsky equation is investigated using the modified Kudryashov method for the new exact solutions. The modified Kudryashov method converts the given nonlinear partial differential equation to algebraic equations, as a result of various steps, which upon solving the so-obtained equation systems yields the analytical solution. By this way, various exact solutions including complex structures are found, and their behavior is drawn in the 2D plane by Maple to compare the uniqueness and wave traveling of the solutions. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

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13 pages, 257 KiB

Open AccessArticle

On the Existence of the Solutions of a Fredholm Integral Equation with a Modified Argument in Hölder Spaces

by Merve Temizer Ersoy and Hasan Furkan

Symmetry 2018, 10(10), 522; https://doi.org/10.3390/sym10100522 - 19 Oct 2018

Cited by 9 | Viewed by 2224

Abstract

This article concerns the entity of solutions of a quadratic integral equation of the Fredholm type with an altered argument, [...] Read more.

This article concerns the entity of solutions of a quadratic integral equation of the Fredholm type with an altered argument,

x (t) = p (t) + x (t) \int_{0}^{1} k (t, τ) (T x) (τ) d τ,

where

p, k

are given functions, T is the given operator satisfying conditions specified later and x is an unknown function. Through the classical Schauder fixed point theorem and a new conclusion about the relative compactness in Hölder spaces, we obtain the existence of solutions under certain assumptions. Our work is more general than the previous works in the Conclusion section. At the end, we introduce several tangible examples where our entity result can be adopted. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

13 pages, 278 KiB

Open AccessArticle

A Class of Nonlinear Boundary Value Problems for an Arbitrary Fractional-Order Differential Equation with the Riemann-Stieltjes Functional Integral and Infinite-Point Boundary Conditions

by Hari M. Srivastava, Ahmed M. A. El-Sayed and Fatma M. Gaafar

Symmetry 2018, 10(10), 508; https://doi.org/10.3390/sym10100508 - 16 Oct 2018

Cited by 26 | Viewed by 2739

Abstract

In this paper, we investigate the existence of an absolute continuous solution to a class of first-order nonlinear differential equation with integral boundary conditions (BCs) or with infinite-point BCs. The Liouville-Caputo fractional derivative is involved in the nonlinear function. We first consider the [...] Read more.

In this paper, we investigate the existence of an absolute continuous solution to a class of first-order nonlinear differential equation with integral boundary conditions (BCs) or with infinite-point BCs. The Liouville-Caputo fractional derivative is involved in the nonlinear function. We first consider the existence of a solution for the first-order nonlinear differential equation with m-point nonlocal BCs. The existence of solutions of our problems is investigated by applying the properties of the Riemann sum for continuous functions. Several examples are given in order to illustrate our results. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

8 pages, 280 KiB

Open AccessArticle

Third-Order Hankel Determinant for Certain Class of Analytic Functions Related with Exponential Function

by Hai-Yan Zhang, Huo Tang and Xiao-Meng Niu

Symmetry 2018, 10(10), 501; https://doi.org/10.3390/sym10100501 - 15 Oct 2018

Cited by 19 | Viewed by 3120

Abstract

Let

S_{l}^{*}

denote the class of analytic functions f in the open unit disk

D = {z : | z | < 1}

normalized by [...] Read more.

Let

S_{l}^{*}

denote the class of analytic functions f in the open unit disk

D = {z : | z | < 1}

normalized by

f (0) = f^{'} (0) - 1 = 0

, which is subordinate to exponential function,

\frac{z f^{'} (z)}{f (z)} ≺ e^{z} (z \in D) .

In this paper, we aim to investigate the third-order Hankel determinant

H_{3} (1)

for this function class

S_{l}^{*}

associated with exponential function and obtain the upper bound of the determinant

H_{3} (1)

. Meanwhile, we give two examples to illustrate the results obtained. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

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11 pages, 281 KiB

Open AccessArticle

Geometric Properties of Lommel Functions of the First Kind

by Young Jae Sim, Oh Sang Kwon and Nak Eun Cho

Symmetry 2018, 10(10), 455; https://doi.org/10.3390/sym10100455 - 01 Oct 2018

Cited by 4 | Viewed by 2102

Abstract

In the present paper, we find sufficient conditions for starlikeness and convexity of normalized Lommel functions of the first kind using the admissible function methods. Additionally, we investigate some inclusion relationships for various classes associated with the Lommel functions. The functions belonging to [...] Read more.

In the present paper, we find sufficient conditions for starlikeness and convexity of normalized Lommel functions of the first kind using the admissible function methods. Additionally, we investigate some inclusion relationships for various classes associated with the Lommel functions. The functions belonging to these classes are related to the starlike functions, convex functions, close-to-convex functions and quasi-convex functions. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

12 pages, 300 KiB

Open AccessFeature PaperArticle

Symmetric Identities for (P,Q)-Analogue of Tangent Zeta Function

by Cheon Seoung Ryoo

Symmetry 2018, 10(9), 395; https://doi.org/10.3390/sym10090395 - 11 Sep 2018

Cited by 4 | Viewed by 2621

Abstract

The goal of this paper is to define the

(p, q)

-analogue of tangent numbers and polynomials by generalizing the tangent numbers and polynomials and Carlitz-type q-tangent numbers and polynomials. We get some explicit formulas and properties in conjunction [...] Read more.

The goal of this paper is to define the

(p, q)

-analogue of tangent numbers and polynomials by generalizing the tangent numbers and polynomials and Carlitz-type q-tangent numbers and polynomials. We get some explicit formulas and properties in conjunction with

(p, q)

-analogue of tangent numbers and polynomials. We give some new symmetric identities for

(p, q)

-analogue of tangent polynomials by using

(p, q)

-tangent zeta function. Finally, we investigate the distribution and symmetry of the zero of

(p, q)

-analogue of tangent polynomials with numerical methods. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

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13 pages, 751 KiB

Open AccessArticle

Sharp Bounds on the Higher Order Schwarzian Derivatives for Janowski Classes

by Nak Eun Cho, Virendra Kumar and V. Ravichandran

Symmetry 2018, 10(8), 348; https://doi.org/10.3390/sym10080348 - 18 Aug 2018

Cited by 6 | Viewed by 2505

Abstract

Higher order Schwarzian derivatives for normalized univalent functions were first considered by Schippers, and those of convex functions were considered by Dorff and Szynal. In the present investigation, higher order Schwarzian derivatives for the Janowski star-like and convex functions are considered, and sharp [...] Read more.

Higher order Schwarzian derivatives for normalized univalent functions were first considered by Schippers, and those of convex functions were considered by Dorff and Szynal. In the present investigation, higher order Schwarzian derivatives for the Janowski star-like and convex functions are considered, and sharp bounds for the first three consecutive derivatives are investigated. The results obtained in this paper generalize several existing results in this direction. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

15 pages, 484 KiB

Open AccessFeature PaperArticle

On the Convolution Quadrature Rule for Integral Transforms with Oscillatory Bessel Kernels

by Junjie Ma and Huilan Liu

Symmetry 2018, 10(7), 239; https://doi.org/10.3390/sym10070239 - 25 Jun 2018

Cited by 8 | Viewed by 3415

Abstract

Lubich’s convolution quadrature rule provides efficient approximations to integrals with special kernels. Particularly, when it is applied to computing highly oscillatory integrals, numerical tests show it does not suffer from fast oscillation. This paper is devoted to studying the convergence property of the [...] Read more.

Lubich’s convolution quadrature rule provides efficient approximations to integrals with special kernels. Particularly, when it is applied to computing highly oscillatory integrals, numerical tests show it does not suffer from fast oscillation. This paper is devoted to studying the convergence property of the convolution quadrature rule for highly oscillatory problems. With the help of operational calculus, the convergence rate of the convolution quadrature rule with respect to the frequency is derived. Furthermore, its application to highly oscillatory integral equations is also investigated. Numerical results are presented to verify the effectiveness of the convolution quadrature rule in solving highly oscillatory problems. It is found from theoretical and numerical results that the convolution quadrature rule for solving highly oscillatory problems is efficient and high-potential. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

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