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9 pages, 257 KiB

Open AccessFeature PaperArticle

On a Conjecture of Alzer, Berg, and Koumandos

by Ladislav Matejíčka

Mathematics 2020, 8(6), 1031; https://0-doi-org.brum.beds.ac.uk/10.3390/math8061031 - 23 Jun 2020

Viewed by 1507

Abstract

In this paper, we find a solution of an open problem posed by Alzer, Berg, and Koumandos: determine

(α, m) \in R^{+} \times N

such that the function [...] Read more.

In this paper, we find a solution of an open problem posed by Alzer, Berg, and Koumandos: determine

(α, m) \in R^{+} \times N

such that the function

x^{α} | ψ^{(m)} (x) |

is completely monotonic on

(0, \infty),

where

ψ (x)

denotes the logarithmic derivative of Euler’s gamma function. Full article

(This article belongs to the Special Issue Special Functions and Applications)

14 pages, 302 KiB

Open AccessArticle

On the Solutions of a Class of Integral Equations Pertaining to Incomplete H-Function and Incomplete H-Function

by Manish Kumar Bansal, Devendra Kumar, Jagdev Singh and Kottakkaran Sooppy Nisar

Mathematics 2020, 8(5), 819; https://0-doi-org.brum.beds.ac.uk/10.3390/math8050819 - 19 May 2020

Cited by 19 | Viewed by 1830

Abstract

The main aim of this article is to study the Fredholm-type integral equation involving the incomplete H-function (IHF) and incomplete H-function in the kernel. Firstly, we solve an integral equation associated with the IHF with the aid of the theory of fractional [...] Read more.

The main aim of this article is to study the Fredholm-type integral equation involving the incomplete H-function (IHF) and incomplete H-function in the kernel. Firstly, we solve an integral equation associated with the IHF with the aid of the theory of fractional calculus and Mellin transform. Next, we examine an integral equation pertaining to the incomplete H-function with the help of theory of fractional calculus and Mellin transform. Further, we indicate some known results by specializing the parameters of IHF and incomplete H-function. The results computed in this article are very general in nature and capable of giving many new and known results connected with integral equations and their solutions hitherto scattered in the literature. The derived results are very useful in solving various real world problems. Full article

(This article belongs to the Special Issue Special Functions and Applications)

9 pages, 237 KiB

Open AccessArticle

A Rational Approximation for the Complete Elliptic Integral of the First Kind

by Zhen-Hang Yang, Jing-Feng Tian and Ya-Ru Zhu

Mathematics 2020, 8(4), 635; https://0-doi-org.brum.beds.ac.uk/10.3390/math8040635 - 21 Apr 2020

Cited by 12 | Viewed by 2708

Abstract

Let

K (r)

be the complete elliptic integral of the first kind. We present an accurate rational lower approximation for

K (r)

. More precisely, we establish the inequality [...] Read more.

Let

K (r)

be the complete elliptic integral of the first kind. We present an accurate rational lower approximation for

K (r)

. More precisely, we establish the inequality

\frac{2}{π} K (r) > \frac{5 {(r^{'})}^{2} + 126 r^{'} + 61}{61 {(r^{'})}^{2} + 110 r^{'} + 21}

for

r \in (0, 1)

, where

r^{'} = \sqrt{1 - r^{2}}

. The lower bound is sharp. Full article

(This article belongs to the Special Issue Special Functions and Applications)

18 pages, 14803 KiB

Open AccessFeature PaperArticle

Various Structures of the Roots and Explicit Properties of q-cosine Bernoulli Polynomials and q-sine Bernoulli Polynomials

by Jung Yoog Kang and Chen Seoung Ryoo

Mathematics 2020, 8(4), 463; https://0-doi-org.brum.beds.ac.uk/10.3390/math8040463 - 25 Mar 2020

Cited by 10 | Viewed by 2099

Abstract

In this paper, we define cosine Bernoulli polynomials and sine Bernoulli polynomials related to the q-number. Furthermore, we intend to find the properties of these polynomials and check the structure of the roots. Through numerical experimentation, we look for various assumptions about [...] Read more.

In this paper, we define cosine Bernoulli polynomials and sine Bernoulli polynomials related to the q-number. Furthermore, we intend to find the properties of these polynomials and check the structure of the roots. Through numerical experimentation, we look for various assumptions about the polynomials above. Full article

(This article belongs to the Special Issue Special Functions and Applications)

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20 pages, 741 KiB

Open AccessArticle

A Numerical Computation of Zeros of q-Generalized Tangent-Appell Polynomials

by Ghazala Yasmin, Cheon Seoung Ryoo and Hibah Islahi

Mathematics 2020, 8(3), 383; https://0-doi-org.brum.beds.ac.uk/10.3390/math8030383 - 09 Mar 2020

Cited by 3 | Viewed by 1520

Abstract

The intended objective of this study is to define and investigate a new class of q-generalized tangent-based Appell polynomials by combining the families of 2-variable q-generalized tangent polynomials and q-Appell polynomials. The investigation includes derivations of generating functions, series definitions, [...] Read more.

The intended objective of this study is to define and investigate a new class of q-generalized tangent-based Appell polynomials by combining the families of 2-variable q-generalized tangent polynomials and q-Appell polynomials. The investigation includes derivations of generating functions, series definitions, and several important properties and identities of the hybrid q-special polynomials. Further, the analogous study for the members of this q-hybrid family are illustrated. The graphical representation of its members is shown, and the distributions of zeros are displayed. Full article

(This article belongs to the Special Issue Special Functions and Applications)

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17 pages, 296 KiB

Open AccessArticle

Special Functions of Mathematical Physics: A Unified Lagrangian Formalism

by Zdzislaw E. Musielak, Niyousha Davachi and Marialis Rosario-Franco

Mathematics 2020, 8(3), 379; https://0-doi-org.brum.beds.ac.uk/10.3390/math8030379 - 09 Mar 2020

Cited by 15 | Viewed by 3771

Abstract

Lagrangian formalism is established for differential equations with special functions of mathematical physics as solutions. Formalism is based on either standard or non-standard Lagrangians. This work shows that the procedure of deriving the standard Lagrangians leads to Lagrangians for which the Euler–Lagrange equation [...] Read more.

Lagrangian formalism is established for differential equations with special functions of mathematical physics as solutions. Formalism is based on either standard or non-standard Lagrangians. This work shows that the procedure of deriving the standard Lagrangians leads to Lagrangians for which the Euler–Lagrange equation vanishes identically, and that only some of these Lagrangians become the null Lagrangians with the well-defined gauge functions. It is also demonstrated that the non-standard Lagrangians require that the Euler–Lagrange equations are amended by the auxiliary conditions, which is a new phenomenon in the calculus of variations. The existence of the auxiliary conditions has profound implications on the validity of the Helmholtz conditions. The obtained results are used to derive the Lagrangians for the Airy, Bessel, Legendre and Hermite equations. The presented examples clearly demonstrate that the developed Lagrangian formalism is applicable to all considered differential equations, including the Airy (and other similar) equations, and that the regular and modified Bessel equations are the only ones with the gauge functions. Possible implications of the existence of the gauge functions for these equations are discussed. Full article

(This article belongs to the Special Issue Special Functions and Applications)

43 pages, 1041 KiB

Open AccessArticle

On Degenerate Truncated Special Polynomials

by Ugur Duran and Mehmet Acikgoz

Mathematics 2020, 8(1), 144; https://0-doi-org.brum.beds.ac.uk/10.3390/math8010144 - 20 Jan 2020

Cited by 12 | Viewed by 3084

Abstract

The main aim of this paper is to introduce the degenerate truncated forms of multifarious special polynomials and numbers and is to investigate their various properties and relationships by using the series manipulation method and diverse special proof techniques. The degenerate truncated exponential [...] Read more.

The main aim of this paper is to introduce the degenerate truncated forms of multifarious special polynomials and numbers and is to investigate their various properties and relationships by using the series manipulation method and diverse special proof techniques. The degenerate truncated exponential polynomials are first considered and their several properties are given. Then the degenerate truncated Stirling polynomials of the second kind are defined and their elementary properties and relations are proved. Also, the degenerate truncated forms of the bivariate Fubini and Bell polynomials and numbers are introduced and various relations and formulas for these polynomials and numbers, which cover several summation formulas, addition identities, recurrence relationships, derivative property and correlations with the degenerate truncated Stirling polynomials of the second kind, are acquired. Thereafter, the truncated degenerate Bernoulli and Euler polynomials are considered and multifarious correlations and formulas including summation formulas, derivation rules and correlations with the degenerate truncated Stirling numbers of the second are derived. In addition, regarding applications, by introducing the degenerate truncated forms of the classical Bernstein polynomials, we obtain diverse correlations and formulas. Some interesting surface plots of these polynomials in the special cases are provided. Full article

(This article belongs to the Special Issue Special Functions and Applications)

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19 pages, 307 KiB

Open AccessArticle

Bounds of Generalized Proportional Fractional Integrals in General Form via Convex Functions and Their Applications

by Gauhar Rahman, Thabet Abdeljawad, Fahd Jarad and Kottakkaran Sooppy Nisar

Mathematics 2020, 8(1), 113; https://0-doi-org.brum.beds.ac.uk/10.3390/math8010113 - 11 Jan 2020

Cited by 26 | Viewed by 2266

Abstract

In this paper, our objective is to apply a new approach to establish bounds of sums of left and right proportional fractional integrals of a general type and obtain some related inequalities. From the obtained results, we deduce some new inequalities for classical [...] Read more.

In this paper, our objective is to apply a new approach to establish bounds of sums of left and right proportional fractional integrals of a general type and obtain some related inequalities. From the obtained results, we deduce some new inequalities for classical generalized proportional fractional integrals as corollaries. These inequalities have a connection with some known and existing inequalities which are mentioned in the literature. In addition, some applications of the main results are presented. Full article

(This article belongs to the Special Issue Special Functions and Applications)

11 pages, 244 KiB

Open AccessFeature PaperArticle

The Chebyshev Difference Equation

by Tom Cuchta, Michael Pavelites and Randi Tinney

Mathematics 2020, 8(1), 74; https://0-doi-org.brum.beds.ac.uk/10.3390/math8010074 - 03 Jan 2020

Cited by 5 | Viewed by 3550

Abstract

We define and investigate a new class of difference equations related to the classical Chebyshev differential equations of the first and second kind. The resulting “discrete Chebyshev polynomials” of the first and second kind have qualitatively similar properties to their continuous counterparts, including [...] Read more.

We define and investigate a new class of difference equations related to the classical Chebyshev differential equations of the first and second kind. The resulting “discrete Chebyshev polynomials” of the first and second kind have qualitatively similar properties to their continuous counterparts, including a representation by hypergeometric series, recurrence relations, and derivative relations. Full article

(This article belongs to the Special Issue Special Functions and Applications)

13 pages, 348 KiB

Open AccessArticle

Fractional Integrations of a Generalized Mittag-Leffler Type Function and Its Application

by Kottakkaran Sooppy Nisar

Mathematics 2019, 7(12), 1230; https://0-doi-org.brum.beds.ac.uk/10.3390/math7121230 - 12 Dec 2019

Cited by 6 | Viewed by 2201

Abstract

A generalized form of the Mittag-Leffler function denoted by

_{p} E_{q; δ}^{λ, μ; ν} (x)

is established and studied in this paper. The fractional integrals involving the newly defined function are investigated. As an application, the solutions of [...] Read more.

A generalized form of the Mittag-Leffler function denoted by

_{p} E_{q; δ}^{λ, μ; ν} (x)

is established and studied in this paper. The fractional integrals involving the newly defined function are investigated. As an application, the solutions of a generalized fractional kinetic equation containing this function are derived and the nature of the solution is studied with the help of graphical analysis. Full article

(This article belongs to the Special Issue Special Functions and Applications)

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11 pages, 269 KiB

Open AccessArticle

Certain Unified Integrals Associated with Product of M-Series and Incomplete H-functions

by Manish Kumar Bansal, Devendra Kumar, Ilyas Khan, Jagdev Singh and Kottakkaran Sooppy Nisar

Mathematics 2019, 7(12), 1191; https://0-doi-org.brum.beds.ac.uk/10.3390/math7121191 - 05 Dec 2019

Cited by 19 | Viewed by 2210

Abstract

In this paper, we established some interesting integrals associated with the product of M-series and incomplete H-functions, which are expressed in terms of incomplete H-functions. Next, we give some special cases by specializing the parameters of M-series and incomplete H-functions [...] Read more.

In this paper, we established some interesting integrals associated with the product of M-series and incomplete H-functions, which are expressed in terms of incomplete H-functions. Next, we give some special cases by specializing the parameters of M-series and incomplete H-functions (for example, Fox’s H-Function, Incomplete Fox Wright functions, Fox Wright functions and Incomplete generalized hypergeometric functions) and also listed few known results. The results obtained in this work are general in nature and very useful in science, engineering and finance. Full article

(This article belongs to the Special Issue Special Functions and Applications)

21 pages, 316 KiB

Open AccessArticle

Space of Quasi-Periodic Limit Functions and Its Applications

by Rui Xie, Zhinan Xia and Junwei Liu

Mathematics 2019, 7(11), 1132; https://0-doi-org.brum.beds.ac.uk/10.3390/math7111132 - 19 Nov 2019

Cited by 4 | Viewed by 2229

Abstract

We introduce a class consisting of what we call quasi-periodic limit functions and then establish the relation between quasi-periodic limit functions and asymptotically quasi-periodic functions. At last, these quasi-periodic limit functions are applied to study the existence of asymptotically quasi-periodic solutions of abstract [...] Read more.

We introduce a class consisting of what we call quasi-periodic limit functions and then establish the relation between quasi-periodic limit functions and asymptotically quasi-periodic functions. At last, these quasi-periodic limit functions are applied to study the existence of asymptotically quasi-periodic solutions of abstract Cauchy problems. Full article

(This article belongs to the Special Issue Special Functions and Applications)

12 pages, 298 KiB

Open AccessArticle

Integral Representations for Products of Two Bessel or Modified Bessel Functions

by Dragana Jankov Maširević and Tibor K. Pogány

Mathematics 2019, 7(10), 978; https://0-doi-org.brum.beds.ac.uk/10.3390/math7100978 - 16 Oct 2019

Cited by 2 | Viewed by 7953

Abstract

The first part of the article contains integral expressions for products of two Bessel functions of the first kind having either different integer orders or different arguments. A similar question for a product of modified Bessel functions of the first kind is solved [...] Read more.

The first part of the article contains integral expressions for products of two Bessel functions of the first kind having either different integer orders or different arguments. A similar question for a product of modified Bessel functions of the first kind is solved next, when the input functions are of different integer orders and have different arguments. Full article

(This article belongs to the Special Issue Special Functions and Applications)

27 pages, 399 KiB

Open AccessFeature PaperArticle

δ-Almost Periodic Functions and Applications to Dynamic Equations

by Chao Wang, Ravi P. Agarwal and Donal O’Regan

Mathematics 2019, 7(6), 525; https://doi.org/10.3390/math7060525 - 09 Jun 2019

Cited by 10 | Viewed by 1827

Abstract

In this paper, by employing matched spaces for time scales, we introduce a

δ

-almost periodic function and obtain some related properties. Also the hull equation for homogeneous dynamic equation is introduced and results of the existence are presented. In the sense of [...] Read more.

In this paper, by employing matched spaces for time scales, we introduce a

δ

-almost periodic function and obtain some related properties. Also the hull equation for homogeneous dynamic equation is introduced and results of the existence are presented. In the sense of admitting exponential dichotomy for the homogeneous equation, the expression of a

δ

-almost periodic solution for a type of nonhomogeneous dynamic equation is obtained and the existence of

δ

-almost periodic solutions for new delay dynamic equations is considered. The results in this paper are valid for delay q-difference equations and delay dynamic equations whose delays may be completely separated from the time scale

T

. Full article

(This article belongs to the Special Issue Special Functions and Applications)

9 pages, 749 KiB

Open AccessArticle

Certain Chebyshev-Type Inequalities Involving Fractional Conformable Integral Operators

by Gauhar Rahman, Zafar Ullah, Aftab Khan, Erhan Set and Kottakkaran Sooppy Nisar

Mathematics 2019, 7(4), 364; https://0-doi-org.brum.beds.ac.uk/10.3390/math7040364 - 21 Apr 2019

Cited by 43 | Viewed by 2639

Abstract

Since an interesting functional by P.L. Chebyshev was presented in the year 1882, many results, which are called Chebyshev-type inequalities, have been established. Some of these inequalities were obtained by using fractional integral operators. Very recently, a new variant of the fractional conformable [...] Read more.

Since an interesting functional by P.L. Chebyshev was presented in the year 1882, many results, which are called Chebyshev-type inequalities, have been established. Some of these inequalities were obtained by using fractional integral operators. Very recently, a new variant of the fractional conformable integral operator was introduced by Jarad et al. Motivated by this operator, we aim at establishing novel inequalities for a class of differentiable functions, which are associated with Chebyshev’s functional, by employing a fractional conformable integral operator. We also aim at showing important connections of the results here with those including Riemann–Liouville fractional and classical integrals. Full article

(This article belongs to the Special Issue Special Functions and Applications)

10 pages, 246 KiB

Open AccessArticle

The Generalized Quadratic Gauss Sum and Its Fourth Power Mean

by Shimeng Shen and Wenpeng Zhang

Mathematics 2019, 7(3), 258; https://0-doi-org.brum.beds.ac.uk/10.3390/math7030258 - 12 Mar 2019

Cited by 2 | Viewed by 2632

Abstract

In this article, our main purpose is to introduce a new and generalized quadratic Gauss sum. By using analytic methods, the properties of classical Gauss sums, and character sums, we consider the calculating problem of its fourth power mean and give two interesting [...] Read more.

In this article, our main purpose is to introduce a new and generalized quadratic Gauss sum. By using analytic methods, the properties of classical Gauss sums, and character sums, we consider the calculating problem of its fourth power mean and give two interesting computational formulae for it. Full article

(This article belongs to the Special Issue Special Functions and Applications)

12 pages, 256 KiB

Open AccessArticle

Certain Geometric Properties of Lommel and Hyper-Bessel Functions

by Saima Mushtaq, Mohsan Raza and Muhey U Din

Mathematics 2019, 7(3), 240; https://0-doi-org.brum.beds.ac.uk/10.3390/math7030240 - 06 Mar 2019

Cited by 4 | Viewed by 2072

Abstract

In this article, we are mainly interested in finding the sufficient conditions under which Lommel functions and hyper-Bessel functions are close-to-convex with respect to the certain starlike functions. Strongly starlikeness and convexity of Lommel functions and hyper-Bessel functions are also discussed. Some applications [...] Read more.

In this article, we are mainly interested in finding the sufficient conditions under which Lommel functions and hyper-Bessel functions are close-to-convex with respect to the certain starlike functions. Strongly starlikeness and convexity of Lommel functions and hyper-Bessel functions are also discussed. Some applications are also the part of our investigation. Full article

(This article belongs to the Special Issue Special Functions and Applications)

9 pages, 246 KiB

Open AccessArticle

Delannoy Numbers and Preferential Arrangements

by Kwang-Wu Chen

Mathematics 2019, 7(3), 238; https://0-doi-org.brum.beds.ac.uk/10.3390/math7030238 - 06 Mar 2019

Cited by 3 | Viewed by 2125

Abstract

A preferential arrangement on

[[n]] = {1, 2, \dots, n}

is a ranking of the elements of

[[n]]

where ties are allowed. The number of preferential arrangements on [...] Read more.

A preferential arrangement on

[[n]] = {1, 2, \dots, n}

is a ranking of the elements of

[[n]]

where ties are allowed. The number of preferential arrangements on

[[n]]

is denoted by

r_{n}

. The Delannoy number

D (m, n)

is the number of lattice paths from

(0, 0)

to

(m, n)

in which only east

(1, 0)

, north

(0, 1)

, and northeast

(1, 1)

steps are allowed. We establish a symmetric identity among the numbers

r_{n}

and

D (p, q)

by means of algebraic and combinatorial methods. Full article

(This article belongs to the Special Issue Special Functions and Applications)

14 pages, 252 KiB

Open AccessArticle

A Study of Generalized Laguerre Poly-Genocchi Polynomials

by Nabiullah Khan, Talha Usman and Kottakkaran Sooppy Nisar

Mathematics 2019, 7(3), 219; https://0-doi-org.brum.beds.ac.uk/10.3390/math7030219 - 26 Feb 2019

Cited by 6 | Viewed by 2076

Abstract

A variety of polynomials, their extensions, and variants, have been extensively investigated, mainly due to their potential applications in diverse research areas. Motivated by their importance and potential for applications in a variety of research fields, numerous polynomials and their extensions have recently [...] Read more.

A variety of polynomials, their extensions, and variants, have been extensively investigated, mainly due to their potential applications in diverse research areas. Motivated by their importance and potential for applications in a variety of research fields, numerous polynomials and their extensions have recently been introduced and investigated. In this paper, we introduce generalized Laguerre poly-Genocchi polynomials and investigate some of their properties and identities, which were found to extend some known results. Among them, an implicit summation formula and addition-symmetry identities for generalized Laguerre poly-Genocchi polynomials are derived. The results presented here, being very general, are pointed out to reduce to yield formulas and identities for relatively simple polynomials and numbers. Full article

(This article belongs to the Special Issue Special Functions and Applications)

8 pages, 242 KiB

Open AccessArticle

Generalized Fractional Integral Operators Pertaining to the Product of Srivastava’s Polynomials and Generalized Mathieu Series

by K.S. Nisar, D.L. Suthar, M. Bohra and S.D. Purohit

Mathematics 2019, 7(2), 206; https://0-doi-org.brum.beds.ac.uk/10.3390/math7020206 - 23 Feb 2019

Cited by 7 | Viewed by 2501

Abstract

Fractional calculus image formulas involving various special functions are important for evaluation of generalized integrals and to obtain the solution of differential and integral equations. In this paper, the Saigo’s fractional integral operators involving hypergeometric function in the kernel are applied to the [...] Read more.

Fractional calculus image formulas involving various special functions are important for evaluation of generalized integrals and to obtain the solution of differential and integral equations. In this paper, the Saigo’s fractional integral operators involving hypergeometric function in the kernel are applied to the product of Srivastava’s polynomials and the generalized Mathieu series, containing the factor

x^{λ} {(x^{k} + c^{k})}^{- ρ}

in its argument. The results are expressed in terms of the generalized hypergeometric function and Hadamard product of the generalized Mathieu series. Corresponding special cases related to the Riemann–Liouville and Erdélyi–Kober fractional integral operators are also considered. Full article

(This article belongs to the Special Issue Special Functions and Applications)

11 pages, 2414 KiB

Open AccessArticle

Orthogonality Properties of the Pseudo-Chebyshev Functions (Variations on a Chebyshev’s Theme)

by Clemente Cesarano and Paolo Emilio Ricci

Mathematics 2019, 7(2), 180; https://0-doi-org.brum.beds.ac.uk/10.3390/math7020180 - 15 Feb 2019

Cited by 8 | Viewed by 2579

Abstract

The third and fourth pseudo-Chebyshev irrational functions of half-integer degree are defined. Their definitions are connected to those of the first- and second-kind pseudo-Chebyshev functions. Their orthogonality properties are shown, with respect to classical weights. Full article

(This article belongs to the Special Issue Special Functions and Applications)

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9 pages, 262 KiB

Open AccessArticle

A Subclass of Bi-Univalent Functions Based on the Faber Polynomial Expansions and the Fibonacci Numbers

by Şahsene Altınkaya, Sibel Yalçın and Serkan Çakmak

Mathematics 2019, 7(2), 160; https://0-doi-org.brum.beds.ac.uk/10.3390/math7020160 - 11 Feb 2019

Cited by 13 | Viewed by 2274

Abstract

In this investigation, by using the Komatu integral operator, we introduce the new class of bi-univalent functions based on the rule of subordination. Moreover, we use the Faber polynomial expansions and Fibonacci numbers to derive bounds for the general coefficient of the bi-univalent [...] Read more.

In this investigation, by using the Komatu integral operator, we introduce the new class of bi-univalent functions based on the rule of subordination. Moreover, we use the Faber polynomial expansions and Fibonacci numbers to derive bounds for the general coefficient of the bi-univalent function class. Full article

(This article belongs to the Special Issue Special Functions and Applications)

20 pages, 271 KiB

Open AccessArticle

Families of Integrals of Polylogarithmic Functions

by Anthony Sofo

Mathematics 2019, 7(2), 143; https://0-doi-org.brum.beds.ac.uk/10.3390/math7020143 - 03 Feb 2019

Cited by 4 | Viewed by 2080

Abstract

We give an overview of the representation and many connections between integrals of products of polylogarithmic functions and Euler sums. We shall consider polylogarithmic functions with linear, quadratic, and trigonometric arguments, thereby producing new results and further reinforcing the well-known connection between Euler [...] Read more.

We give an overview of the representation and many connections between integrals of products of polylogarithmic functions and Euler sums. We shall consider polylogarithmic functions with linear, quadratic, and trigonometric arguments, thereby producing new results and further reinforcing the well-known connection between Euler sums and polylogarithmic functions. Many examples of integrals of products of polylogarithmic functions in terms of Riemann zeta values and Dirichlet values will be given. Suggestions for further research are also suggested, including a study of polylogarithmic functions with inverse trigonometric functions. Full article

(This article belongs to the Special Issue Special Functions and Applications)

13 pages, 288 KiB

Open AccessArticle

A New Representation of the k-Gamma Functions

by Asifa Tassaddiq

Mathematics 2019, 7(2), 133; https://0-doi-org.brum.beds.ac.uk/10.3390/math7020133 - 01 Feb 2019

Cited by 16 | Viewed by 4792

Abstract

The products of the form

z (z + l) (z + 2 l) \dots (z + (k - 1) l)

are of interest for a wide-ranging audience. In particular, they frequently arise in diverse situations, [...] Read more.

The products of the form

z (z + l) (z + 2 l) \dots (z + (k - 1) l)

are of interest for a wide-ranging audience. In particular, they frequently arise in diverse situations, such as computation of Feynman integrals, combinatory of creation, annihilation operators and in fractional calculus. These expressions can be successfully applied for stated applications by using a mathematical notion of k-gamma functions. In this paper, we develop a new series representation of k-gamma functions in terms of delta functions. It led to a novel extension of the applicability of k-gamma functions that introduced them as distributions defined for a specific set of functions. Full article

(This article belongs to the Special Issue Special Functions and Applications)

10 pages, 252 KiB

Open AccessArticle

A Further Extension for Ramanujan’s Beta Integral and Applications

by Gao-Wen Xi and Qiu-Ming Luo

Mathematics 2019, 7(2), 118; https://0-doi-org.brum.beds.ac.uk/10.3390/math7020118 - 23 Jan 2019

Viewed by 1908

Abstract

In 1915, Ramanujan stated the following formula [...] Read more.

In 1915, Ramanujan stated the following formula

\int_{0}^{\infty} t^{x - 1} \frac{{(- a t; q)}_{\infty}}{{(- t; q)}_{\infty}} d t = \frac{π}{sin π x} \frac{{(q^{1 - x}, a; q)}_{\infty}}{{(q, a q^{- x}; q)}_{\infty}},

where

0 < q < 1

,

x > 0

, and

0 < a < q^{x}

. The above formula is called Ramanujan’s beta integral. In this paper, by using q-exponential operator, we further extend Ramanujan’s beta integral. As some applications, we obtain some new integral formulas of Ramanujan and also show some new representation with gamma functions and q-gamma functions. Full article

(This article belongs to the Special Issue Special Functions and Applications)

8 pages, 241 KiB

Open AccessArticle

Further Extension of the Generalized Hurwitz-Lerch Zeta Function of Two Variables

by Kottakkaran Sooppy Nisar

Mathematics 2019, 7(1), 48; https://0-doi-org.brum.beds.ac.uk/10.3390/math7010048 - 06 Jan 2019

Cited by 4 | Viewed by 2323

Abstract

The main aim of this paper is to provide a new generalization of Hurwitz-Lerch Zeta function of two variables. We also investigate several interesting properties such as integral representations, summation formula, and a connection with the generalized hypergeometric function. To strengthen the main [...] Read more.

The main aim of this paper is to provide a new generalization of Hurwitz-Lerch Zeta function of two variables. We also investigate several interesting properties such as integral representations, summation formula, and a connection with the generalized hypergeometric function. To strengthen the main results we also consider some important special cases. Full article

(This article belongs to the Special Issue Special Functions and Applications)

15 pages, 1375 KiB

Open AccessArticle

Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function

by Dejan Brkić and Pavel Praks

Mathematics 2019, 7(1), 34; https://0-doi-org.brum.beds.ac.uk/10.3390/math7010034 - 31 Dec 2018

Cited by 33 | Viewed by 6722 | Correction

Abstract

The Colebrook equation is a popular model for estimating friction loss coefficients in water and gas pipes. The model is implicit in the unknown flow friction factor,

f

. To date, the captured flow friction factor,

f

, can be extracted from the [...] Read more.

The Colebrook equation is a popular model for estimating friction loss coefficients in water and gas pipes. The model is implicit in the unknown flow friction factor,

f

. To date, the captured flow friction factor,

f

, can be extracted from the logarithmic form analytically only in the term of the Lambert

W

-function. The purpose of this study is to find an accurate and computationally efficient solution based on the shifted Lambert

W

-function also known as the Wright ω-function. The Wright ω-function is more suitable because it overcomes the problem with the overflow error by switching the fast growing term,

y = W (e^{x})

, of the Lambert

W

-function to series expansions that further can be easily evaluated in computers without causing overflow run-time errors. Although the Colebrook equation transformed through the Lambert

W

-function is identical to the original expression in terms of accuracy, a further evaluation of the Lambert

W

-function can be only approximate. Very accurate explicit approximations of the Colebrook equation that contain only one or two logarithms are shown. The final result is an accurate explicit approximation of the Colebrook equation with a relative error of no more than 0.0096%. The presented approximations are in a form suitable for everyday engineering use, and are both accurate and computationally efficient. Full article

(This article belongs to the Special Issue Special Functions and Applications)

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11 pages, 256 KiB

Open AccessArticle

Mean Values of Products of L-Functions and Bernoulli Polynomials

by Abdelmejid Bayad and Daeyeoul Kim

Mathematics 2018, 6(12), 337; https://0-doi-org.brum.beds.ac.uk/10.3390/math6120337 - 19 Dec 2018

Cited by 2 | Viewed by 2269

Abstract

Let

m_{1}, \dots, m_{r}

be nonnegative integers, and set:

M_{r} = m_{1} + \dots + m_{r} .

In this paper, first we establish an explicit linear decomposition of: [...] Read more.

Let

m_{1}, \dots, m_{r}

be nonnegative integers, and set:

M_{r} = m_{1} + \dots + m_{r} .

In this paper, first we establish an explicit linear decomposition of:

\prod_{i = 1}^{r} \frac{B_{m_{i}} (x)}{m_{i}!}

in terms of Bernoulli polynomials

B_{k} (x)

with

0 \leq k \leq M_{r}

. Second, for any integer

q \geq 2

, we study the mean values of the Dirichlet L-functions at negative integers:

\sum_{χ_{1}, \dots, χ_{r} (\mod q); χ_{1} \dots χ_{r} = 1} \prod_{i = 1}^{r} L (- m_{i}, χ_{i})

where the summation is over Dirichlet characters

χ_{i}

modulo q. Incidentally, a part of our work recovers Nielsen’s theorem, Nörlund’s formula, and its generalization by Hu, Kim, and Kim. Full article

(This article belongs to the Special Issue Special Functions and Applications)

10 pages, 267 KiB

Open AccessArticle

Some Symmetric Identities Involving the Stirling Polynomials Under the Finite Symmetric Group

by Dongkyu Lim and Feng Qi

Mathematics 2018, 6(12), 332; https://0-doi-org.brum.beds.ac.uk/10.3390/math6120332 - 17 Dec 2018

Cited by 1 | Viewed by 2910

Abstract

In the paper, the authors present some symmetric identities involving the Stirling polynomials and higher order Bernoulli polynomials under all permutations in the finite symmetric group of degree n. These identities extend and generalize some known results. Full article

(This article belongs to the Special Issue Special Functions and Applications)

14 pages, 306 KiB

Open AccessArticle

Higher-Order Convolutions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi Polynomials

by Yuan He, Serkan Araci, Hari M. Srivastava and Mahmoud Abdel-Aty

Mathematics 2018, 6(12), 329; https://0-doi-org.brum.beds.ac.uk/10.3390/math6120329 - 14 Dec 2018

Cited by 15 | Viewed by 2472

Abstract

In this paper, we present a systematic and unified investigation for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. By applying the generating-function methods and summation-transform techniques, we establish some higher-order convolutions for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the [...] Read more.

In this paper, we present a systematic and unified investigation for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. By applying the generating-function methods and summation-transform techniques, we establish some higher-order convolutions for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. Some results presented here are the corresponding extensions of several known formulas. Full article

(This article belongs to the Special Issue Special Functions and Applications)

11 pages, 274 KiB

Open AccessArticle

Some Properties of the Fuss–Catalan Numbers

by Feng Qi and Pietro Cerone

Mathematics 2018, 6(12), 277; https://0-doi-org.brum.beds.ac.uk/10.3390/math6120277 - 24 Nov 2018

Cited by 13 | Viewed by 3752

Abstract

In the paper, the authors express the Fuss–Catalan numbers as several forms in terms of the Catalan–Qi function, find some analytic properties, including the monotonicity, logarithmic convexity, complete monotonicity, and minimality, of the Fuss–Catalan numbers, and derive a double inequality for bounding the [...] Read more.

In the paper, the authors express the Fuss–Catalan numbers as several forms in terms of the Catalan–Qi function, find some analytic properties, including the monotonicity, logarithmic convexity, complete monotonicity, and minimality, of the Fuss–Catalan numbers, and derive a double inequality for bounding the Fuss–Catalan numbers. Full article

(This article belongs to the Special Issue Special Functions and Applications)

15 pages, 288 KiB

Open AccessArticle

Fourier Series for Functions Related to Chebyshev Polynomials of the First Kind and Lucas Polynomials

by Taekyun Kim, Dae San Kim, Lee-Chae Jang and Gwan-Woo Jang

Mathematics 2018, 6(12), 276; https://0-doi-org.brum.beds.ac.uk/10.3390/math6120276 - 23 Nov 2018

Cited by 13 | Viewed by 2644

Abstract

In this paper, we derive Fourier series expansions for functions related to sums of finite products of Chebyshev polynomials of the first kind and of Lucas polynomials. From the Fourier series expansions, we are able to express those two kinds of sums of [...] Read more.

In this paper, we derive Fourier series expansions for functions related to sums of finite products of Chebyshev polynomials of the first kind and of Lucas polynomials. From the Fourier series expansions, we are able to express those two kinds of sums of finite products of polynomials as linear combinations of Bernoulli polynomials. Full article

(This article belongs to the Special Issue Special Functions and Applications)

12 pages, 280 KiB

Open AccessArticle

The Coefficients of Powers of Bazilević Functions

by Nak Eun Cho, Virendra Kumar and Ji Hyang Park

Mathematics 2018, 6(11), 263; https://0-doi-org.brum.beds.ac.uk/10.3390/math6110263 - 18 Nov 2018

Cited by 2 | Viewed by 2269

Abstract

In the present work, a sharp bound on the modulus of the initial coefficients for powers of strongly Bazilević functions is obtained. As an application of these results, certain conditions are investigated under which the Littlewood-Paley conjecture holds for strongly Bazilević functions for [...] Read more.

In the present work, a sharp bound on the modulus of the initial coefficients for powers of strongly Bazilević functions is obtained. As an application of these results, certain conditions are investigated under which the Littlewood-Paley conjecture holds for strongly Bazilević functions for large values of the parameters involved therein. Further, sharp estimate on the generalized Fekete-Szegö functional is also derived. Relevant connections of our results with the existing ones are also made. Full article

(This article belongs to the Special Issue Special Functions and Applications)

16 pages, 359 KiB

Open AccessArticle

Determinant Forms, Difference Equations and Zeros of the q-Hermite-Appell Polynomials

by Subuhi Khan and Tabinda Nahid

Mathematics 2018, 6(11), 258; https://0-doi-org.brum.beds.ac.uk/10.3390/math6110258 - 17 Nov 2018

Cited by 13 | Viewed by 2400

Abstract

The present paper intends to introduce the hybrid form of q-special polynomials, namely q-Hermite-Appell polynomials by means of generating function and series definition. Some significant properties of q-Hermite-Appell polynomials such as determinant definitions, q-recurrence relations and q-difference equations [...] Read more.

The present paper intends to introduce the hybrid form of q-special polynomials, namely q-Hermite-Appell polynomials by means of generating function and series definition. Some significant properties of q-Hermite-Appell polynomials such as determinant definitions, q-recurrence relations and q-difference equations are established. Examples providing the corresponding results for certain members belonging to this q-Hermite-Appell family are considered. In addition, graphs of certain q-special polynomials are demonstrated using computer experiment. Thereafter, distribution of zeros of these q-special polynomials is displayed. Full article

(This article belongs to the Special Issue Special Functions and Applications)

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8 pages, 245 KiB

Open AccessArticle

A New Version of the Generalized Krätzel–Fox Integral Operators

by Shrideh K. Q. Al-Omari, Ghalib Jumah, Jafar Al-Omari and Deepali Saxena

Mathematics 2018, 6(11), 222; https://0-doi-org.brum.beds.ac.uk/10.3390/math6110222 - 28 Oct 2018

Cited by 8 | Viewed by 2324

Abstract

This article deals with some variants of Krätzel integral operators involving Fox’s H-function and their extension to classes of distributions and spaces of Boehmians. For real numbers a and

b > 0

, the Fréchet space

H_{a, b}

of testing [...] Read more.

This article deals with some variants of Krätzel integral operators involving Fox’s H-function and their extension to classes of distributions and spaces of Boehmians. For real numbers a and

b > 0

, the Fréchet space

H_{a, b}

of testing functions has been identified as a subspace of certain Boehmian spaces. To establish the Boehmian spaces, two convolution products and some related axioms are established. The generalized variant of the cited Krätzel-Fox integral operator is well defined and is the operator between the Boehmian spaces. A generalized convolution theorem has also been given. Full article

(This article belongs to the Special Issue Special Functions and Applications)

Other

Jump to: Research

6 pages, 574 KiB

Open AccessReply

Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function: Reply to the Discussion by Majid Niazkar

by Pavel Praks and Dejan Brkić

Mathematics 2020, 8(5), 796; https://0-doi-org.brum.beds.ac.uk/10.3390/math8050796 - 14 May 2020

Cited by 1 | Viewed by 1618

Abstract

In this reply, we present updated approximations to the Colebrook equation for flow friction. The equations are equally computational simple, but with increased accuracy thanks to the optimization procedure, which was proposed by the discusser, Dr. Majid Niazkar. Our large-scale quasi-Monte Carlo verifications [...] Read more.

In this reply, we present updated approximations to the Colebrook equation for flow friction. The equations are equally computational simple, but with increased accuracy thanks to the optimization procedure, which was proposed by the discusser, Dr. Majid Niazkar. Our large-scale quasi-Monte Carlo verifications confirm that the here presented novel optimized numerical parameters further significantly increase accuracy of the estimated flow friction factor. Full article

(This article belongs to the Special Issue Special Functions and Applications)

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6 pages, 218 KiB

Open AccessComment

Discussion of “Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function” by Dejan Brkić and Pavel Praks, Mathematics 2019, 7, 34; doi:10.3390/math7010034

by Majid Niazkar

Mathematics 2020, 8(5), 793; https://0-doi-org.brum.beds.ac.uk/10.3390/math8050793 - 14 May 2020

Cited by 1 | Viewed by 1456

Abstract

Estimating the Darcy–Weisbach friction factor is crucial to various engineering applications. Although the literature has accepted the Colebrook–White formula as a standard approach for this prediction, its implicit structure brings about an active field of research seeking for alternatives more suitable in practice. [...] Read more.

Estimating the Darcy–Weisbach friction factor is crucial to various engineering applications. Although the literature has accepted the Colebrook–White formula as a standard approach for this prediction, its implicit structure brings about an active field of research seeking for alternatives more suitable in practice. This study mainly attempts to increase the precision of two explicit equations proposed by Brkić and Praks. The results obviously demonstrate that the modified relations outperformed the original ones from nine out of 10 accuracy evaluation criteria. Finally, one of the improved equations estimates closer friction factors to those obtained by the Colebrook–White formula among 18 one-step explicit equations available in the literature based on three of the considered criteria. Full article

(This article belongs to the Special Issue Special Functions and Applications)

2 pages, 183 KiB

Open AccessCorrection

Correction: Brkić, D., and Praks, P. Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function. Mathematics 2019, 7, 34

by Dejan Brkić and Pavel Praks

Mathematics 2019, 7(10), 951; https://0-doi-org.brum.beds.ac.uk/10.3390/math7100951 - 12 Oct 2019

Viewed by 1474

Abstract

Having in mind that in the title of the article contains term “Wright ω-function” and not its cognate “Lambert W-function”, the authors would like to change Equation (2) of [1] in order to contain both expressions, as follows [...] Full article

(This article belongs to the Special Issue Special Functions and Applications)

7 pages, 618 KiB

Open AccessReply

Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function: Reply to Discussion

by Dejan Brkić and Pavel Praks

Mathematics 2019, 7(5), 410; https://0-doi-org.brum.beds.ac.uk/10.3390/math7050410 - 08 May 2019

Cited by 14 | Viewed by 2353

Abstract

This reply gives two corrections of typographical errors in respect to the commented article, and then provides few comments in respect to the discussion and one improved version of the approximation of the Colebrook equation for flow friction, based on the Wright ω-function. [...] Read more.

This reply gives two corrections of typographical errors in respect to the commented article, and then provides few comments in respect to the discussion and one improved version of the approximation of the Colebrook equation for flow friction, based on the Wright ω-function. Finally, this reply gives an exact explicit version of the Colebrook equation expressed through the Wright ω-function, which does not introduce any additional errors in respect to the original equation. All mentioned approximations are computationally efficient and also very accurate. Results are verified using more than 2 million of Quasi Monte-Carlo samples. Full article

(This article belongs to the Special Issue Special Functions and Applications)

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7 pages, 927 KiB

Open AccessComment

Discussion of “Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function” by DejanBrkić; and Pavel Praks, Mathematics 2019, 7, 34; doi:10.3390/math7010034

by Lotfi Zeghadnia, Bachir Achour and Jean Loup Robert

Mathematics 2019, 7(3), 253; https://0-doi-org.brum.beds.ac.uk/10.3390/math7030253 - 12 Mar 2019

Cited by 1 | Viewed by 1864

Abstract

The Colebrook-White equation is often used for calculation of the friction factor in turbulent regimes; it has succeeded in attracting a great deal of attention from researchers. The Colebrook–White equation is a complex equation where the computation of the friction factor is not [...] Read more.

The Colebrook-White equation is often used for calculation of the friction factor in turbulent regimes; it has succeeded in attracting a great deal of attention from researchers. The Colebrook–White equation is a complex equation where the computation of the friction factor is not direct, and there is a need for trial-error methods or graphical solutions; on the other hand, several researchers have attempted to alter the Colebrook-White equation by explicit formulas with the hope of achieving zero-percent (0%) maximum deviation, among them Dejan Brkić and Pavel Praks. The goal of this paper is to discuss the results proposed by the authors in their paper:” Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function” and to propose more accurate formulas. Full article

(This article belongs to the Special Issue Special Functions and Applications)

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