Special Issue "Special Functions and Applications"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: 31 December 2019

Special Issue Editors

Guest Editor
Prof. Junesang Choi

Department of Mathematics, Dongguk University, Gyeongju 38066, Republic of Korea
Website | E-Mail
Interests: complex analysis; real analysis; special functions; number theory
Guest Editor
Prof. Ilya Shilin

1. Department of Higher Mathematics, National Research University MPEI, Krasnokazarmennaya 14, Moscow 111250, Russia
2. Depart-ment of Algebra, Moscow State Pedagogical University, Malaya Pirogovskaya 1, Moscow 119991, Russia
Website | E-Mail
Interests: special functions; group theoretical methods

Special Issue Information

Dear Colleagues,

Due mainly to their remarkable properties, for centuries, a surprisingly large number of special functions have been developed and applied in a variety of fields, such as combinatorics, astronomy, applied mathematics, physics, and engineering. The main purpose of this Special Issue is to be a forum of recently-developed theories and formulas of special functions with their possible applications to some other research areas. This Special Issue includes certain theories, formulas, and applications of gamma function, beta function, multiple gamma functions, and their q-extensions and other extensions; confluent hypergeometric functions, hypergeometric function, generalized hypergeometric functions, multiple hypergeometric functions, their various extensions and associated polynomials; Bernoulli numbers and polynomials, Euler numbers and polynomials, other classical numbers and polynomials, a variety of recently developed numbers and polynomials; Riemann zeta function, generalized (Hurwitz zeta) function, multiple Hurwitz zeta functions, multiple zeta values, and their extensions; special functions and the theory of group representations; and their applications. However, this Special Issue is not limited to the above list, when the content of a paper is related to some special functions and their applications.

Prof. Junesang Choi
Prof. Ilya Shilin
Guest Editors

Manuscript Submission Information

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Keywords

  • Gamma and related functions, and their extenstions
  • Generalized hypergeometric functions and their extenstions
  • Classical polynomials and their extensions
  • Recently-developed and new polynomials
  • Zeta functions

Published Papers (22 papers)

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Open AccessArticle
Certain Chebyshev-Type Inequalities Involving Fractional Conformable Integral Operators
Mathematics 2019, 7(4), 364; https://doi.org/10.3390/math7040364
Received: 2 April 2019 / Revised: 15 April 2019 / Accepted: 16 April 2019 / Published: 21 April 2019
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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)
Open AccessArticle
The Generalized Quadratic Gauss Sum and Its Fourth Power Mean
Mathematics 2019, 7(3), 258; https://doi.org/10.3390/math7030258
Received: 15 January 2019 / Revised: 8 March 2019 / Accepted: 11 March 2019 / Published: 12 March 2019
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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)
Open AccessArticle
Certain Geometric Properties of Lommel and Hyper-Bessel Functions
Mathematics 2019, 7(3), 240; https://doi.org/10.3390/math7030240
Received: 31 December 2018 / Revised: 26 February 2019 / Accepted: 1 March 2019 / Published: 6 March 2019
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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)
Open AccessArticle
Delannoy Numbers and Preferential Arrangements
Mathematics 2019, 7(3), 238; https://doi.org/10.3390/math7030238
Received: 25 January 2019 / Revised: 1 March 2019 / Accepted: 2 March 2019 / Published: 6 March 2019
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Abstract
A preferential arrangement on [[n]]={1,2,,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 , , 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)
Open AccessArticle
A Study of Generalized Laguerre Poly-Genocchi Polynomials
Mathematics 2019, 7(3), 219; https://doi.org/10.3390/math7030219
Received: 25 December 2018 / Revised: 2 February 2019 / Accepted: 22 February 2019 / Published: 26 February 2019
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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)
Open AccessArticle
Generalized Fractional Integral Operators Pertaining to the Product of Srivastava’s Polynomials and Generalized Mathieu Series
Mathematics 2019, 7(2), 206; https://doi.org/10.3390/math7020206
Received: 31 December 2018 / Revised: 16 February 2019 / Accepted: 20 February 2019 / Published: 23 February 2019
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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)
Open AccessArticle
Orthogonality Properties of the Pseudo-Chebyshev Functions (Variations on a Chebyshev’s Theme)
Mathematics 2019, 7(2), 180; https://doi.org/10.3390/math7020180
Received: 24 January 2019 / Revised: 10 February 2019 / Accepted: 12 February 2019 / Published: 15 February 2019
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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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Open AccessArticle
A Subclass of Bi-Univalent Functions Based on the Faber Polynomial Expansions and the Fibonacci Numbers
Mathematics 2019, 7(2), 160; https://doi.org/10.3390/math7020160
Received: 10 December 2018 / Revised: 8 January 2019 / Accepted: 9 January 2019 / Published: 11 February 2019
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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)
Open AccessArticle
Families of Integrals of Polylogarithmic Functions
Mathematics 2019, 7(2), 143; https://doi.org/10.3390/math7020143
Received: 30 December 2018 / Revised: 23 January 2019 / Accepted: 26 January 2019 / Published: 3 February 2019
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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)
Open AccessArticle
A New Representation of the k-Gamma Functions
Mathematics 2019, 7(2), 133; https://doi.org/10.3390/math7020133
Received: 8 December 2018 / Revised: 16 January 2019 / Accepted: 19 January 2019 / Published: 1 February 2019
Cited by 1 | PDF Full-text (288 KB) | HTML Full-text | XML Full-text
Abstract
The products of the form z(z+l)(z+2l)(z+(k1)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 ) ( 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)
Open AccessArticle
A Further Extension for Ramanujan’s Beta Integral and Applications
Mathematics 2019, 7(2), 118; https://doi.org/10.3390/math7020118
Received: 26 December 2018 / Revised: 19 January 2019 / Accepted: 21 January 2019 / Published: 23 January 2019
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Abstract
In 1915, Ramanujan stated the following formula 0tx1(at;q)(t;q)dt=πsinπx(q1x,a [...] Read more.
In 1915, Ramanujan stated the following formula 0 t x 1 ( a t ; q ) ( t ; q ) d t = π sin π x ( q 1 x , a ; q ) ( q , a q x ; q ) , 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)
Open AccessArticle
Further Extension of the Generalized Hurwitz-Lerch Zeta Function of Two Variables
Mathematics 2019, 7(1), 48; https://doi.org/10.3390/math7010048
Received: 12 December 2018 / Revised: 1 January 2019 / Accepted: 3 January 2019 / Published: 6 January 2019
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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)
Open AccessArticle
Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function
Mathematics 2019, 7(1), 34; https://doi.org/10.3390/math7010034
Received: 23 October 2018 / Revised: 21 December 2018 / Accepted: 24 December 2018 / Published: 31 December 2018
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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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Open AccessArticle
Mean Values of Products of L-Functions and Bernoulli Polynomials
Mathematics 2018, 6(12), 337; https://doi.org/10.3390/math6120337
Received: 17 November 2018 / Revised: 12 December 2018 / Accepted: 14 December 2018 / Published: 19 December 2018
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Abstract
Let m1,,mr be nonnegative integers, and set: Mr=m1++mr. In this paper, first we establish an explicit linear decomposition of: i=1rBmi [...] Read more.
Let m 1 , , m r be nonnegative integers, and set: M r = m 1 + + m r . In this paper, first we establish an explicit linear decomposition of: i = 1 r B m i ( x ) m i ! in terms of Bernoulli polynomials B k ( x ) with 0 k M r . Second, for any integer q 2 , we study the mean values of the Dirichlet L-functions at negative integers: χ 1 , , χ r ( mod q ) ; χ 1 χ r = 1 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)
Open AccessArticle
Some Symmetric Identities Involving the Stirling Polynomials Under the Finite Symmetric Group
Mathematics 2018, 6(12), 332; https://doi.org/10.3390/math6120332
Received: 6 November 2018 / Revised: 6 December 2018 / Accepted: 15 December 2018 / Published: 17 December 2018
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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)
Open AccessArticle
Higher-Order Convolutions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi Polynomials
Mathematics 2018, 6(12), 329; https://doi.org/10.3390/math6120329
Received: 20 November 2018 / Revised: 6 December 2018 / Accepted: 10 December 2018 / Published: 14 December 2018
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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)
Open AccessArticle
Some Properties of the Fuss–Catalan Numbers
Mathematics 2018, 6(12), 277; https://doi.org/10.3390/math6120277
Received: 31 October 2018 / Revised: 12 November 2018 / Accepted: 14 November 2018 / Published: 24 November 2018
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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)
Open AccessArticle
Fourier Series for Functions Related to Chebyshev Polynomials of the First Kind and Lucas Polynomials
Mathematics 2018, 6(12), 276; https://doi.org/10.3390/math6120276
Received: 23 October 2018 / Revised: 19 November 2018 / Accepted: 20 November 2018 / Published: 23 November 2018
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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)
Open AccessArticle
The Coefficients of Powers of Bazilević Functions
Mathematics 2018, 6(11), 263; https://doi.org/10.3390/math6110263
Received: 30 September 2018 / Revised: 12 November 2018 / Accepted: 13 November 2018 / Published: 18 November 2018
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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)
Open AccessArticle
Determinant Forms, Difference Equations and Zeros of the q-Hermite-Appell Polynomials
Mathematics 2018, 6(11), 258; https://doi.org/10.3390/math6110258
Received: 26 October 2018 / Revised: 12 November 2018 / Accepted: 13 November 2018 / Published: 17 November 2018
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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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Open AccessArticle
A New Version of the Generalized Krätzel–Fox Integral Operators
Mathematics 2018, 6(11), 222; https://doi.org/10.3390/math6110222
Received: 12 October 2018 / Revised: 25 October 2018 / Accepted: 26 October 2018 / Published: 28 October 2018
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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 Ha,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)

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Open AccessReply
Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function: Reply to Discussion
Mathematics 2019, 7(5), 410; https://doi.org/10.3390/math7050410
Received: 25 April 2019 / Revised: 3 May 2019 / Accepted: 6 May 2019 / Published: 8 May 2019
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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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