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Symmetry
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Special Functions and Polynomials
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Special Functions and Polynomials

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 August 2021) | Viewed by 31341

Special Issue Editor

Prof. Dr. Paolo Emilio Ricci

E-Mail Website
Guest Editor
Department of Mathematics, International Telematic University UniNettuno, Rome, Italy
Interests: special functions; matrix functions; eigenvalues; differential and integral equations; number theory
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Special functions are used in many applications of physics, engineering, and applied mathematics, such as electrodynamics, classical and modern physics, quantum mechanics, classical mechanics, and statistics, and, more recently, in the biological sciences and many other fields.

Furthermore, special polynomials are not only used in mathematical physics; they are also connected to deep problems of number theory.

You could say that every mathematical problem could be solved efficiently if the special functions that fit the solution were known. For example, certain equations that, like Airy's, have a symmetry connected to the third roots of the unity could be solved in a natural way by looking for the solution within particular expansions connected to the Chebyshev polynomials in two variables that have the same symmetry properties.

The hypergeometric functions constitute an important class within special functions that unify, through the introduction of appropriate parameters, most (if not all) parts of special functions, including elliptic integrals, beta functions, the incomplete Gamma function, Bessel functions, Legendre functions, classical orthogonal polynomials, Kummer confluent functions, and so on.

However, the broad range of special functions includes many other extensions and techniques, such as the multivariate generalizations of hypergeometric functions such as the Meijer G-function, the Lie and operational methods, and, in particular, G. Dattoli's monomiality principle.

The operational methods that can be found in earlier works of Euler and Lagrange, in relation to the construction of the generating functions of numerical sequences, were considered by G. Boole and O. Heaviside, who showed the connection between the derivative and the difference operator.

Their methods are at present included in the umbral calculus, a term introduced by J.J. Sylvester, since the exponent, for example that of $a^n$, is transformed into his ``shade'', which appears in $a_n$, so that powers are considered as sequences. The operational methods of umbral calculus, according to the notation of E. Lucas, have been made rigorous by G-C. Rota and S.M. Roman.

The efficiency of operational methods has been shown even in relation to the study of new classes of special functions, which include the multi-dimensional and multi-index cases. Among other things, their use allows us to build, in a simple way, formal solutions for a wide class of boundary value problems for partial differential equations.

Keywords

  • special functions
  • matrix functions
  • eigenvalues
  • differential and integral equations
  • number theory

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Published Papers (15 papers)

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Editorial

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2 pages, 180 KiB  
Editorial
Special Issue Editorial “Special Functions and Polynomials”
by Paolo Emilio Ricci
Symmetry 2022, 14(8), 1503; https://doi.org/10.3390/sym14081503 - 22 Jul 2022
Viewed by 953
Abstract
This Special Issue contains 14 articles from the MDPI journal Symmetry on the general subject area of “Special Functions and Polynomials”, written by scholars belonging to different countries of the world. A similar number of submitted articles was not accepted for publication. Several [...] Read more.
This Special Issue contains 14 articles from the MDPI journal Symmetry on the general subject area of “Special Functions and Polynomials”, written by scholars belonging to different countries of the world. A similar number of submitted articles was not accepted for publication. Several successful Special Issues on the same or closely related topics have already appeared in MDPI’s Symmetry, Mathematics and Axioms journals, in particular those edited by illustrious colleagues such as Hari Mohan Srivastava, Charles F. Dunkl, Junesang Choi, Taekyun Kim, Gradimir Milovanović, and many others, who testify to the importance of this matter for its applications in every field of mathematical, physical, chemical, engineering and statistical sciences. The subjects treated in this Special Issue include, in particular, the following Keywords. Full article
(This article belongs to the Special Issue Special Functions and Polynomials)

Research

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19 pages, 355 KiB  
Article
Clausen’s Series 3F2(1) with Integral Parameter Differences
by Kwang-Wu Chen
Symmetry 2021, 13(10), 1783; https://doi.org/10.3390/sym13101783 - 25 Sep 2021
Cited by 5 | Viewed by 1766
Abstract
Ebisu and Iwassaki proved that there are three-term relations for 3F2(1) with a group symmetry of order 72. In this paper, we apply some specific three-term relations for 3F2(1) to partially answer the [...] Read more.
Ebisu and Iwassaki proved that there are three-term relations for 3F2(1) with a group symmetry of order 72. In this paper, we apply some specific three-term relations for 3F2(1) to partially answer the open problem raised by Miller and Paris in 2012. Given a known value 3F2((a,b,x),(c,x+1),1), if f−x is an integer, then we construct an algorithm to obtain 3F2((a,b,f),(c,f+n),1) in an explicit closed form, where n is a positive integer and a,b,c and f are arbitrary complex numbers. We also extend our results to evaluate some specific forms of p+1Fp(1), for any positive integer p≥2. Full article
(This article belongs to the Special Issue Special Functions and Polynomials)
11 pages, 271 KiB  
Article
Distance Fibonacci Polynomials—Part II
by Urszula Bednarz and Małgorzata Wołowiec-Musiał
Symmetry 2021, 13(9), 1723; https://doi.org/10.3390/sym13091723 - 17 Sep 2021
Cited by 3 | Viewed by 1623
Abstract
In this paper we use a graph interpretation of distance Fibonacci polynomials to get a new generalization of Lucas polynomials in the distance sense. We give a direct formula, a generating function and we prove some identities for generalized Lucas polynomials. We present [...] Read more.
In this paper we use a graph interpretation of distance Fibonacci polynomials to get a new generalization of Lucas polynomials in the distance sense. We give a direct formula, a generating function and we prove some identities for generalized Lucas polynomials. We present Pascal-like triangles with left-justified rows filled with coefficients of these polynomials, in which one can observe some symmetric patterns. Using a general Q-matrix and a symmetric matrix of initial conditions we also define matrix generators for generalized Lucas polynomials. Full article
(This article belongs to the Special Issue Special Functions and Polynomials)
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7 pages, 248 KiB  
Article
The Third-Order Hermitian Toeplitz Determinant for Alpha-Convex Functions
by Anna Dobosz
Symmetry 2021, 13(7), 1274; https://doi.org/10.3390/sym13071274 - 16 Jul 2021
Cited by 3 | Viewed by 1881
Abstract
Sharp lower and upper bounds of the second- and third-order Hermitian Toeplitz determinants for the class of α-convex functions were found. The symmetry properties of the arithmetic mean underlying the definition of α-convexity and the symmetry properties of Hermitian matrices were [...] Read more.
Sharp lower and upper bounds of the second- and third-order Hermitian Toeplitz determinants for the class of α-convex functions were found. The symmetry properties of the arithmetic mean underlying the definition of α-convexity and the symmetry properties of Hermitian matrices were used. Full article
(This article belongs to the Special Issue Special Functions and Polynomials)
16 pages, 431 KiB  
Article
Ordered Structures of Polynomials over Max-Plus Algebra
by Cailu Wang, Yuanqing Xia and Yuegang Tao
Symmetry 2021, 13(7), 1137; https://doi.org/10.3390/sym13071137 - 25 Jun 2021
Cited by 3 | Viewed by 1958
Abstract
The ordered structures of polynomial idempotent algebras over max-plus algebra are investigated in this paper. Based on the antisymmetry, the partial orders on the sets of formal polynomials and polynomial functions are introduced to generate two partially ordered idempotent algebras (POIAs). Based on [...] Read more.
The ordered structures of polynomial idempotent algebras over max-plus algebra are investigated in this paper. Based on the antisymmetry, the partial orders on the sets of formal polynomials and polynomial functions are introduced to generate two partially ordered idempotent algebras (POIAs). Based on the symmetry, the quotient POIA of formal polynomials is then obtained. The order structure relationships among these three POIAs are described: the POIA of polynomial functions and the POIA of formal polynomials are orderly homomorphic; the POIA of polynomial functions and the quotient POIA of formal polynomials are orderly isomorphic. By using the partial order on formal polynomials, an algebraic method is provided to determine the upper and lower bounds of an equivalence class in the quotient POIA of formal polynomials. The criterion for a formal polynomial to be the minimal element of an equivalence class is derived. Furthermore, it is proven that any equivalence class is either an uncountable set with cardinality of the continuum or a finite set with a single element. Full article
(This article belongs to the Special Issue Special Functions and Polynomials)
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10 pages, 263 KiB  
Article
On Certain Differential Subordination of Harmonic Mean Related to a Linear Function
by Anna Dobosz, Piotr Jastrzębski and Adam Lecko
Symmetry 2021, 13(6), 966; https://doi.org/10.3390/sym13060966 - 29 May 2021
Cited by 2 | Viewed by 1550
Abstract
In this paper we study a certain differential subordination related to the harmonic mean and its symmetry properties, in the case where a dominant is a linear function. In addition to the known general results for the differential subordinations of the harmonic mean [...] Read more.
In this paper we study a certain differential subordination related to the harmonic mean and its symmetry properties, in the case where a dominant is a linear function. In addition to the known general results for the differential subordinations of the harmonic mean in which the dominant was any convex function, one can study such differential subordinations for the selected convex function. In this case, a reasonable and difficult issue is to look for the best dominant or one that is close to it. This paper is devoted to this issue, in which the dominant is a linear function, and the differential subordination of the harmonic mean is a generalization of the Briot–Bouquet differential subordination. Full article
(This article belongs to the Special Issue Special Functions and Polynomials)
25 pages, 347 KiB  
Article
Hermite Functions and Fourier Series
by Enrico Celeghini, Manuel Gadella and Mariano A. del Olmo
Symmetry 2021, 13(5), 853; https://doi.org/10.3390/sym13050853 - 11 May 2021
Cited by 14 | Viewed by 4003
Abstract
Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle (L2(C)) and in l2(Z), which are related to each other by means of the [...] Read more.
Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle (L2(C)) and in l2(Z), which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm–Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both L2(C) and l2(Z), so that all the mentioned operators are continuous. Full article
(This article belongs to the Special Issue Special Functions and Polynomials)
18 pages, 302 KiB  
Article
Inverse Derivative Operator and Umbral Methods for the Harmonic Numbers and Telescopic Series Study
by Giuseppe Dattoli, Silvia Licciardi and Rosa Maria Pidatella
Symmetry 2021, 13(5), 781; https://doi.org/10.3390/sym13050781 - 1 May 2021
Cited by 3 | Viewed by 1864
Abstract
The formalism of differ-integral calculus, initially developed to treat differential operators of fractional order, realizes a complete symmetry between differential and integral operators. This possibility has opened new and interesting scenarios, once extended to positive and negative order derivatives. The associated rules offer [...] Read more.
The formalism of differ-integral calculus, initially developed to treat differential operators of fractional order, realizes a complete symmetry between differential and integral operators. This possibility has opened new and interesting scenarios, once extended to positive and negative order derivatives. The associated rules offer an elegant, yet powerful, tool to deal with integral operators, viewed as derivatives of order-1. Although it is well known that the integration is the inverse of the derivative operation, the aforementioned rules offer a new mean to obtain either an explicit iteration of the integration by parts or a general formula to obtain the primitive of any infinitely differentiable function. We show that the method provides an unexpected link with generalized telescoping series, yields new useful tools for the relevant treatment, and allows a practically unexhausted tool to derive identities involving harmonic numbers and the associated generalized forms. It is eventually shown that embedding the differ-integral point of view with techniques of umbral algebraic nature offers a new insight into, and the possibility of, establishing a new and more powerful formalism. Full article
(This article belongs to the Special Issue Special Functions and Polynomials)
15 pages, 287 KiB  
Article
Investigation of the k-Analogue of Gauss Hypergeometric Functions Constructed by the Hadamard Product
by Mohamed Abdalla and Muajebah Hidan
Symmetry 2021, 13(4), 714; https://doi.org/10.3390/sym13040714 - 18 Apr 2021
Cited by 5 | Viewed by 2250
Abstract
Traditionally, the special function theory has many applications in various areas of mathematical physics, economics, statistics, engineering, and many other branches of science. Inspired by certain recent extensions of the k-analogue of gamma, the Pochhammer symbol, and hypergeometric functions, this work is devoted [...] Read more.
Traditionally, the special function theory has many applications in various areas of mathematical physics, economics, statistics, engineering, and many other branches of science. Inspired by certain recent extensions of the k-analogue of gamma, the Pochhammer symbol, and hypergeometric functions, this work is devoted to the study of the k-analogue of Gauss hypergeometric functions by the Hadamard product. We give a definition of the Hadamard product of k-Gauss hypergeometric functions (HPkGHF) associated with the fourth numerator and two denominator parameters. In addition, convergence properties are derived from this function. We also discuss interesting properties such as derivative formulae, integral representations, and integral transforms including beta transform and Laplace transform. Furthermore, we investigate some contiguous function relations and differential equations connecting the HPkGHF. The current results are more general than previous ones. Moreover, the proposed results are useful in the theory of k-special functions where the hypergeometric function naturally occurs. Full article
(This article belongs to the Special Issue Special Functions and Polynomials)
12 pages, 324 KiB  
Article
Certain Matrix Riemann–Liouville Fractional Integrals Associated with Functions Involving Generalized Bessel Matrix Polynomials
by Mohamed Abdalla, Mohamed Akel and Junesang Choi
Symmetry 2021, 13(4), 622; https://doi.org/10.3390/sym13040622 - 8 Apr 2021
Cited by 17 | Viewed by 2166
Abstract
The fractional integrals involving a number of special functions and polynomials have significant importance and applications in diverse areas of science; for example, statistics, applied mathematics, physics, and engineering. In this paper, we aim to introduce a slightly modified matrix of Riemann–Liouville fractional [...] Read more.
The fractional integrals involving a number of special functions and polynomials have significant importance and applications in diverse areas of science; for example, statistics, applied mathematics, physics, and engineering. In this paper, we aim to introduce a slightly modified matrix of Riemann–Liouville fractional integrals and investigate this matrix of Riemann–Liouville fractional integrals associated with products of certain elementary functions and generalized Bessel matrix polynomials. We also consider this matrix of Riemann–Liouville fractional integrals with a matrix version of the Jacobi polynomials. Furthermore, we point out that a number of Riemann–Liouville fractional integrals associated with a variety of functions and polynomials can be presented, which are presented as problems for further investigations. Full article
(This article belongs to the Special Issue Special Functions and Polynomials)
18 pages, 2150 KiB  
Article
Tricomi’s Method for the Laplace Transform and Orthogonal Polynomials
by Paolo Emilio Ricci, Diego Caratelli and Francesco Mainardi
Symmetry 2021, 13(4), 589; https://doi.org/10.3390/sym13040589 - 2 Apr 2021
Cited by 2 | Viewed by 1935
Abstract
Tricomi’s method for computing a set of inverse Laplace transforms in terms of Laguerre polynomials is revisited. By using the more recent results about the inversion and the connection coefficients for the series of orthogonal polynomials, we find the possibility to extend the [...] Read more.
Tricomi’s method for computing a set of inverse Laplace transforms in terms of Laguerre polynomials is revisited. By using the more recent results about the inversion and the connection coefficients for the series of orthogonal polynomials, we find the possibility to extend the Tricomi method to more general series expansions. Some examples showing the effectiveness of the considered procedure are shown. Full article
(This article belongs to the Special Issue Special Functions and Polynomials)
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15 pages, 400 KiB  
Article
Application of the Efros Theorem to the Function Represented by the Inverse Laplace Transform of s−μ exp(−sν)
by Alexander Apelblat and Francesco Mainardi
Symmetry 2021, 13(2), 354; https://doi.org/10.3390/sym13020354 - 22 Feb 2021
Cited by 10 | Viewed by 2574
Abstract
Using a special case of the Efros theorem which was derived by Wlodarski, and operational calculus, it was possible to derive many infinite integrals, finite integrals and integral identities for the function represented by the inverse Laplace transform. The integral identities are mainly [...] Read more.
Using a special case of the Efros theorem which was derived by Wlodarski, and operational calculus, it was possible to derive many infinite integrals, finite integrals and integral identities for the function represented by the inverse Laplace transform. The integral identities are mainly in terms of convolution integrals with the Mittag–Leffler and Volterra functions. The integrands of determined integrals include elementary functions (power, exponential, logarithmic, trigonometric and hyperbolic functions) and the error functions, the Mittag–Leffler functions and the Volterra functions. Some properties of the inverse Laplace transform of s−μexp(−sν) with μ≥0 and 0<ν<1 are presented. Full article
(This article belongs to the Special Issue Special Functions and Polynomials)
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17 pages, 326 KiB  
Article
Legendre-Gould Hopper-Based Sheffer Polynomials and Operational Methods
by Nabiullah Khan, Mohd Aman, Talha Usman and Junesang Choi
Symmetry 2020, 12(12), 2051; https://doi.org/10.3390/sym12122051 - 10 Dec 2020
Cited by 13 | Viewed by 1773
Abstract
A remarkably large of number of polynomials have been presented and studied. Among several important polynomials, Legendre polynomials, Gould-Hopper polynomials, and Sheffer polynomials have been intensively investigated. In this paper, we aim to incorporate the above-referred three polynomials to introduce the Legendre-Gould Hopper-based [...] Read more.
A remarkably large of number of polynomials have been presented and studied. Among several important polynomials, Legendre polynomials, Gould-Hopper polynomials, and Sheffer polynomials have been intensively investigated. In this paper, we aim to incorporate the above-referred three polynomials to introduce the Legendre-Gould Hopper-based Sheffer polynomials by modifying the classical generating function of the Sheffer polynomials. In addition, we investigate diverse properties and formulas for these newly introduced polynomials. Full article
(This article belongs to the Special Issue Special Functions and Polynomials)
15 pages, 756 KiB  
Article
Rational Approximation on Exponential Meshes
by Umberto Amato and Biancamaria Della Vecchia
Symmetry 2020, 12(12), 1999; https://doi.org/10.3390/sym12121999 - 4 Dec 2020
Cited by 3 | Viewed by 1647
Abstract
Error estimates of pointwise approximation, that are not possible by polynomials, are obtained by simple rational operators based on exponential-type meshes, improving previous results. Rational curves deduced from such operators are analyzed by Discrete Fourier Transform and a CAGD modeling technique for Shepard-type [...] Read more.
Error estimates of pointwise approximation, that are not possible by polynomials, are obtained by simple rational operators based on exponential-type meshes, improving previous results. Rational curves deduced from such operators are analyzed by Discrete Fourier Transform and a CAGD modeling technique for Shepard-type curves by truncated DFT and the PIA algorithm is developed. Full article
(This article belongs to the Special Issue Special Functions and Polynomials)
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Review

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28 pages, 474 KiB  
Review
Entropy-Like Properties and Lq-Norms of Hypergeometric Orthogonal Polynomials: Degree Asymptotics
by Jesús S. Dehesa
Symmetry 2021, 13(8), 1416; https://doi.org/10.3390/sym13081416 - 3 Aug 2021
Cited by 7 | Viewed by 1835
Abstract
In this work, the spread of hypergeometric orthogonal polynomials (HOPs) along their orthogonality interval is examined by means of the main entropy-like measures of their associated Rakhmanov’s probability density—so, far beyond the standard deviation and its generalizations, the ordinary moments. The Fisher information, [...] Read more.
In this work, the spread of hypergeometric orthogonal polynomials (HOPs) along their orthogonality interval is examined by means of the main entropy-like measures of their associated Rakhmanov’s probability density—so, far beyond the standard deviation and its generalizations, the ordinary moments. The Fisher information, the Rényi and Shannon entropies, and their corresponding spreading lengths are analytically expressed in terms of the degree and the parameter(s) of the orthogonality weight function. These entropic quantities are closely related to the gradient functional (Fisher) and the Lq-norms (Rényi, Shannon) of the polynomials. In addition, the degree asymptotics for these entropy-like functionals of the three canonical families of HPOs (i.e., Hermite, Laguerre, and Jacobi polynomials) are given and briefly discussed. Finally, a number of open related issues are identified whose solutions are both physico-mathematically and computationally relevant. Full article
(This article belongs to the Special Issue Special Functions and Polynomials)
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