Polynomials: Theory and Applications, 2nd Edition

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "A: Algebra and Logic".

Deadline for manuscript submissions: 31 July 2025 | Viewed by 5841

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Department of Mathematics, Hannam University, Daejeon 34430, Republic of Korea
Interests: numerical verification method; scientific computing; differential equations; dynamical systems; quantum calculus; special functions
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Special Issue Information

Dear Colleagues,

The importance of polynomials in the interdisciplinary field of mathematics, engineering, and science is well known. Over the past several decades, research on polynomials has been conducted extensively in many disciplines.

This Special Issue welcomes all research papers related to polynomials in mathematics, science, and industry.

Potential topics include but are not limited to the following:

  • The modern umbral calculus (binomial, Appell, and Sheffer polynomial sequences)
  • Orthogonal polynomials, matrix orthogonal polynomials, multiple orthogonal polynomials
  • Matrix and determinant approach to special polynomial sequences
  • Applications of special polynomial sequences
  • Number theory and special functions
  • Asymptotic methods in orthogonal polynomials
  • Fractional calculus and special functions
  • Symbolic computations and special functions
  • Applications of special functions to statistics, physical sciences, and engineering.

Prof. Dr. Cheon-Seoung Ryoo
Guest Editor

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Keywords

  • umbral calculus
  • orthogonal polynomials
  • matrix orthogonal polynomials
  • special polynomial sequences
  • applications of special polynomial sequences
  • number theory and special functions
  • fractional calculus and special functions
  • symbolic computations and special functions
  • applications of special functions to statistics, physical sciences, and engineering

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Related Special Issue

Published Papers (8 papers)

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Research

21 pages, 1519 KiB  
Article
Two-Variable q-General-Appell Polynomials Within the Context of the Monomiality Principle
by Noor Alam, Waseem Ahmad Khan, Can Kızılateş and Cheon Seoung Ryoo
Mathematics 2025, 13(5), 765; https://doi.org/10.3390/math13050765 - 26 Feb 2025
Viewed by 133
Abstract
In this study, we consider the two-variable q-general polynomials and derive some properties. By using these polynomials, we introduce and study the theory of two-variable q-general Appell polynomials (2VqgAP) using q-operators. The effective use of the q-multiplicative [...] Read more.
In this study, we consider the two-variable q-general polynomials and derive some properties. By using these polynomials, we introduce and study the theory of two-variable q-general Appell polynomials (2VqgAP) using q-operators. The effective use of the q-multiplicative operator of the base polynomial produces the generating equation for 2VqgAP involving the q-exponential function. Furthermore, we establish the q-multiplicative and q-derivative operators and the corresponding differential equations. Then, we obtain the operational, explicit and determinant representations for these polynomials. Some examples are constructed in terms of the two-variable q-general Appell polynomials to illustrate the main results. Finally, graphical representations are provided to illustrate the behavior of some special cases of the two-variable q-general Appell polynomials and their potential applications. Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications, 2nd Edition)
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27 pages, 370 KiB  
Article
New Results for Certain Jacobsthal-Type Polynomials
by Waleed Mohamed Abd-Elhameed, Omar Mazen Alqubori and Amr Kamel Amin
Mathematics 2025, 13(5), 715; https://doi.org/10.3390/math13050715 - 22 Feb 2025
Viewed by 220
Abstract
This paper investigates a class of Jacobsthal-type polynomials (JTPs) that involves one parameter. We present several new formulas for these polynomials, including expressions for their derivatives, moments, and linearization formulas. The key idea behind the derivation of these formulas is based on developing [...] Read more.
This paper investigates a class of Jacobsthal-type polynomials (JTPs) that involves one parameter. We present several new formulas for these polynomials, including expressions for their derivatives, moments, and linearization formulas. The key idea behind the derivation of these formulas is based on developing a new connection formula that expresses the shifted Chebyshev polynomials of the third kind in terms of the JTPs. This connection formula is used to deduce a new inversion formula of the JTPs. Therefore, by utilizing the power form representation of these polynomials and their corresponding inversion formula, we can derive additional expressions for them. Additionally, we compute some definite integrals based on some formulas of these polynomials. Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications, 2nd Edition)
18 pages, 335 KiB  
Article
On Symmetrical Sonin Kernels in Terms of Hypergeometric-Type Functions
by Yuri Luchko
Mathematics 2024, 12(24), 3943; https://doi.org/10.3390/math12243943 - 15 Dec 2024
Viewed by 531
Abstract
In this paper, a new class of kernels of integral transforms of the Laplace convolution type that we named symmetrical Sonin kernels is introduced and investigated. For a symmetrical Sonin kernel given in terms of elementary or special functions, its associated kernel has [...] Read more.
In this paper, a new class of kernels of integral transforms of the Laplace convolution type that we named symmetrical Sonin kernels is introduced and investigated. For a symmetrical Sonin kernel given in terms of elementary or special functions, its associated kernel has the same form with possibly different parameter values. In the paper, several new kernels of this type are derived by means of the Sonin method in the time domain and using the Laplace integral transform in the frequency domain. Moreover, for the first time in the literature, a class of Sonin kernels in terms of the convolution series, which are a far-reaching generalization of the power series, is constructed. The new symmetrical Sonin kernels derived in the paper are represented in terms of the Wright function and the new special functions of the hypergeometric type that are extensions of the corresponding Horn functions in two variables. Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications, 2nd Edition)
13 pages, 290 KiB  
Article
Some Symmetry and Duality Theorems on Multiple Zeta(-Star) Values
by Kwang-Wu Chen, Minking Eie and Yao Lin Ong
Mathematics 2024, 12(20), 3292; https://doi.org/10.3390/math12203292 - 20 Oct 2024
Viewed by 839
Abstract
In this paper, we provide a symmetric formula and a duality formula relating multiple zeta values and zeta-star values. We find that the summation [...] Read more.
In this paper, we provide a symmetric formula and a duality formula relating multiple zeta values and zeta-star values. We find that the summation a+b=r1(1)aζ(a+2,{2}p1)ζ({1}b+1,{2}q) equals ζ({2}p,{1}r,{2}q)+(1)r+1ζ({2}q,r+2,{2}p1). With the help of this equation and Zagier’s ζ({2}p,3,{2}q) formula, we can easily determine ζ({2}p,1,{2}q) and several interesting expressions. Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications, 2nd Edition)
20 pages, 1441 KiB  
Article
On the Containment of the Unit Disc Image by Analytical Functions in the Lemniscate and Nephroid Domains
by Saiful R. Mondal
Mathematics 2024, 12(18), 2869; https://doi.org/10.3390/math12182869 - 14 Sep 2024
Viewed by 777
Abstract
Suppose that A1 is a class of analytic functions f:D={zC:|z|<1}C with normalization f(0)=1. Consider two functions [...] Read more.
Suppose that A1 is a class of analytic functions f:D={zC:|z|<1}C with normalization f(0)=1. Consider two functions Pl(z)=1+z and ΦNe(z)=1+zz3/3, which map the boundary of D to a cusp of lemniscate and to a twi-cusped kidney-shaped nephroid curve in the right half plane, respectively. In this article, we aim to construct functions fA0 for which (i) f(D)Pl(D)ΦNe(D) (ii) f(D)Pl(D), but f(D)ΦNe(D) (iii) f(D)ΦNe(D), but f(D)Pl(D). We validate the results graphically and analytically. To prove the results analytically, we use the concept of subordination. In this process, we establish the connection lemniscate (and nephroid) domain and functions, including gα(z):=1+αz2, |α|1, the polynomial gα,β(z):=1+αz+βz3, α,βR, as well as Lerch’s transcendent function, Incomplete gamma function, Bessel and Modified Bessel functions, and confluent and generalized hypergeometric functions. Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications, 2nd Edition)
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11 pages, 260 KiB  
Article
Absolute Monotonicity of Normalized Tail of Power Series Expansion of Exponential Function
by Feng Qi
Mathematics 2024, 12(18), 2859; https://doi.org/10.3390/math12182859 - 14 Sep 2024
Cited by 5 | Viewed by 1080
Abstract
In this work, the author reviews the origination of normalized tails of the Maclaurin power series expansions of infinitely differentiable functions, presents that the ratio between two normalized tails of the Maclaurin power series expansion of the exponential function is decreasing on the [...] Read more.
In this work, the author reviews the origination of normalized tails of the Maclaurin power series expansions of infinitely differentiable functions, presents that the ratio between two normalized tails of the Maclaurin power series expansion of the exponential function is decreasing on the positive axis, and proves that the normalized tail of the Maclaurin power series expansion of the exponential function is absolutely monotonic on the whole real axis. Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications, 2nd Edition)
11 pages, 453 KiB  
Article
On Polar Jacobi Polynomials
by Roberto S. Costas-Santos
Mathematics 2024, 12(17), 2767; https://doi.org/10.3390/math12172767 - 6 Sep 2024
Viewed by 666
Abstract
In the present work, we investigate certain algebraic and differential properties of the orthogonal polynomials with respect to a discrete–continuous Sobolev-type inner product defined in terms of the Jacobi measure. Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications, 2nd Edition)
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15 pages, 292 KiB  
Article
Generating Functions for Binomial Series Involving Harmonic-like Numbers
by Chunli Li and Wenchang Chu
Mathematics 2024, 12(17), 2685; https://doi.org/10.3390/math12172685 - 29 Aug 2024
Cited by 3 | Viewed by 895
Abstract
By employing the coefficient extraction method, a class of binomial series involving harmonic numbers will be reviewed through three hypergeometric F12(y2)-series. Numerous closed-form generating functions for infinite series containing binomial coefficients and harmonic numbers will be [...] Read more.
By employing the coefficient extraction method, a class of binomial series involving harmonic numbers will be reviewed through three hypergeometric F12(y2)-series. Numerous closed-form generating functions for infinite series containing binomial coefficients and harmonic numbers will be established, including several conjectured ones. Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications, 2nd Edition)
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