Mathematics

Research

16 pages, 316 KiB

Open AccessArticle

Asymptotic for Orthogonal Polynomials with Respect to a Rational Modification of a Measure Supported on the Semi-Axis

by Carlos Féliz-Sánchez, Héctor Pijeira-Cabrera and Javier Quintero-Roba

Mathematics 2024, 12(7), 1082; https://doi.org/10.3390/math12071082 - 3 Apr 2024

Viewed by 925

Abstract

Given a sequence of orthogonal polynomials

{L_{n}}_{n = 0}^{\infty}

, orthogonal with respect to a positive Borel

ν

measure supported on

R_{+}

, let

{Q_{n}}_{n = 0}^{\infty}

be the the sequence of [...] Read more.

Given a sequence of orthogonal polynomials

{L_{n}}_{n = 0}^{\infty}

, orthogonal with respect to a positive Borel

ν

measure supported on

R_{+}

, let

{Q_{n}}_{n = 0}^{\infty}

be the the sequence of orthogonal polynomials with respect to the modified measure

r (x) d ν (x)

, where r is certain rational function. This work is devoted to the proof of the relative asymptotic formula

\frac{Q_{n}^{(d)} (z)}{L_{n}^{(d)} (z)} ⇉_{n} \prod_{k = 1}^{N_{1}} {(\frac{\sqrt{a_{k}} + i}{\sqrt{z} + \sqrt{a_{k}}})}^{A_{k}} \prod_{j = 1}^{N_{2}} {(\frac{\sqrt{z} + \sqrt{b_{j}}}{\sqrt{b_{j}} + i})}^{B_{j}}

, on compact subsets of

C ∖ R_{+}

, where

a_{k}

and

b_{j}

are the zeros and poles of r, and the

A_{k}

,

B_{j}

are their respective multiplicities. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions: Recent Trends and Their Applications)

15 pages, 737 KiB

Open AccessArticle

Some Properties of Generalized Apostol-Type Frobenius–Euler–Fibonacci Polynomials

by Maryam Salem Alatawi, Waseem Ahmad Khan, Can Kızılateş and Cheon Seoung Ryoo

Mathematics 2024, 12(6), 800; https://doi.org/10.3390/math12060800 - 8 Mar 2024

Cited by 1 | Viewed by 519

Abstract

In this paper, by using the Golden Calculus, we introduce the generalized Apostol-type Frobenius–Euler–Fibonacci polynomials and numbers; additionally, we obtain various fundamental identities and properties associated with these polynomials and numbers, such as summation theorems, difference equations, derivative properties, recurrence relations, and more. [...] Read more.

In this paper, by using the Golden Calculus, we introduce the generalized Apostol-type Frobenius–Euler–Fibonacci polynomials and numbers; additionally, we obtain various fundamental identities and properties associated with these polynomials and numbers, such as summation theorems, difference equations, derivative properties, recurrence relations, and more. Subsequently, we present summation formulas, Stirling–Fibonacci numbers of the second kind, and relationships for these polynomials and numbers. Finally, we define the new family of the generalized Apostol-type Frobenius–Euler–Fibonacci matrix and obtain some factorizations of this newly established matrix. Using Mathematica, the computational formulae and graphical representation for the mentioned polynomials are obtained. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions: Recent Trends and Their Applications)

► Show Figures

Figure 1

16 pages, 417 KiB

Open AccessArticle

Mixed-Type Hypergeometric Bernoulli–Gegenbauer Polynomials

by Dionisio Peralta, Yamilet Quintana and Shahid Ahmad Wani

Mathematics 2023, 11(18), 3920; https://doi.org/10.3390/math11183920 - 15 Sep 2023

Viewed by 1227

Abstract

In this paper, we consider a novel family of the mixed-type hypergeometric Bernoulli–Gegenbauer polynomials. This family represents a fascinating fusion between two distinct categories of special functions: hypergeometric Bernoulli polynomials and Gegenbauer polynomials. We focus our attention on some algebraic and differential properties [...] Read more.

In this paper, we consider a novel family of the mixed-type hypergeometric Bernoulli–Gegenbauer polynomials. This family represents a fascinating fusion between two distinct categories of special functions: hypergeometric Bernoulli polynomials and Gegenbauer polynomials. We focus our attention on some algebraic and differential properties of this class of polynomials, including its explicit expressions, derivative formulas, matrix representations, matrix-inversion formulas, and other relations connecting it with the hypergeometric Bernoulli polynomials. Furthermore, we show that unlike the hypergeometric Bernoulli polynomials and Gegenbauer polynomials, the mixed-type hypergeometric Bernoulli–Gegenbauer polynomials do not fulfill either Hanh or Appell conditions. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions: Recent Trends and Their Applications)

► Show Figures

Figure 1

11 pages, 280 KiB

Open AccessArticle

The Recurrence Coefficients of Orthogonal Polynomials with a Weight Interpolating between the Laguerre Weight and the Exponential Cubic Weight

by Chao Min and Pixin Fang

Mathematics 2023, 11(18), 3842; https://doi.org/10.3390/math11183842 - 7 Sep 2023

Cited by 1 | Viewed by 653

Abstract

In this paper, we consider the orthogonal polynomials with respect to the weight [...] Read more.

In this paper, we consider the orthogonal polynomials with respect to the weight

w (x) = w (x; s) : = x^{λ} e^{- N [x + s (x^{3} - x)]}, x \in R^{+},

where

λ > 0

,

N > 0

and

0 \leq s \leq 1

. By using the ladder operator approach, we obtain a pair of second-order nonlinear difference equations and a pair of differential–difference equations satisfied by the recurrence coefficients

α_{n} (s)

and

β_{n} (s)

. We also establish the relation between the associated Hankel determinant and the recurrence coefficients. From Dyson’s Coulomb fluid approach, we prove that the recurrence coefficients converge and the limits are derived explicitly when

q : = n / N

is fixed as

n \to \infty

. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions: Recent Trends and Their Applications)

17 pages, 321 KiB

Open AccessArticle

Properties of Multivariate Hermite Polynomials in Correlation with Frobenius–Euler Polynomials

by Mohra Zayed, Shahid Ahmad Wani and Yamilet Quintana

Mathematics 2023, 11(16), 3439; https://doi.org/10.3390/math11163439 - 8 Aug 2023

Cited by 3 | Viewed by 730

Abstract

A comprehensive framework has been developed to apply the monomiality principle from mathematical physics to various mathematical concepts from special functions. This paper presents research on a novel family of multivariate Hermite polynomials associated with Apostol-type Frobenius–Euler polynomials. The study derives the generating [...] Read more.

A comprehensive framework has been developed to apply the monomiality principle from mathematical physics to various mathematical concepts from special functions. This paper presents research on a novel family of multivariate Hermite polynomials associated with Apostol-type Frobenius–Euler polynomials. The study derives the generating expression, operational rule, differential equation, and other defining characteristics for these polynomials. Additionally, the monomiality principle for these polynomials is verified. Moreover, the research establishes series representations, summation formulae, and operational and symmetric identities, as well as recurrence relations satisfied by these polynomials. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions: Recent Trends and Their Applications)

20 pages, 383 KiB

Open AccessArticle

Differential Properties of Jacobi-Sobolev Polynomials and Electrostatic Interpretation

by Héctor Pijeira-Cabrera, Javier Quintero-Roba and Juan Toribio-Milane

Mathematics 2023, 11(15), 3420; https://doi.org/10.3390/math11153420 - 6 Aug 2023

Cited by 1 | Viewed by 2171

Abstract

We study the sequence of monic polynomials

{S_{n}}_{n ⩾ 0}

, orthogonal with respect to the Jacobi-Sobolev inner product [...] Read more.

We study the sequence of monic polynomials

{S_{n}}_{n ⩾ 0}

, orthogonal with respect to the Jacobi-Sobolev inner product

{⟨ f, g ⟩}_{s} = \int_{- 1}^{1} f (x) g (x) d μ^{α, β} (x) + \sum_{j = 1}^{N} \sum_{k = 0}^{d_{j}} λ_{j, k} f^{(k)} (c_{j}) g^{(k)} (c_{j}),

where

N, d_{j} \in Z_{+}

,

λ_{j, k} ⩾ 0

,

d μ^{α, β} (x) = {(1 - x)}^{α} {(1 + x)}^{β} d x

,

α, β > - 1

, and

c_{j} \in R ∖ (- 1, 1)

. A connection formula that relates the Sobolev polynomials

S_{n}

with the Jacobi polynomials is provided, as well as the ladder differential operators for the sequence

{S_{n}}_{n ⩾ 0}

and a second-order differential equation with a polynomial coefficient that they satisfied. We give sufficient conditions under which the zeros of a wide class of Jacobi-Sobolev polynomials can be interpreted as the solution of an electrostatic equilibrium problem of n unit charges moving in the presence of a logarithmic potential. Several examples are presented to illustrate this interpretation. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions: Recent Trends and Their Applications)

15 pages, 306 KiB

Open AccessArticle

A Look at Generalized Degenerate Bernoulli and Euler Matrices

by Juan Hernández, Dionisio Peralta and Yamilet Quintana

Mathematics 2023, 11(12), 2731; https://doi.org/10.3390/math11122731 - 16 Jun 2023

Cited by 1 | Viewed by 781

Abstract

In this paper, we consider the generalized degenerate Bernoulli/Euler polynomial matrices and study some algebraic properties for them. In particular, we focus our attention on some matrix-inversion formulae involving these matrices. Furthermore, we provide analytic properties for the so-called generalized degenerate Pascal matrix [...] Read more.

In this paper, we consider the generalized degenerate Bernoulli/Euler polynomial matrices and study some algebraic properties for them. In particular, we focus our attention on some matrix-inversion formulae involving these matrices. Furthermore, we provide analytic properties for the so-called generalized degenerate Pascal matrix of the first kind, and some factorizations for the generalized degenerate Euler polynomial matrix. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions: Recent Trends and Their Applications)

15 pages, 349 KiB

Open AccessArticle

Sequentially Ordered Sobolev Inner Product and Laguerre–Sobolev Polynomials

by Abel Díaz-González, Juan Hernández and Héctor Pijeira-Cabrera

Mathematics 2023, 11(8), 1956; https://doi.org/10.3390/math11081956 - 20 Apr 2023

Cited by 1 | Viewed by 1086

Abstract

We study the sequence of polynomials

{S_{n}}_{n \geq 0}

that are orthogonal with respect to the general discrete Sobolev-type inner product [...] Read more.

We study the sequence of polynomials

{S_{n}}_{n \geq 0}

that are orthogonal with respect to the general discrete Sobolev-type inner product

{⟨ f, g ⟩}_{s} = \int f (x) g (x) d μ (x) + \sum_{j = 1}^{N} \sum_{k = 0}^{d_{j}} λ_{j, k} f^{(k)} (c_{j}) g^{(k)} (c_{j}),

where

μ

is a finite Borel measure whose support

supp (μ)

is an infinite set of the real line,

λ_{j, k} \geq 0

, and the mass points

c_{i}

,

i = 1, \dots, N

are real values outside the interior of the convex hull of

supp (μ)

(

c_{i} \in R \ C_{h} {(supp (μ))}^{\circ})

. Under some restriction of order in the discrete part of

{⟨ \cdot, \cdot ⟩}_{s}

, we prove that

S_{n}

has at least

n - d^{*}

zeros on

C_{h} {(supp (μ))}^{\circ}

, being

d^{*}

the number of terms in the discrete part of

{⟨ \cdot, \cdot ⟩}_{s}

. Finally, we obtain the outer relative asymptotic for

{S_{n}}

in the case that the measure

μ

is the classical Laguerre measure, and for each mass point, only one order derivative appears in the discrete part of

{⟨ \cdot, \cdot ⟩}_{s}

. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions: Recent Trends and Their Applications)

13 pages, 285 KiB

Open AccessArticle

On Apostol-Type Hermite Degenerated Polynomials

by Clemente Cesarano, William Ramírez, Stiven Díaz, Adnan Shamaoon and Waseem Ahmad Khan

Mathematics 2023, 11(8), 1914; https://doi.org/10.3390/math11081914 - 18 Apr 2023

Cited by 5 | Viewed by 918

Abstract

This article presents a generalization of new classes of degenerated Apostol–Bernoulli, Apostol–Euler, and Apostol–Genocchi Hermite polynomials of level m. We establish some algebraic and differential properties for generalizations of new classes of degenerated Apostol–Bernoulli polynomials. These results are shown using generating function [...] Read more.

This article presents a generalization of new classes of degenerated Apostol–Bernoulli, Apostol–Euler, and Apostol–Genocchi Hermite polynomials of level m. We establish some algebraic and differential properties for generalizations of new classes of degenerated Apostol–Bernoulli polynomials. These results are shown using generating function methods for Apostol–Euler and Apostol–Genocchi Hermite polynomials of level m. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions: Recent Trends and Their Applications)

18 pages, 325 KiB

Open AccessArticle

The Cauchy Exponential of Linear Functionals on the Linear Space of Polynomials

by Francisco Marcellán and Ridha Sfaxi

Mathematics 2023, 11(8), 1895; https://doi.org/10.3390/math11081895 - 17 Apr 2023

Cited by 1 | Viewed by 840

Abstract

In this paper, we introduce the notion of the Cauchy exponential of a linear functional on the linear space of polynomials in one variable with real or complex coefficients using a functional equation by using the so-called moment equation. It seems that this [...] Read more.

In this paper, we introduce the notion of the Cauchy exponential of a linear functional on the linear space of polynomials in one variable with real or complex coefficients using a functional equation by using the so-called moment equation. It seems that this notion hides several properties and results. Our purpose is to explore some of these properties and to compute the Cauchy exponential of some special linear functionals. Finally, a new characterization of the positive-definiteness of a linear functional is given. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions: Recent Trends and Their Applications)

14 pages, 287 KiB

Open AccessArticle

Integral Inequalities Involving Strictly Monotone Functions

by Mohamed Jleli and Bessem Samet

Mathematics 2023, 11(8), 1873; https://doi.org/10.3390/math11081873 - 14 Apr 2023

Viewed by 912

Abstract

Functional inequalities involving special functions are very useful in mathematical analysis, and several interesting results have been obtained in this topic. Several methods have been used by many authors in order to derive upper or lower bounds of certain special functions. In this [...] Read more.

Functional inequalities involving special functions are very useful in mathematical analysis, and several interesting results have been obtained in this topic. Several methods have been used by many authors in order to derive upper or lower bounds of certain special functions. In this paper, we establish some general integral inequalities involving strictly monotone functions. Next, some special cases are discussed. In particular, several estimates of trigonometric and hyperbolic functions are deduced. For instance, we show that Mitrinović-Adamović inequality, Lazarevic inequality, and Cusa-Huygens inequality are special cases of our obtained results. Moreover, an application to integral equations is provided. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions: Recent Trends and Their Applications)

9 pages, 1062 KiB

Open AccessArticle

A New Discretization Scheme for the Non-Isotropic Stockwell Transform

by Hari M. Srivastava, Azhar Y. Tantary and Firdous A. Shah

Mathematics 2023, 11(8), 1839; https://doi.org/10.3390/math11081839 - 12 Apr 2023

Cited by 3 | Viewed by 734

Abstract

To avoid the undesired angular expansion of the sampling grid in the discrete non-isotropic Stockwell transform, in this communication we propose a scale-dependent discretization scheme that controls both the radial and angular expansions in unison. Based on the new discretization scheme, we derive [...] Read more.

To avoid the undesired angular expansion of the sampling grid in the discrete non-isotropic Stockwell transform, in this communication we propose a scale-dependent discretization scheme that controls both the radial and angular expansions in unison. Based on the new discretization scheme, we derive a sufficient condition for the construction of Stockwell frames in

L^{2} (R^{2})

. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions: Recent Trends and Their Applications)

► Show Figures

Figure 1

17 pages, 329 KiB

Open AccessArticle

Redheffer-Type Bounds of Special Functions

by Reem Alzahrani and Saiful R. Mondal

Mathematics 2023, 11(2), 379; https://doi.org/10.3390/math11020379 - 11 Jan 2023

Viewed by 1051

Abstract

In this paper, we aim to construct inequalities of the Redheffer type for certain functions defined by the infinite product involving the zeroes of these functions. The key tools used in our proofs are classical results on the monotonicity of the ratio of [...] Read more.

In this paper, we aim to construct inequalities of the Redheffer type for certain functions defined by the infinite product involving the zeroes of these functions. The key tools used in our proofs are classical results on the monotonicity of the ratio of differentiable functions. The results are proved using the

n^{th}

positive zero, denoted by

b_{n} (ν)

. Special cases lead to several examples involving special functions, namely, Bessel, Struve, and Hurwitz functions, as well as several other trigonometric functions. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions: Recent Trends and Their Applications)

Journal Menu

Journal Browser

Orthogonal Polynomials and Special Functions: Recent Trends and Their Applications

Share This Special Issue

Special Issue Editor

Special Issue Information

Keywords

Published Papers (13 papers)

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI