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21 pages, 1519 KiB

Open AccessArticle

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

Cited by 2 | Viewed by 254

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

Open AccessArticle

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 338

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

Open AccessArticle

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

Cited by 2 | Viewed by 597

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

Open AccessFeature PaperArticle

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 887

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

\sum_{a + b = r - 1} {(- 1)}^{a} ζ^{★} (a + 2, {2}^{p - 1}) ζ^{★} ({1}^{b + 1}, {2}^{q})

equals

ζ^{★} ({2}^{p}, {1}^{r}, {2}^{q}) + {(- 1)}^{r + 1} ζ^{★} ({2}^{q}, r + 2, {2}^{p - 1})

. 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

Open AccessArticle

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 848

Abstract

Suppose that

A_{1}

is a class of analytic functions

f : D = {z \in C : | z | < 1} \to C

with normalization

f (0) = 1

. Consider two functions [...] Read more.

Suppose that

A_{1}

is a class of analytic functions

f : D = {z \in C : | z | < 1} \to C

with normalization

f (0) = 1

. Consider two functions

P_{l} (z) = \sqrt{1 + z}

and

Φ_{N_{e}} (z) = 1 + z - z^{3} / 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

f \in A_{0}

for which (i)

f (D) \subset P_{l} (D) \cap Φ_{N_{e}} (D)

(ii)

f (D) \subset P_{l} (D)

, but

f (D) ⊄ Φ_{N_{e}} (D)

(iii)

f (D) \subset Φ_{N_{e}} (D)

, but

f (D) ⊄ P_{l} (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

g_{α} (z) : = \sqrt{1 + α z^{2}}

,

| α | \leq 1

, the polynomial

g_{α, β} (z) : = 1 + α z + β z^{3}

,

α, β \in R

, 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

Open AccessFeature PaperArticle

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 6 | Viewed by 1160

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

Open AccessArticle

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 700

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

Open AccessFeature PaperArticle

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 958

Abstract

By employing the coefficient extraction method, a class of binomial series involving harmonic numbers will be reviewed through three hypergeometric

{}_{2}F_{1} (y^{2})

-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

{}_{2}F_{1} (y^{2})

-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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