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Open AccessArticle

Novel Error Bounds of Milne Formula Type Inequalities via Quantum Calculus with Computational Analysis and Applications

by Amjad E. Hazma, Abdul Mateen, Talha Anwar and Ghada AlNemer

Mathematics 2025, 13(22), 3698; https://doi.org/10.3390/math13223698 - 18 Nov 2025

Viewed by 347

Quantum calculus is a powerful extension of classical calculus, providing novel tools for deriving sharper and more efficient analytical results without relying on limits. This study investigates error estimations for Milne formula-type inequalities within the framework of quantum calculus, offering a fresh perspective [...] Read more.

Quantum calculus is a powerful extension of classical calculus, providing novel tools for deriving sharper and more efficient analytical results without relying on limits. This study investigates error estimations for Milne formula-type inequalities within the framework of quantum calculus, offering a fresh perspective on numerical integration theory. New variants of Milne’s formula-type inequalities are established for q-differentiable convex functions by first deriving a key quantum integral identity. The primary aim of this work is to obtain sharper and more accurate bounds for Milne’s formula compared to existing results in the literature. The validity of the proposed results is demonstrated through illustrative examples and graphical analysis. Furthermore, applications to special means of real numbers, the Mittag–Leffler function, and numerical integration formulas are presented to emphasize the practical significance of the findings. This study contributes to advancing the theoretical foundations of both classical and quantum calculus and enhances the understanding of integral inequality theory. Full article

(This article belongs to the Section C: Mathematical Analysis)

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26 pages, 565 KB

Open AccessArticle

Efficient Scheme for Solving Tempered Fractional Quantum Differential Problem

by Lakhlifa Sadek and Ali Algefary

Fractal Fract. 2025, 9(11), 709; https://doi.org/10.3390/fractalfract9110709 - 3 Nov 2025

Viewed by 454

Abstract

This study presents a novel family of operators, referred to as tempered quantum fractional operators, and investigates the well-posedness of related tempered quantum fractional differential equations. The q-Adams predictor–corrector method is employed to conduct the analysis. By carefully adjusting the scheme’s parameters, [...] Read more.

This study presents a novel family of operators, referred to as tempered quantum fractional operators, and investigates the well-posedness of related tempered quantum fractional differential equations. The q-Adams predictor–corrector method is employed to conduct the analysis. By carefully adjusting the scheme’s parameters, the convergence rate can be controlled, while computational expenses increase linearly with time. Numerical simulations confirm the effectiveness and precision of the introduced algorithm. Full article

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20 pages, 699 KB

Open AccessArticle

Analysis of Quantum Multiplicative Calculus and Related Inequalities

by Muhammad Nasim Aftab, Saad Ihsan Butt, Mohammed Alammar and Youngsoo Seol

Mathematics 2025, 13(21), 3381; https://doi.org/10.3390/math13213381 - 23 Oct 2025

Cited by 1 | Viewed by 450

Abstract

This article investigates the exact meaning of a quantum derivative result and the corresponding definition of a quantum definite integral in multiplicative calculus from a geometrical viewpoint. After this critical analysis, we give an accurate definition of the

q

-multiplicative definite integral and [...] Read more.

This article investigates the exact meaning of a quantum derivative result and the corresponding definition of a quantum definite integral in multiplicative calculus from a geometrical viewpoint. After this critical analysis, we give an accurate definition of the

q

-multiplicative definite integral and the corresponding derivative result. Additionally, an example pertaining to

q

-multiplicative definite integrals is presented, and rigorous analysis to prove several fundamental results is provided. In addition, two other concepts are defined: the left

q

-multiplicative derivative and definite integral and the right

q

-multiplicative derivative and definite integral. Finally, several

q

-multiplicative Hermite–Hadamard-type inequalities are constructed, and related examples are shown to support our recent findings. Full article

(This article belongs to the Special Issue Current Topics in Optimization, Inequalities and Convex Function Theory)

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18 pages, 724 KB

Open AccessArticle

Coefficient Estimates and Symmetry Analysis for Certain Families of Bi-Univalent Functions Defined by the q-Bernoulli Polynomial

by Abbas Kareem Wanas, Qasim Ali Shakir and Adriana Catas

Symmetry 2025, 17(9), 1532; https://doi.org/10.3390/sym17091532 - 13 Sep 2025

Viewed by 663

Abstract

In the present work, we define certain families,

M_{Σ} (μ, Υ, ℷ, q; x)

and

N_{Σ} (μ, Υ, ℷ, q; x)

, of normalized holomorphic and bi-univalent functions associated with Bazilevič [...] Read more.

In the present work, we define certain families,

M_{Σ} (μ, Υ, ℷ, q; x)

and

N_{Σ} (μ, Υ, ℷ, q; x)

, of normalized holomorphic and bi-univalent functions associated with Bazilevič functions and

ℷ

-pseudo functions involving the

q

-Bernoulli polynomial, which is defined by the symmetric nature of quantum calculus in the open unit disk

U

. We determine the upper bounds for the initial symmetry Taylor–Maclaurin coefficients and the Fekete–Szegö-type inequalities of functions in the families we have introduced here. In addition, we indicate certain special cases and consequences for our results. Full article

(This article belongs to the Special Issue Applications of Special Functions in Complex Analysis and Their Symmetries)

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12 pages, 1622 KB

Open AccessArticle

Symmetry and Quantum Calculus in Defining New Classes of Analytic Functions

by Fuad Alsarari, Abdulbasit Darem, Muflih Alhazmi and Alaa Awad Alzulaibani

Mathematics 2025, 13(14), 2317; https://doi.org/10.3390/math13142317 - 21 Jul 2025

Viewed by 776

Abstract

This paper introduces a novel class of analytic functions that integrates q-calculus, Janowski-type functions, and (a, b)-symmetrical functions. By exploring convolution operations and quantum calculus, we establish essential convolution conditions that lay the groundwork for subsequent research. Building on [...] Read more.

This paper introduces a novel class of analytic functions that integrates q-calculus, Janowski-type functions, and (a, b)-symmetrical functions. By exploring convolution operations and quantum calculus, we establish essential convolution conditions that lay the groundwork for subsequent research. Building on a new conceptual framework, we also define analogous neighborhoods for the classes

{\bar{F}}_{q}^{a, b} (F, H)

and investigate related neighborhood properties. These developments provide a deeper understanding of the structural and analytical behavior of these functions, opening up avenues for future study. Full article

(This article belongs to the Special Issue Geometric Function Theory and Applications―Festschrift for Grigore-Stefan Salagean's 75th Birthday)

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24 pages, 1601 KB

Open AccessArticle

Finding the q-Appell Convolution of Certain Polynomials Within the Context of Quantum Calculus

by Waseem Ahmad Khan, Khidir Shaib Mohamed, Francesco Aldo Costabile, Can Kızılateş and Cheon Seoung Ryoo

Mathematics 2025, 13(13), 2073; https://doi.org/10.3390/math13132073 - 23 Jun 2025

Cited by 5 | Viewed by 526

Abstract

This article introduces the theory of three-variable q-truncated exponential Gould–Hopper-based Appell polynomials by employing a generating function approach that incorporates q-calculus functions. This study further explores these polynomials by using a computational algebraic approach. The determinant form, recurrences, and differential equations [...] Read more.

This article introduces the theory of three-variable q-truncated exponential Gould–Hopper-based Appell polynomials by employing a generating function approach that incorporates q-calculus functions. This study further explores these polynomials by using a computational algebraic approach. The determinant form, recurrences, and differential equations are proven. Relationships with the monomiality principle are given. Finally, graphical representations are presented to illustrate the behavior and potential applications of the three-variable q-truncated exponential Gould–Hopper-based Appell polynomials. Full article

(This article belongs to the Special Issue Polynomials: Theory and Applications, 2nd Edition)

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19 pages, 330 KB

Open AccessArticle

On the Existence of (p,q)-Solutions for the Post-Quantum Langevin Equation: A Fixed-Point-Based Approach

by Mohammed Jasim Mohammed, Ali Ghafarpanah, Sina Etemad, Sotiris K. Ntouyas and Jessada Tariboon

Axioms 2025, 14(6), 474; https://doi.org/10.3390/axioms14060474 - 19 Jun 2025

Viewed by 657

Abstract

The two-parameter

(p, q)

-operators are a new family of operators in calculus that have shown their capabilities in modeling various systems in recent years. Following this path, in this paper, we present a new construction of the Langevin equation [...] Read more.

The two-parameter

(p, q)

-operators are a new family of operators in calculus that have shown their capabilities in modeling various systems in recent years. Following this path, in this paper, we present a new construction of the Langevin equation using two-parameter

(p, q)

-Caputo derivatives. For this new Langevin equation, equivalently, we obtain the solution structure as a post-quantum integral equation and then conduct an existence analysis via a fixed-point-based approach. The use of theorems such as the Krasnoselskii and Leray–Schauder fixed-point theorems will guarantee the existence of solutions to this equation, whose uniqueness is later proven by Banach’s contraction principle. Finally, we provide three examples in different structures and validate the results numerically. Full article

(This article belongs to the Special Issue Fractional Order Functional Differential Equations and Fixed Point Theory)

19 pages, 1266 KB

Open AccessArticle

A New Generalization of q-Truncated Polynomials Associated with q-General Polynomials

by Waseem Ahmad Khan, Khidir Shaib Mohamed, Francesco Aldo Costabile, Can Kızılateş and Cheon Seoung Ryoo

Mathematics 2025, 13(12), 1964; https://doi.org/10.3390/math13121964 - 14 Jun 2025

Cited by 1 | Viewed by 500

Abstract

This article presents the theory of trivariate q-truncated Gould–Hopper polynomials through a generating function approach utilizing q-calculus functions. These polynomials are subsequently examined within the framework of quasi-monomiality, leading to the establishment of fundamental operational identities. Operational representations are then derived, [...] Read more.

This article presents the theory of trivariate q-truncated Gould–Hopper polynomials through a generating function approach utilizing q-calculus functions. These polynomials are subsequently examined within the framework of quasi-monomiality, leading to the establishment of fundamental operational identities. Operational representations are then derived, and q-differential and partial differential equations are formulated for the trivariate q-truncated Gould–Hopper polynomials. Summation formulae are presented to elucidate the analytical properties of these polynomials. Finally, graphical representations are provided to illustrate the behavior of trivariate q-truncated Gould–Hopper polynomials and their potential applications. Full article

(This article belongs to the Section E: Applied Mathematics)

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18 pages, 422 KB

Open AccessArticle

On Higher-Order Generalized Fibonacci Hybrid Numbers with q-Integer Components: New Properties, Recurrence Relations, and Matrix Representations

by Can Kızılateş, Emrah Polatlı, Nazlıhan Terzioğlu and Wei-Shih Du

Symmetry 2025, 17(4), 584; https://doi.org/10.3390/sym17040584 - 11 Apr 2025

Cited by 2 | Viewed by 847

Abstract

Many properties of special numbers, such as sum formulas, symmetric properties, and their relationships with each other, have been studied in the literature with the help of the Binet formula and generating function. In this paper, higher-order generalized Fibonacci hybrid numbers with q [...] Read more.

Many properties of special numbers, such as sum formulas, symmetric properties, and their relationships with each other, have been studied in the literature with the help of the Binet formula and generating function. In this paper, higher-order generalized Fibonacci hybrid numbers with q-integer components are defined through the utilization of q-integers and higher-order generalized Fibonacci numbers. Several special cases of these newly established hybrid numbers are presented. The article explores the integration of q-calculus and hybrid numbers, resulting in the derivation of a Binet-like formula, novel identities, a generating function, a recurrence relation, an exponential generating function, and sum properties of hybrid numbers with quantum integer coefficients. Furthermore, new identities for these types of hybrids are obtained using two novel special matrices. To substantiate the findings, numerical examples are provided, generated with the assistance of Maple. Full article

(This article belongs to the Special Issue Symmetry in Integrable Systems and Soliton Theories)

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24 pages, 652 KB

Open AccessArticle

Fundamentals of Dual Basic Symmetric Quantum Calculus and Its Fractional Perspectives

by Muhammad Nasim Aftab, Saad Ihsan Butt and Youngsoo Seol

Fractal Fract. 2025, 9(4), 237; https://doi.org/10.3390/fractalfract9040237 - 10 Apr 2025

Cited by 2 | Viewed by 1031

Abstract

Taylor expansion is a remarkable tool with broad applications in analysis, science, engineering, and mathematics. In this manuscript, we derive a proof of generalized Taylor expansion for polynomials and write its particular case in symmetric quantum calculus. In addition, we define a novel [...] Read more.

Taylor expansion is a remarkable tool with broad applications in analysis, science, engineering, and mathematics. In this manuscript, we derive a proof of generalized Taylor expansion for polynomials and write its particular case in symmetric quantum calculus. In addition, we define a novel type of calculus that is called symmetric

(p, q)

- or dual basic symmetric quantum calculus. Moreover, we derive a symmetric

(p, q)

-Taylor expansion for polynomials based on this calculus. After that, we investigate Taylor’s formulae through an example. Furthermore, we define symmetric definite

(p, q)

-integral and derive a fundamental law of symmetric

(p, q)

-calculus. Finally, we derive the symmetric

(p, q)

-Cauchy formula for integrals that enables us to construct the fractional perspectives of

(p, q)

-symmetric integrals. Full article

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22 pages, 378 KB

Open AccessArticle

A Novel Family of Starlike Functions Involving Quantum Calculus and a Special Function

by Baseer Gul, Daniele Ritelli, Reem K. Alhefthi and Muhammad Arif

Fractal Fract. 2025, 9(3), 179; https://doi.org/10.3390/fractalfract9030179 - 14 Mar 2025

Cited by 2 | Viewed by 953

Abstract

The intent of quantum calculus, briefly q-calculus, is to find q-analogues of mathematical entities so that the original object is achieved when a certain limit is taken. In the case of q-analogue of the ordinary derivative, the limit is [...] Read more.

The intent of quantum calculus, briefly q-calculus, is to find q-analogues of mathematical entities so that the original object is achieved when a certain limit is taken. In the case of q-analogue of the ordinary derivative, the limit is

q \to 1^{-}

. Also, the study of integral as well as differential operators has remained a significant field of inquiry from the early developments of function theory. In the present article, a subclass

S_{s c} (μ, q)

of functions being analytic in

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

is introduced. The definition of

S_{s c} (μ, q)

involves the concepts of subordination, that of q-derivative and q-Ruscheweyh operators. Since coefficient estimates and coefficient functionals provide insights into different geometric properties of analytic functions, for this newly defined subclass, we investigate coefficient estimates up to

a_{4}

, in which both bounds for

| a_{2} |

and

| a_{3} |

are sharp, while that of

| a_{4} |

is sharp in one case. We also discuss the sharp Fekete–Szegö functional for the said class. In addition, Toeplitz determinant bounds up to

T_{3} (2)

(sharp in some cases) and sufficient condition are obtained. Several consequences derived from our above-mentioned findings are also part of the discussion. Full article

(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)

21 pages, 1519 KB

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 9 | Viewed by 848

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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12 pages, 264 KB

Open AccessArticle

Subclasses of q-Uniformly Starlike Functions Obtained via the q-Carlson–Shaffer Operator

by Qiuxia Hu, Rizwan Salim Badar and Muhammad Ghaffar Khan

Axioms 2024, 13(12), 842; https://doi.org/10.3390/axioms13120842 - 29 Nov 2024

Viewed by 829

Abstract

This article investigates the applications of the q-Carlson–Shaffer operator on subclasses of q-uniformly starlike functions, introducing the class

S T_{q} (m, c, d, β)

. The study establishes a necessary condition for membership in this class [...] Read more.

This article investigates the applications of the q-Carlson–Shaffer operator on subclasses of q-uniformly starlike functions, introducing the class

S T_{q} (m, c, d, β)

. The study establishes a necessary condition for membership in this class and examines its behavior within conic domains. The article delves into properties such as coefficient bounds, the Fekete–Szegö inequality, and criteria defined via the Hadamard product, providing both necessary and sufficient conditions for these properties. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory, 3rd Edition)

13 pages, 300 KB

Open AccessArticle

A Novel Family of q-Mittag-Leffler-Based Bessel and Tricomi Functions via Umbral Approach

by Waseem Ahmad Khan, Mofareh Alhazmi and Tabinda Nahid

Symmetry 2024, 16(12), 1580; https://doi.org/10.3390/sym16121580 - 26 Nov 2024

Cited by 4 | Viewed by 1022

Abstract

Many properties of special polynomials, such as recurrence relations, sum formulas, integral transforms and symmetric identities, have been studied in the literature with the help of generating functions and their functional equations. In this paper, we introduce hybrid forms of q-Mittag-Leffler functions. [...] Read more.

Many properties of special polynomials, such as recurrence relations, sum formulas, integral transforms and symmetric identities, have been studied in the literature with the help of generating functions and their functional equations. In this paper, we introduce hybrid forms of q-Mittag-Leffler functions. The q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are constructed using a q-symbolic operator. The generating functions, series definitions, q-derivative formulas and q-recurrence formulas for q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are obtained. The

N_{q}

-transforms and

{}_{q}N

-transforms of q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are obtained. These hybrid q-special functions are also studied by plotting their graphs for specific values of the indices and parameters. Full article

(This article belongs to the Special Issue Special Functions, Integral Transforms and Polynomial Sequences in Real World with Symmetry)

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17 pages, 294 KB

Open AccessArticle

New Properties and Matrix Representations on Higher-Order Generalized Fibonacci Quaternions with q-Integer Components

by Can Kızılateş, Wei-Shih Du, Nazlıhan Terzioğlu and Ren-Chuen Chen

Axioms 2024, 13(10), 677; https://doi.org/10.3390/axioms13100677 - 30 Sep 2024

Cited by 2 | Viewed by 1192

Abstract

In this paper, by using q-integers and higher-order generalized Fibonacci numbers, we define the higher-order generalized Fibonacci quaternions with q-integer components. We give some special cases of these newly established quaternions. This article examines q-calculus and quaternions together. We obtain [...] Read more.

In this paper, by using q-integers and higher-order generalized Fibonacci numbers, we define the higher-order generalized Fibonacci quaternions with q-integer components. We give some special cases of these newly established quaternions. This article examines q-calculus and quaternions together. We obtain a Binet-like formula, some new identities, a generating function, a recurrence relation, an exponential generating function, and sum properties of quaternions with quantum integer coefficients. In addition, we obtain some new identities for these types of quaternions by using three new special matrices. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities)

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