Search Results (65)

Mock theta functions, introduced by Ramanujan in his last letter to Hardy, play a significant role in q-series theory and have natural connections to fractional q-calculus. In this paper, we study bilateral hypergeometric series of the form ${}_2{\psi }_2$= ${\sum }_{n=-\infty }^{\infty }{\displaystyle \frac{{(a,b;q)}_n}{{(c,d;q)}_n}}{z}^n,$ where ${(a;q)}_n$ denotes the q-shifted factorial. Using Slater’s three-term transformation formula for bilateral ${}_2{\psi }_2$ series, we derive new identities for Ramanujan’s mock theta functions of orders 2, 3, 6, and 8. These transformations reveal previously unknown relationships between different q-series representations and extend the classical theory of mock theta functions within the framework of q-special functions. Full article

In this paper, we introduce a novel version of the Legendre polynomials in the bicomplex system. We investigate the essential properties of the Legendre polynomial, focusing on its bicomplex structure, generating functions, orthogonality, and recurrence relations. We present a solution to the Legendre differential equation in bicomplex space. Additionally, we discuss both theoretical and practical contributions, especially in bicomplex Riemann Liouville fractional calculus. We numerically study the construction of bicomplex Legendre polynomials, orthogonality, spectral projection, coefficient decay, and spectral convergence in bicomplex space. The findings contribute to a deeper insight into bicomplex functions, paving the way for further developments in science and mathematical analysis, and providing a foundation for future research on special functions and fractional operators within the bicomplex setting. Full article

This paper introduces novel subfamilies of analytic and bi-univalent functions in $\mathrm{\Omega }=\left\{\varsigma \in \mathbb{C}:\left|\varsigma \right|<1\right\}$, defined by applying a linear operator associated with the Mittag–Leffler function and requiring subordination to domains related to generalized bivariate Fibonacci polynomials. The proposed framework provides a unified treatment that generalizes numerous earlier studies by incorporating parameters controlling both the operator’s fractional calculus features and the domain’s combinatorial geometry. For these subfamilies, we establish initial coefficient bounds ($\left|d_2\right|$, $\left|d_3\right|$) and solve the Fekete–Szegö problem ($\left|d_3-\xi d_2^2\right|$). The derived inequalities are interesting, and their proofs leverage the intricate interplay between the series expansions of the Mittag–Leffler function and the generating function of the Fibonacci polynomials. By specializing the parameters governing the operator and the polynomial domain, we show how our main theorems systematically recover and extend a wide range of known results from the literature, thereby demonstrating the generality and unifying power of our approach. Full article

In this paper, we evaluate a general class of finite integrals involving the error function, generalized Mittag-Leffler functions, and incomplete Aleph functions. The main result provides a unified framework that extends several known formulas related to the incomplete Gamma, I-, and H-functions. Under suitable conditions, these results reduce to many classical special cases. We discuss convergence conditions that justify the validity of the obtained formulas and include explicit corollaries that highlight connections with earlier results in the literature. To illustrate applicability, we present numerical examples and graphs, demonstrating the behavior of the error function integral and Mittag-Leffler functions for specific parameter values. These integrals arise naturally in fractional calculus, probability theory, viscoelasticity, and anomalous diffusion, underscoring the importance of the present work in both mathematical analysis and applications. Full article

The aim of this manuscript is to introduce the fractional integral inequalities of H-H types via multiplicative (Antagana-Baleanu) A-B fractional operators. We also provide the fractional version of the H-H type of the product and quotient of multiplicative superquadratic and multiplicative subquadratic functions via the same operators. Superquadratic functions, have stronger convexity-like behavior. They provide sharper bounds and more refined inequalities, which are valuable in optimization, information theory, and related fields. The use of multiplicative fractional operators establishes a nonlinear fractional structure, enhancing the analytical tools available for studying dynamic and nonlinear systems. The authenticity of the obtained results are verified by graphical and numerical illustrations by taking into account some examples. Additionally, the study explores applications involving special means, special functions and moments of random variables resulting in new fractional recurrence relations within the multiplicative calculus framework. These contributions not only generalize existing inequalities but also pave the way for future research in both theoretical mathematics and real-world modeling scenarios. Full article

In this study, we examine the error bounds related to Milne-type inequalities and a widely recognized Newton–Cotes method, originally developed for three-times-differentiable convex functions within the context of Jensen–Mercer inequalities. Expanding on this foundation, we explore Milne–Mercer-type inequalities and their application to a more refined class of three-times-differentiable s-convex functions. This work introduces a new identity involving such functions and Jensen–Mercer inequalities, which is then used to improve the error bounds for Milne-type inequalities in both Jensen–Mercer and classical calculus frameworks. Our research highlights the importance of convexity principles and incorporates the power mean inequality to derive novel inequalities. Furthermore, we provide a new lemma using Caputo–Fabrizio fractional integral operators and apply it to derive several results of Milne–Mercer-type inequalities pertaining to $(\alpha ,m)$-convex functions. Additionally, we extend our findings to various classes of functions, including bounded and Lipschitzian functions, and explore their applications to special means, the q-digamma function, the modified Bessel function, and quadrature formulas. We also provide clear mathematical examples to demonstrate the effectiveness of the newly derived bounds for Milne–Mercer-type inequalities. Full article

The Riemann-Liuoville fractional integrals are the simplest and most popular operators of the classical fractional calculus. But their variants, the Erdélyi-Kober operators of fractional integration, have many more applications due to the freedom to choose the additional (three) parameters. We introduce and study a generalization of the Erdélyi-Kober and Riemann-Liuoville fractional integrals, where the elementary kernel function is replaced by a suitably chosen $I_{1,1}^{1,0}$-function. The I-functions introduced by Rathie in 1997 are generalized hypergeometric functions extending the Fox H-functions and the Meijer G-functions. Note that till recently this new class of special functions has not been popular because of their too complicated structure involving fractional powers of the Gamma functions and their multi-valued behavior. However, the I-functions happened to arise not only for the needs of statistical physics, but also since they included important special functions in mathematics that were not covered by the H- and G-functions. In our previous works, as Kiryakova and Paneva-Konovska, we have shown the relations of such functions, among which are the Mittag-Leffler and Le Roy type, their multi-index variants, and others related to fractional calculus, to the I-functions. Here, we propose a new theory of generalization of the Erdélyi-Kober fractional integrals, based on the use of an I-function as a kernel. This will serve next as a base to extend our generalized multi-order fractional calculus with operators involving $I_{m,m}^{m,0}$. In this paper, we also evaluate the images under these new generalized fractional integrals of special functions of very general form. Finally, in the Conclusion section, we comment on some earlier discussions on the relations between fractal geometry and fractional calculus, nowadays already without any doubts. Full article

In this paper, we derive novel results concerning third-order differential subordinations for meromorphic functions, utilizing a newly defined linear operator that involves the inverse of the Legendre chi function in conjunction with the Mittag-Leffler identity. To establish these results, we introduce several families of admissible functions tailored to this operator and formulate sufficient conditions under which the subordinations hold. Our study presents three fundamental theorems that extend and generalize known results in the literature. Each theorem is accompanied by rigorous proofs and further supported by corollaries and illustrative examples that validate the applicability and sharpness of the derived results. In particular, we highlight special cases and discuss their implications through both analytical evaluations and graphical interpretations, demonstrating the strength and flexibility of our framework. This work contributes meaningfully to the field of geometric function theory by offering new insights into the behavior of third-order differential operators acting on p-valent meromorphic functions. Furthermore, the involvement of the Mittag-Leffler function positions the results within the broader context of fractional calculus, suggesting potential for applications in the mathematical modeling of complex and nonlinear phenomena. We hope this study stimulates further research in related domains. Full article

This study investigates the stability of fractional-order systems with infinite delay, which are prevalent in many fields due to their effectiveness in modeling complex dynamic behaviors. Recent advancements concerning the existence and various categories of stability for solutions to the given problem are also highlighted. This investigation utilizes tools such as the Picard operator approach, the Banach fixed-point theorem, an extended form of Gronwall’s inequality, and several well-known special functions. We establish key stability criteria for fractional differential equations using Hadamard fractional derivatives and illustrate these concepts using a numerical example. Specifically, graphical representations of the system’s responses demonstrate how fractional-order control enhances stability compared to traditional integer-order approaches. Our results emphasize the value of fractional systems in improving system performance and robustness. Full article

This paper introduces a new class of harmonic functions defined through a generalized symmetric q-differential that acts on both the analytic and co-analytic parts of the function. By combining concepts from symmetric q-calculus and geometric function theory, we develop a framework that extends several well-known operators as special cases. The main contributions of this study include new criteria for harmonic univalence, sharp coefficient bounds, distortion theorems, and covering results. Our operator offers increased flexibility in modeling symmetric structures, with potential applications in complex analysis, fractional calculus, and mathematical physics. To support these theoretical developments, we provide concrete examples and highlight potential directions for future research, including extensions to higher-dimensional settings. Full article

The primary objective of this study is to explore sufficient conditions for the existence, uniqueness, and optimal stability of positive solutions to a finite system of Hilfer–Hadamard fractional differential equations with two-point boundary conditions. Our analysis centers around transforming fractional differential equations into fractional integral equations under minimal requirements. This investigation employs several well-known special control functions, including the Mittag–Leffler function, the Wright function, and the hypergeometric function. The results are obtained by constructing upper and lower control functions for nonlinear expressions without any monotonous conditions, utilizing fixed point theorems, such as Banach and Schauder, and applying techniques from nonlinear functional analysis. To demonstrate the practical implications of the theoretical findings, a pertinent example is provided, which validates the results obtained. Full article

On the one hand, convex functions are important to derive rigorous convergence rates, and on the other, synchronous functions are significant to solve statistical problems using Chebyshev inequalities. Therefore, fractional integral inequalities involving such functions play a crucial role in creating new models and methods. Although a large class of fractional operators have been used to establish inequalities, nevertheless, these operators having the Fox-H and the Meijer-G functions in their kernel have been applied to establish fractional integral inequalities for such important classes of functions. Taking motivation from these facts, the primary objective of this work is to develop fractional inequalities involving the Fox-H function for convex and synchronous functions. Since the Fox-H function generalizes several important special functions of fractional calculus, our results are significant to innovate the existing literature. The inventive features of these functions compel researchers to formulate deeper results involving them. Therefore, compared with the ongoing research in this field, our results are general enough to yield novel and inventive fractional inequalities. For instance, new inequalities involving the Meijer-G function are obtained as the special cases of these outcomes, and certain generalizations of Chebyshev inequality are also included in this article. Full article

We introduce a novel class of coherent states, termed ${\mathcal{W}}_{\alpha ,\beta ,\nu }$-coherent states, constructed using a deformed boson algebra based on the generalised factorial ${\left[n\right]}_{\alpha ,\beta ,\nu }!$. This algebra extends conventional factorials, incorporating advanced special functions such as the Mittag-Leffler and Wright functions, enabling the exploration of a broader class of quantum states. The mathematical properties of these states, including their continuity, completeness, and quantum fluctuations, are analysed. A key aspect of this work is the resolution of the Stieltjes moment problem associated with these states, achieved through the inverse Mellin transformation method. The framework provides insights into the interplay between the classical and quantum regimes, with potential applications in quantum optics and fractional quantum mechanics. By extending the theoretical landscape of coherent states, this study opens avenues for further exploration in mathematical physics and quantum technologies. Full article

In this effort, we extend the fractal–fractional operators into the complex plane together with the quantum calculus derivative to obtain a quantum–fractal–fractional operators (QFFOs). Using this newly created operator, we create an entirely novel subclass of analytical functions in the unit disk. Motivated by the concept of differential subordination, we explore the most important geometric properties of this novel operator. This leads to a study on a set of differential inequalities in the open unit disk. We focus on the conditions to obtain a bounded turning function of QFFOs. Some examples are considered, involving special functions like Bessel and generalized hypergeometric functions. Full article

Special functions have been widely used in fractional calculus, particularly for addressing the symmetric behavior of the function. This paper provides improved delta Mittag–Leffler and exponential functions to establish new types of fractional difference operators in the setting of Riemann–Liouville and Liouville–Caputo. We give some properties of these discrete functions and use them as the kernel of the new fractional operators. In detail, we propose the construction of the new fractional sums and differences. We also find the Laplace transform of them. Finally, the relationship between the Riemann–Liouville and Liouville–Caputo operators are examined to verify the feasibility and effectiveness of the new fractional operators. Full article