Special Functions Research Articles

Analytic study and statistical enforcement of extended beta functions imposed by Mittag-Leffler and Hurwitz-Lerch Zeta functions.

Special Function Theory is used in many mathematical fields to model scientific progress, from theoretical to practical. This helps efficiently analyze the newly expanded Beta class of functions on a complicated domain. We use Mittag-Leffler and Hurwitz Lerch zeta (HLZ) kernels to produce the Beta function using the convolution tool. This special function advances a statistical implementation research approach. This unique function also discusses and gives analytical benefits, including functional and summation relations, Mellin transformations, and integral representations. Additionally, many derivative formulae are obtained. The statistical implementation of expanded Beta distribution using the suggested beta function was also conducted. We use the extended Beta function to create the new extended ordinary hypergeometric function and Kummer function. Derivative formulae, integral representations, generating functions, and fractional derivatives are also investigated.•Developed utilizing Mittag-Leffler and Hurwitz Lerch Zeta functions as kernels, delivering increased analytical and computational capabilities.•Comprises derivative formulae, integral representations, Mellin transformations, and generating functions, offering a comprehensive mathematical foundation.•Illustrates the use of the extended Beta function inside the Beta distribution, highlighting its statistical importance.

MiR-125/let-7 cluster orchestrates neuronal cell fate determination and cortical layer formation during neurogenesis.

MiR-125/let-7 cluster orchestrates neuronal cell fate determination and cortical layer formation during neurogenesis.

Monotonicity of three functions involving the confluent hypergeometric functions of the second kind

Abstract In this paper, we approach the monotonicity of three functions involving the confluent hypergeometric function of the second kind from the perspective of using the monotonicity rules. By employing the monotonicity of these functions, we establish bounds for $U^\prime(a,b,x)/U(a,b,x)$ and $U(a,b,y)/U(a,b,x)$, which are shown to be extremely tight for large values of $x$ and $y$. Furthermore, by using the relationships between the confluent hypergeometric function of the second kind and other special functions, we derive a series of results concerning the incomplete gamma function and the modified Bessel function of the second kind, including the bounds for $\Gamma(\gamma,x)$, $K_{\nu+1}(x)/K_{\nu}(x)$, and $K_{\nu}^\prime(x)/K_{\nu}(x)$.

On Fixed Point Theorems for Self-Mappings in Complex Metric Spaces with Special Functions

This paper delves into the forefront of fixed point theory, focusing on recent advancements within the context of contraction mappings in complex metric spaces. The study introduces a novel perspective by incorporating the pivotal role of control functions in elucidating the behavior and properties of fixed points. We investigate the interplay between contraction mappings and complex metric spaces via control function. We provide an example to illustrate our findings.

Diversity and functional specialization of oyster immune cells uncovered by integrative single-cell level investigations.

Mollusks are a major component of animal biodiversity and play a critical role in ecosystems and global food security. The Pacific oyster, Crassostrea (Magallana) gigas, is the most farmed bivalve mollusk in the world and is becoming a model species for invertebrate biology. Despite the extensive research on hemocytes, the immune cells of bivalves, their characterization remains elusive. Here, we were able to extensively characterize the diverse hemocytes and identified at least seven functionally distinct cell types and three hematopoietic lineages. A combination of single-cell RNA sequencing, quantitative cytology, cell sorting, functional assays, and pseudo-time analyses was used to deliver a comprehensive view of the distinct hemocyte types. This integrative analysis enabled us to reconcile molecular and cellular data and identify distinct cell types performing specialized immune functions, such as phagocytosis, reactive oxygen species production, copper accumulation, and expression of antimicrobial peptides. This study emphasized the need for more in depth studies of cellular immunity in mollusks and non-model invertebrates and set the ground for further comparative immunology studies at the cellular level.

Diversity and functional specialization of oyster immune cells uncovered by integrative single-cell level investigations

Mollusks are a major component of animal biodiversity and play a critical role in ecosystems and global food security. The Pacific oyster, Crassostrea (Magallana) gigas, is the most farmed bivalve mollusk in the world and is becoming a model species for invertebrate biology. Despite the extensive research on hemocytes, the immune cells of bivalves, their characterization remains elusive. Here, we were able to extensively characterize the diverse hemocytes and identified at least seven functionally distinct cell types and three hematopoietic lineages. A combination of single-cell RNA sequencing, quantitative cytology, cell sorting, functional assays, and pseudo-time analyses was used to deliver a comprehensive view of the distinct hemocyte types. This integrative analysis enabled us to reconcile molecular and cellular data and identify distinct cell types performing specialized immune functions, such as phagocytosis, reactive oxygen species production, copper accumulation, and expression of antimicrobial peptides. This study emphasized the need for more in depth studies of cellular immunity in mollusks and non-model invertebrates and set the ground for further comparative immunology studies at the cellular level.

1D fluids with repulsive nearest-neighbour interactions: low-temperature anomalies

Abstract A limited number of two-dimensional and three-dimensional materials under a constant pressure contract in volume upon heating isobarically; this anomalous phenomenon is known as the negative thermal expansion (NTE). In this paper, the NTE anomaly is observed in 1D fluids of classical particles interacting pairwisely with two competing length scales: the hard-core diameter a and the finite range a ′ > a of a soft repulsive potential component. If a ′ ⩽ 2 a , the pair interactions reduce themselves to nearest neighbours which permits a closed-form solution of thermodynamics in the isothermal–isobaric ensemble characterized by temperature T and pressure p. We focus on the equation of state (EoS) which relates the average distance between particles (reciprocal density) l to T and p ⩾ 0 . The EoS is expressible explicitly in terms of elementary or special functions for specific, already known and new, cases like the square shoulder, the linear and quadratic ramps as well as certain types of logarithmic interaction potentials. The emphasis is put on low-T anomalies of the EoS. Firstly, the equidistant ground state as the function of the pressure can exhibit, at some ‘compressibility’ pressures, a jump in chain spacing from a′ to a. Secondly, the analytical structure of the low-T expansion of l ( T , p ) depends on ranges of p-values. Thirdly, the presence of the NTE anomaly depends very much on the shape of the core-softened potential.

Fractional Laguerre derivatives and associated differential equations

Fractional versions of the Laguerre derivative were introduced in 2021 by Yakubovich, who defined both fractional integrals and fractional derivatives (of Riemann–Liouville type) within the Laguerre setting. We continue with the mathematical study of these operators: proving many equivalent formulae for them, introducing the corresponding Caputo-type Laguerre fractional derivatives, and finding eigenfunctions for the new operators. Applications of our work are given by solving several fractional differential equations involving Laguerre-type operators, their solutions often being expressible in terms of special functions such as the multi-index Mittag-Leffler function.

On Spirallikeness of Entire Functions

In this article, we establish conditions under which certain entire functions represented as infinite products of their positive zeros are α-spirallike of order cos(α)/2. The discussion includes several examples featuring special functions such as Bessel, Struve, Lommel, and q-Bessel functions.

Доповнення до статті Дуак, Мароні (2020) про новий клас 2-ортогональних поліномів

UDC 510 We consider some issues related to the 2-orthogonal polynomials (2-OP). The answers to these issues can be regarded a supplement to the article [K. Douak, P. Maroni, On a new class of 2-orthogonal polynomials, I: The recurrence relations and some properties, Integral Transform and Special Functions (2020)]. The conditions imposed on the parameters of two original recurrence relations (the first of these conditions is for the 2-OP, while the second condition is for their normalized derivatives) and guaranteeing the ``"classical" nature of the 2-OP in Hahn's sense are clarified. It is constructively proved that these recurrence relations do not cover all possible "classical" 2-OPs. An example of ``"classical" 2-OP generated by the generating function constructed by using a Bessel function of the first kind of order zero is presented. These OPs are unique because their properties are similar to the classical OPs. In particular, this concerns the fact that their zeros are real and their location. The analysis of the available literature and our own numerous numerical-analytic experiments reveals the absence of other examples of the ``"classical" 2-OPs with similar properties.

Wavelet transforms based on the special affine convolution

In this paper, we mainly study the wavelet construction based on the special affine Fourier transform, including the continuous wavelet transform, dyadic wavelet transform and discrete wavelet transform. First, we give a new definition of the continuous special affine wavelets based on the special affine convolution and show that all admissible wavelets in the Fourier transform domain can also be used as special affine wavelet function. Second, the dyadic special affine wavelet transform is defined and the construction of the dyadic dual wavelets for reconstruction is given. Then, the special affine multiresolution analysis and the corresponding discrete special affine wavelets are proposed. Moreover, fast decomposition and reconstruction algorithms based on the special affine convolution are established. Finally, numerical results demonstrate that the proposed special affine wavelet transform has good performance for denoising non-stationary signals and color images.

Sufficient Conditions for Optimal Stability in Hilfer–Hadamard Fractional Differential Equations

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.

Efficient Numerical Quadrature for Highly Oscillatory Integrals with Bessel Function Kernels

In this paper, we investigate efficient numerical methods for highly oscillatory integrals with Bessel function kernels over finite and infinite domains. Initially, we decompose the two types of integrals into the sum of two integrals. For one of these integrals, we reformulate the Bessel function Jν(z) as a linear combination of the modified Bessel function of the second kind Kν(z), subsequently transforming it into a line integral over an infinite interval on the complex plane. This transformation allows for efficient approximation using the Cauchy residue theorem and appropriate Gaussian quadrature rules. For the other integral, we achieve efficient computation by integrating special functions with Gaussian quadrature rules. Furthermore, we conduct an error analysis of the proposed methods and validate their effectiveness through numerical experiments. The proposed methods are applicable for any real number ν and require only the first ⌊ν⌋ derivatives of f at 0, rendering them more efficient than existing methods that typically necessitate higher-order derivatives.

WKB Energy levels in gapped graphene under crossed electromagnetic fields

We consider a single layer of graphene subjected to a magnetic field H applied perpendicular to the layer and an in-plane constant radial electric field E. The Dirac equation for this configuration does not admit analytical solutions in terms of known special functions. Using the Wentzel–Kramers–Brillouin (WKB) approximation, we demonstrate that for gapped graphene the Bohr–Sommerfeld quantization condition for eigenenergies includes an additional valley-dependent geometrical phase. When this term is accounted for, the WKB approximation exhibits good agreement with results from the exact diagonalization method except for the lowest Landau level.

Evolutionary duplication of the leishmanial adaptor protein α-SNAP plays a role in its pathogenicity.

Essential-gene duplication during evolution promotes specialized functions beyond the typical role. Our in silico study unveiled two α-SNAP paralogs in Leishmania, a crucial component that, along with NSF, triggers disassembly of the cis-SNARE complex, formed during vesicle fusion with target membranes. While multiple α-SNAPs are common in many flagellated protists, including the trypanosomatids, they are unusual among other eukaryotes. This study explores the evolutionary and functional relevance of α-SNAP gene duplication in Leishmania donovani, emphasizing both subfunctionalization and neofunctionalization. We discovered that L.donovani α-SNAP (Ldα-SNAP) genes are transcribed in promastigote and amastigote stages, indicating they are not pseudogenes. Although the two paralogs share essential residues and structural features, only Ldα-SNAP1660 (Ldα-SNAP1) can effectively substitute the function of its yeast counterpart, while Ldα-SNAP3040 (Ldα-SNAP2) cannot. This functional difference is attributed to a replacement of alanine with phosphorylatable-serine in Ldα-SNAP1 during evolution from the most common ancestral ortholog. This modification is rarely observed in corresponding orthologs of other trypanosomatids. Incidentally, Ldα-SNAP paralogs exhibit differential localization in the ER and flagellar pocket. However, both paralogs, either actively or passively, regulate the secretion of exosomes and PM blebs, containing the virulence protein GP63. This indicates functional division and their indirect participation in the host's macrophage inactivation. Moreover, a small fraction of Ldα-SNAP1's presence in the flagellum hints at a potential role in sensing environmental cues and aiding the parasite's attachment to the sandfly's hindgut. Our findings underscore that duplicated Ldα-SNAPs have retained ancestral functions through subfunctionalization, and subsequently, they acquired parasite-specific neofunction(s) through the accumulation of natural mutation(s).

Glycobiology of IgE.

Antibodies play a vital role in the immune system, with distinct isotypes having unique tropisms and performing specialized functions. Of these isotypes, IgE is the least abundant in circulation yet plays a critical role in defense against parasitic infection and allergic reactions. IgE is also heavily N-linked glycosylated, a posttranslational modification that influences receptor interactions of effector responses. The importance of glycosylation on IgG function is well established, and the roles of IgE glycans are emerging. This review examines the relationship between IgE glycosylation and its biological function. IgE glycosylation, specifically the oligomannosidic glycan, is necessary for IgE binding to its high-affinity receptor FcεRI on mast cells and basophils. Recent evidence suggests that terminal sialic acid residues on complex biantennary glycans significantly enhance IgE's allergic potential, with sialylation of IgE demonstrating reduced capacity to trigger degranulation and anaphylaxis. Glycosylation also influences IgE's interaction with its low-affinity receptor FcεRII/CD23, affecting serum clearance and antigen presentation. Beyond allergy, this review also covers IgE's impacts on its roles in autoimmunity, parasite defense, and protection against venoms. Current therapeutic approaches targeting IgE include monoclonal antibodies like omalizumab, with emerging therapeutics looking to target systemic IgE production mechanisms also covered. Although the understanding of IgG glycosylation is known, there is much to uncover in terms of IgE glycosylation, which may open new avenues for developing more precise interventions that modulate its effector functions.

New Extension of Inequalities through Extended Version of Fractional Operators for s-Convexity with Applications

Fractional integral inequalities play a significant role in both pure and applied mathematics, contributing to the advancement and extension of various mathematical techniques. An accurate formulation of such inequalities is essential to establish the existence and uniqueness of fractional methods. Additionally, convexity theory serves as a fundamental component in the study of fractional integral inequalities due to its defining characteristics and properties. Moreover, there is a strong interconnection between convexity and symmetric theories, allowing results from one to be effectively applied to the other. This correlation has become particularly evident in recent decades, further enhancing their importance in mathematical research. This article investigates the Hermite-Hadamard inequalities and their refinements by implementation of generalized fractional operators through the $s$-convex functions, which are considered in both single and double differentiable forms. The study aims to extend and refine existing inequalities with fractional operator having extended Bessel-Maitland functions as a kernel, providing a more generalized framework. By incorporating these special functions, the results encompass and improve numerous classical inequalities found in the literature, offering deeper insights and broader applicability in mathematical analysis.

Stepwise neofunctionalization of the NF-κB family member Rel during vertebrate evolution

Adaptive immunity and the five vertebrate NF-κB family members first emerged in cartilaginous fish, suggesting that NF-κB family divergence helped to facilitate adaptive immunity. One specialized function of the NF-κB Rel protein in macrophages is activation of Il12b, which encodes a key regulator of T cell development. We found that Il12b exhibits much greater Rel dependence than inducible innate immunity genes in macrophages, with the unique function of Rel dimers depending on a heightened intrinsic DNA-binding affinity. Chromatin immunoprecipitation followed by sequencing experiments defined differential DNA-binding preferences of NF-κB family members genome-wide, and X-ray crystallography revealed a key residue that supports the heightened DNA-binding affinity of Rel dimers. Unexpectedly, this residue, the heightened affinity of Rel dimers, and the portion of the Il12b promoter bound by Rel dimers were largely restricted to mammals. Our findings reveal major structural transitions in an NF-κB family member and one of its key target promoters at a late stage of vertebrate evolution that apparently contributed to immunoregulatory rewiring in mammalian species.

Current State of Knowledge Regarding the Treatment of Cranial Bone Defects: An Overview.

In this article, an analysis of the problem of treating bone defects using cranial bone disorders as an example is presented. The study was performed in the context of the development of various implant biomaterials used to fill bone defects. An analysis of the requirements for modern materials is undertaken, indicating the need for their further development. The article focuses particular attention on these biomaterial properties, which have an influence on bioresorbability and promote osteointegration and bone growth. The analysis showed the need for further development of biomaterials, the characteristics of which may be multifunctionality. Multifunctional scaffolds are those that simultaneously fill and stabilize the defect and contribute to the proper process of regeneration and reconstruction of cranial bones. Due to the complex structure of the skull and special protective functions, there is a need to develop innovative implants. Implants with complex geometries can be successfully manufactured using additive technologies.

Deep Neural Network for Cure Fraction Survival Analysis Using Pseudo Values

Introduction: The hidden assumption in most of survival analysis models is the occurrence of the event of interest for all study units. The violation of this assumption occurs in several situations. For example, in medicine, some patients may never have cancer, and some may never face Alzheimer. Ignoring such information and analyzing the data with traditional survival models may lead to misleading results. Analyzing long term survivals can be performed using both traditional and neural networks. There has been an increasing interest in modeling lifetime data using neural network due to its ability to handle complex covariates if any. Also, in several numerical results it provides a better prediction. However, for long-term survivors only one neural network was introduced to estimate the uncured proportion together with the EM algorithm to account for the latency part. Neural network in survival analysis requires special cost function to account for censoring. Methods: In this paper, we extend the neural network using pseudo values to analyze cure fraction model. It neither requires the use of special cost function nor the EM algorithm Results: The network is applied on both synthetic and Melanoma real datasets to evaluate its performance. We compared the results using goodness of fit methods in both datasets with cox proportional model using EM algorithm. Conclusion: The proposed neural network has the flexibility of analyzing data without parametric assumption or special cost function. Also, it has the advantage of analyzing the data without the need of EM algorithm. Comparing the results with cox proportional model using EM algorithm, the proposed neural network performed better