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Geometric function theory has extensively explored the geometric characteristics of analytic functions within symmetric domains. This study analyzes the geometric properties of a specific class of analytic functions employing confluent hypergeometric functions and generalized Bessel functions of the first kind. Specific constraints are imposed on the parameters to ensure the inclusion of the confluent hypergeometric function within the analytic function class. The coefficient bound of the class is used to determine the inclusion properties of integral operators involving generalized Bessel functions of the first kind. Different results are observed for these operators, depending on the specific values of the parameters. The results presented here include some previously published findings as special cases. [ABSTRACT FROM AUTHOR]

Moments of order statistics (OSs) characterize the Weibull–geometric and half-logistic families of distributions, of which the extended exponential–geometric (EEG) distribution is a particular case. The EEG distribution is used to create the log-extended exponential–geometric (LEEG) distribution, which is bounded in the unit interval (0, 1). In addition to the generalized Stirling numbers of the first kind, a few years ago, the polylogarithm function and the Lerch transcendent function were used to determine the moments of order statistics of the LEEG distributions. As an application based on the L-moments, we expand the features of the LEEG distribution in this work. In terms of the Gauss hypergeometric function, this work presents the precise equations and recurrence relations for the single moments of OSs from the LEEG distribution. Along with recurrence relations between the expectations of function of two OSs from the LEEG distribution, it also displays the truncated and conditional distribution of the OSs. Additionally, we use the L-moments to estimate the parameters of the LEEG distribution. We further fit the LEEG distribution on three practical data sets from medical and environmental sciences areas. It is seen that the estimated parameters through L-moments of the OSs give a superior fit. We finally determine the correspondence between the entropies and the OSs. [ABSTRACT FROM AUTHOR]

In this paper, we present an extension of the Karlsson–Minton summation formula for a generalized hypergeometric function with integral parameter differences. Namely, we extend one single negative difference in Karlsson–Minton formula to a finite number of integral negative differences, some of which will be repeated. Next, we continue our study of the generalized hypergeometric function evaluated at unity and with integral positive differences (IPD hypergeometric function at the unit argument). We obtain a recurrence relation that reduces the IPD hypergeometric function at the unit argument to F 3 4 . Finally, we note that Euler–Pfaff-type transformations are always based on summation formulas for finite hypergeometric functions, and we give a number of examples. [ABSTRACT FROM AUTHOR]

In order to examine various geometric features, we expanded the supertrigonometric function (STF) and superhyperbolic function (SHF) into the open unit disk. A convolution differential operator of the STF provides the formulas. The suggested operator works with both integral and double differential inequalities. As a result, we present a collection of findings that includes recent works. The idea of subordination and superordination serves as a guide for our method, and we then developed the primary conclusion as a double side’s theorem. [ABSTRACT FROM AUTHOR]

I n this work we introduce new subclasses of meromorphic functions in punctured unit disk and analyzes numerous connections and various features of these subclasses by making use of the linear operator that is connected to Gaussian hypergeometric functions. [ABSTRACT FROM AUTHOR]

In this paper, we define the incomplete fourth Appell hypergeometric matrix functions Υ4 and Γ4 through application of the incomplete Pochhammer matrix symbols. We also give certain properties such as matrix differential equation, integral formula, recursion formula, differentiation formula of the incomplete fourth Appell hypergeometric matrix functions Υ4 and Γ4, where not all the matrices involved are commuting matrices. [ABSTRACT FROM AUTHOR]

In this note, two interesting integrals involving the product of two generalized hypergeometric functions have been evaluated in terms of gamma function. The results are derived with the help of a known integral involving hypergeometric function recorded in the paper of Lavoie and Trottlier. We also give several interesting special cases of our main findings. [ABSTRACT FROM AUTHOR]

Hypergeometric functions of one and many variables play an important role in various branches of modern physics and mathematics. We often encounter hypergeometric functions with indices linearly dependent on a small parameter with respect to which we need to perform Laurent expansions. Moreover, it is desirable that such expansions be expressed in terms of well-known functions that can be evaluated with arbitrary precision. To solve this problem, we use the method of differential equations and the reduction of corresponding differential systems to a canonical basis. In this paper, we are interested in the generalized hypergeometric functions of one variable and in the Appell and Lauricella functions and their expansions in terms of the Goncharov polylogarithms. Particular attention is paid to the case of rational indices of the considered hypergeometric functions when the reduction to the canonical basis involves a nontrivial variable change. The paper comes with a Mathematica package Diogenes, which provides an algorithmic implementation of the required steps. [ABSTRACT FROM AUTHOR]

Pseudo-differential operators (PDO) Q (x , ℒ a , x) and Q (x , ℒ a , x) involving the index Whittaker transform are defined. Estimates for these operators in Hilbert space L2a(ℝ+;ma(x)dx) are obtained. A symbol class Ω is introduced. Later product and commutators for the PDO are investigated and their boundedness results are discussed. [ABSTRACT FROM AUTHOR]

We study the n$n$‐point differentials corresponding to Kadomtsev–Petviashvili (KP) tau functions of hypergeometric type (also known as Orlov–Scherbin partition functions), with an emphasis on their ℏ2$\hbar ^2$‐deformations and expansions. Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We also distinguish two large families of the Orlov–Scherbin partition functions that do satisfy the natural analytic assumptions, and for these families, we prove in addition the so‐called projection property and thus the full statement of the Chekhov–Eynard–Orantin topological recursion. A particular feature of our argument is that it clarifies completely the role of ℏ2$\hbar ^2$‐deformations of the Orlov–Scherbin parameters for the partition functions, whose necessity was known from a variety of earlier obtained results in this direction but never properly understood in the context of topological recursion. As special cases of the results of this paper, one recovers new and uniform proofs of the topological recursion to all previously studied cases of enumerative problems related to weighted double Hurwitz numbers. By virtue of topological recursion and the Grothendieck–Riemann–Roch formula, this, in turn, gives new and uniform proofs of almost all Ekedahl–Lando–Shapiro–Vainshtein (ELSV)‐type formulas discussed in the literature. [ABSTRACT FROM AUTHOR]

The Pearson-type family densities are among the most important classes of distributions, also playing key roles in directional statistics. To model data scattered asymmetrically on non-Euclidean spaces, including spheres, the researchers confined themselves to extending particular distributions fromthe class of the Pearson-type family densities. Those specific distributions are symmetric, but their extended versions are usually heavy-tailed. This paper introduces alternative probability density functions in the class of Pearson-type distributions on the sphere with the spherical Student's t, Fisher, and Chi-square densities as the subfamilies. We show that it is intrinsically asymmetric by investigating various theoretical properties of this new subclass. Intensive simulation studies are conducted to explore various aspects of this subclass. Also, modeling two real-life data using the proposed densities and comparing the results with the fits arising from other common spherical distributions are considered. [ABSTRACT FROM AUTHOR]

The hypergeometric functions occur naturally in variety of problems in analysis, statistics, operational research, theoretical and mathematical physics and engineering sciences. In the physical problems where these special functions occur, sometimes it is required to have their first approximations. In the advance analysis of a given problem, we need to dispose with some simple but sufficiently accurate algorithms for the approxima-. tions of the hypergeometric functions. In this paper we present some series and integral representations of them and discuss several simply-structured single term approximation formulae: for the Hubbell type radiation integrals, generalized hypergeometric functions and Appell's functions. We propose also a new approach to estimate the pFq-functions by relating them to three simple elementary functions. The notions generalized (multiple) fractional integrals and derivatives are used. [ABSTRACT FROM AUTHOR]

A one-dimensional model for fluid and solute transport in poroelastic materials (PEMs) is studied. Although the model was recently derived and some exact solutions, in particular steady-state solutions and their applications, were studied, special cases occurring when some parameters vanish were not analysed earlier. Since the governing equations are nonintegrable in nonstationary cases, the Lie symmetry method and modern tools for solving ODE systems are applied in order to construct time-dependent exact solutions. Depending on parameters arising in the governing equations, several special cases with new Lie symmetries are identified. Some of them have a highly nontrivial structure that cannot be predicted from a physical point of view or using Lie symmetries of other real-world models. Applying the symmetries obtained, multiparameter families of exact solutions are constructed, including those in terms of elementary and special functions (hypergeometric, Whittaker, Bessel and modified Bessel functions). A possible application of the solutions obtained is demonstrated, and it is shown that some exact solutions can describe (at least qualitatively) the solute transport in PEM. The obtained exact solutions can also be used as test problems for estimating the accuracy of approximate analytical and numerical methods for solving relevant boundary value problems. [ABSTRACT FROM AUTHOR]

A boundary value problem for a nonhomogeneous heat equation with a load in the form of a fractional Riemann–Liouville integral of an order β ∈ (0, 1) is considered. By inverting the differential part, the problem is reduced to an integral equation with a kernel with a special function. The special function is presented as a generalized hypergeometric function. The limiting cases of the order β of the fractional derivative are studied: it is shown that the interval for changing the order of the fractional derivative can be expanded to integer values β ∈ [0, 1]. The results of the study remain unchanged. The kernel of the integral equation is estimated. Conditions for the solvability of the integral equation are obtained. [ABSTRACT FROM AUTHOR]

We investigate the joint moments of derivatives of characteristic polynomials over the unitary symplectic group S p (2 N) and the orthogonal ensembles S O (2 N) and O − (2 N) . We prove asymptotic formulae for the joint moments of the n 1th and n 2th derivatives of the characteristic polynomials for all three matrix ensembles. Our results give two explicit formulae for each of the leading order coefficients, one in terms of determinants of hypergeometric functions and the other as combinatorial sums over partitions. We use our results to put forward conjectures on the joint moments of derivatives of L -functions with symplectic and orthogonal symmetry. [ABSTRACT FROM AUTHOR]

We sum in a closed form the Sneddon-Bessel series ∑∞ m=1 Jα(xjm,v)Jβ(yjm,v) /j m,v2n+α+β-2v+2 Jv+1(jm,v)², where 0 < x, 0 < y, x + y < 2, n is an integer, α, β, v ε C\{-1,-2, ... } with 2 Re v < 2n + 1 + Re α + Re β and {jm,v}m≥0 are the zeros of the Bessel function Jv of order v. In most cases, the explicit expressions for these sums involve hypergeometric functions pFq. As an application, we prove some extensions of the Kneser-Sommerfeld expansion. For instance, we show that ∑∞ m=1 j v-β m,v Jv (xjm,v)Jβ (yjm,v) (j²m,v-z²)Jv+1(jm,v)² = πJβ (yz)/4zβ-v Jv (z) (Yv (z)Jv (xz) - Jv (z)Yv (xz)), if Re v < Re β + 1 and 0 < y ≤ x, x + y < 2 (here, Yv denotes the Bessel function of the second kind), which becomes the Kneser-Sommerfeld expansion when β = v. [ABSTRACT FROM AUTHOR]

In this paper, the authors briefly review some closed-form formulas of the Gauss hypergeometric function at specific arguments, alternatively prove four of these formulas, newly extend a closed-form formula of the Gauss hypergeometric function at some specific arguments, successfully apply a special case of the newly extended closed-form formula to derive an alternative form for the Maclaurin power series expansion of the Wilf function, and discover two novel increasing rational approximations to a quarter of the circular constant. [ABSTRACT FROM AUTHOR]

In the paper, the authors introduce two notions, the normalized remainders, or say, the normalized tails, of the Maclaurin power series expansions of the sine and cosine functions, derive two integral representations of the normalized tails, prove the nonnegativity, positivity, decreasing property, and concavity of the normalized tails, compute several special values of the Young function, the Lommel function, and a generalized hypergeometric function, recover two inequalities for the tails of the Maclaurin power series expansions of the sine and cosine functions, propose three open problems about the nonnegativity, positivity, decreasing property, and concavity of a newly introduced function which is a generalization of the normalized tails of the Maclaurin power series expansions of the sine and cosine functions. These results are related to the Riemann–Liouville fractional integrals. [ABSTRACT FROM AUTHOR]

We consider pairs of automorphisms acting on fields of Laurent or Puiseux series: pairs of shift operators .W x 7 x C h1; W x 7 x C h2/, of q-difference operators .W x 7 q1x, W x 7 q2x/, and of Mahler operators .W x 7 xp1 ; W x xp2 /. Given a solution f to a linear -equation and a solution g to an algebraic -equation, both transcendental, we show that f and g are algebraically independent over the field of rational functions, assuming that the corresponding parameters are sufficiently independent. As a consequence, we settle a conjecture about Mahler functions put forward by Loxton and van der Poorten in 1987. We also give an application to the algebraic independence of q-hypergeometric functions. Our approach provides a general strategy to study this kind of question and is based on a suitable Galois theory: the -Galois theory of linear -equations. [ABSTRACT FROM AUTHOR]

We give new explicit formulas as well as new generating functions for the associated Meixner, Charlier, and Krawtchouk polynomials. The obtained results are then used to derive new generating functions and convolution-type formulas of the corresponding classical polynomials. Some consequences of our results are also mentioned. [ABSTRACT FROM AUTHOR]

In this paper, we derive some Fourier transforms of confluent hypergeometric functions. We give generalizations of several well-known results involving Fourier transforms of gamma functions. In particular, the generalizations include some Ramanujan's remarkable formulas. [ABSTRACT FROM AUTHOR]

Our paper introduces a new linear operator using the convolution between a Mittag Leffler Function and basic hypergeometric function. Use of the linear operator creates a new class of meromorphic functions defined in the punctured open unit disk. Consequently, the paper examines different aspects Apps and assets like, extreme points, coefficient inequality, growth and distortion. In conclusion, the work discusses modified Hadamard product and closure theorems. [ABSTRACT FROM AUTHOR]

In recent years, optical systems near the diffraction limit have been widely used in high-end applications. Evidently, an analytical solution of the point spread function (PSF) will help to enhance both understanding and dealing with the imaging process. This paper analyzes the Fresnel diffraction of diffraction-limited optical systems in defocused conditions. For this work, an analytical solution of the defocused PSF was obtained using the series expansion of the confluent hypergeometric functions. The analytical expression of the defocused optical transfer function is also presented herein for comparison with the PSF. Additionally, some characteristic parameters for the PSF are provided, such as the equivalent bandwidth and the Strehl ratio. Comparing the PSF obtained using the fast Fourier transform algorithm of an optical system with known, detailed parameters to the analytical solution derived in this paper using only the typical parameters, the root mean square errors of the two methods were found to be less than 3% in the weak and medium defocus range. The attractive advantages of the universal model, which is independent of design details, objective types, and applications, are discussed. [ABSTRACT FROM AUTHOR]

The Selberg integral has a twin ('the Dotsenko–Fateev integral') of the following form. We replace real variables x k in the integrand ∏ | x k | σ - 1 | 1 - x k | τ - 1 ∏ | x k - x l | 2 θ of the Selberg integral by complex variables z k , integration over a cube we replace by an integration over the whole complex space C n . According to Dotsenko, Fateev, and Aomoto, such integral is a product of Gamma functions. We define and evaluate a family of beta integrals over spaces C m × C m + 1 × ⋯ × C n , which for m = n gives the complex twin of the Selberg integral (with three additional integer parameters). [ABSTRACT FROM AUTHOR]

We prove an evaluation for the stuffle-regularised alternating multiple t value t ∗ , V (1 ¯ , ... , 1 ¯ , 1 , 1 ¯ , ... , 1 ¯) in terms of V , the regularisation parameter, log (2) , ζ (k) and β (k) . This arises by evaluating the corresponding generating series using the Evans-Stanton/Ramanujan asymptotics of a zero-balanced hypergeometric function 3 F 2 , and an evaluation established by Li in an alternative approach to Zagier's evaluation of ζ (2 , ... , 2 , 3 , 2 , ... , 2) . We end with some discussion and conjectures on possible motivic applications. [ABSTRACT FROM AUTHOR]

In this paper, we derive the exact expressions, as well as recurrence relations, for the single and product moments of the order statistics (OSs) and record values of the Topp-Leone Lomax (TLLo) distribution proposed by Oguntunde et al. (Wirel Person Commun 109:349–360, 2019). In addition, we study the corresponding L-moments. We estimate the distribution parameters through these L-moments and compare our results to those obtained in other models. Subsequently, motivated by the study of Okorie and Nadarajah (Wirel Person Commun 115:589–596, 2020), we fit the flood level data to the TLLo distribution. Finally, we reveal a relationship between the entropy and the upper and lower record values, as well as OSs. [ABSTRACT FROM AUTHOR]

This document is a summary of a special issue of the journal Axioms titled "New Developments in Geometric Function Theory II." The special issue contains 14 research papers that explore various topics related to complex-valued functions in the field of Geometric Function Theory. The papers cover subjects such as coefficient estimates, subordination theories, hypergeometric functions, and differential operators. Each paper presents new findings and results that contribute to the development of Geometric Function Theory. The special issue is recommended for researchers and scholars interested in this field of study. The document also acknowledges the authors, reviewers, and editors involved in the creation of the special issue. [Extracted from the article]

We present a canonical series expansion for generalized harmonic functions in the open unit disc in the complex plane that generalizes that recently obtained for the class of $ (p,q) $ (p , q) -harmonic functions. [ABSTRACT FROM AUTHOR]

This document titled "Fractional Calculus and Hypergeometric Functions in Complex Analysis" explores the impact of fractional calculus on various scientific and engineering disciplines. It emphasizes the significance of fractional operators in the study of fractional calculus and their applications in complex analysis research, specifically in the theory of univalent functions. The document also introduces hypergeometric functions and their connection to the theory of univalent functions. It compiles 12 research papers that cover topics such as geometric properties of fractional differential operators, logarithmic-related problems of univalent functions, and the study of generalized bi-subordinate functions. This document serves as a valuable resource for researchers interested in these subjects and their applications in complex analysis. Additionally, it provides a summary of three articles published in the Special Issue on "Fractional Calculus and Hypergeometric Functions in Complex Analysis." The first article explores the use of the Sălăgean q-differential operator for meromorphic multivalent functions, introducing new subclasses of functions. The second article presents three general double-series identities using Whipple transformations for terminating generalized hypergeometric functions, which can be used to derive additional identities. The third article defines a new generalized domain based on the quotient of two analytic functions and investigates the upper bounds of certain coefficients and determinants. The authors anticipate that these findings will inspire further research in the field. [Extracted from the article]

The primary object of this article is to introduce (p, q)-beta logarithmic function with extended beta function by using the logarithmic mean. We evaluate different properties and representations of beta logarithmic function. Further, it is evaluated logarithmic distribution, hypergeometric and confluent hypergeometric functions via logarithmic mean are evaluated and their essential properties are studied. Numerous formulas of (p, q)-beta logarithmic functions such as integral formula, derivative formula, transformation formula and generating function are analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we will establish sharp inequalities of the lemniscate sine function and the so-called weighted (p, q)-exponential type function, of which the latter is an extension of the weighted Hölder mean. These results provide a systematic method for the previous obtained inequalities and give great improvements for bounds of the lemniscate sine function. As applications, several high accuracy approximations for the lemniscate sine function are also derived. [ABSTRACT FROM AUTHOR]

In this paper, we prove two supercongruences conjectured by Z.-W. Sun via the Wilf–Zeilberger method. One of them is, for any prime p > 3, 4 F 3 [ 7 6 1 2 1 2 1 2 1 6 1 1 | - 1 8 ] p - 1 2 = p ( - 2 p ) + p 3 4 ( 2 p ) E p - 3 (mod p 4) , where (· p) stands for the Legendre symbol, and En is the n-th Euler number. [ABSTRACT FROM AUTHOR]

We study the qKZ difference equations with values in the n$n$th tensor power of the vector sl2${\mathfrak {sl}_2}$ representation V$V$, variables z1,⋯,zn$z_1,\dots,z_n$, and integer step κ$\kappa$. For any integer N$N$ relatively prime to the step κ$\kappa$, we construct a family of polynomials fr(z)$f_r(z)$ in variables z1,⋯,zn$z_1,\dots,z_n$ with values in V⊗n$V^{\otimes n}$ such that the coordinates of these polynomials with respect to the standard basis of V⊗n$V^{\otimes n}$ are polynomials with integer coefficients. We show that fr(z)$f_r(z)$ satisfy the qKZ equations modulo N$N$. Polynomials fr(z)$f_r(z)$ are modulo N$N$ analogs of the hypergeometric solutions of the qKZ given in the form of multidimensional Barnes integrals. [ABSTRACT FROM AUTHOR]

The intertwining relationship between sustainability and discrete probability distributions found its significance in decision-making processes and risk assessment frameworks. Count data modeling and its practical applications have gained attention in numerous research studies. This investigation focused on a particular discrete distribution characterized by a single parameter obtained through the survival discretization method. Statistical attributes of this distribution were accurately explicated using generalized hypergeometric functions. The unveiled characteristics highlighted its suitability for analyzing data displaying "right-skewed" asymmetry and possessing extended "heavy" tails. Its failure rate function effectively addressed scenarios marked by a consistent decrease in rates. Furthermore, it proved to be a valuable tool for probabilistic modeling of over-dispersed data. The study introduced various estimation methods such as maximum product of spacings, Anderson-Darling, right-tail Anderson-Darling, maximum likelihood, least-squares, weighted least-squares, percentile, and Cramer-Von-Mises, offering comprehensive explanations. A ranking simulation study was conducted to evaluate the performance of these estimators, employing ranking techniques to identify the most effective estimator across different sample sizes. Finally, real-world sustainability engineering and medical datasets were analyzed to demonstrate the significance and application of the newly introduced model. [ABSTRACT FROM AUTHOR]

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In this paper, we prove two supercongruences conjectured by Z.-W. Sun via the Wilf–Zeilberger method. One of them is, for any prime p > 3, 4 F 3 [ 7 6 1 2 1 2 1 2 1 6 1 1 | - 1 8 ] p - 1 2 = p ( - 2 p ) + p 3 4 ( 2 p ) E p - 3 (mod p 4) , where (· p) stands for the Legendre symbol, and En is the n-th Euler number. [ABSTRACT FROM AUTHOR]

The variable-order fractional Laplacian plays an important role in the study of heterogeneous systems. In this paper, we propose the first numerical methods for the variable-order Laplacian (- Δ) α (x) / 2 with 0 < α (x) ≤ 2 , which will also be referred as the variable-order fractional Laplacian if α (x) is strictly less than 2. We present a class of hypergeometric functions whose variable-order Laplacian can be analytically expressed. Building on these analytical results, we design the meshfree methods based on globally supported radial basis functions (RBFs), including Gaussian, generalized inverse multiquadric, and Bessel-type RBFs, to approximate the variable-order Laplacian (- Δ) α (x) / 2 . Our meshfree methods integrate the advantages of both pseudo-differential and hypersingular integral forms of the variable-order fractional Laplacian, and thus avoid numerically approximating the hypersingular integral. Moreover, our methods are simple and flexible of domain geometry, and their computer implementation remains the same for any dimension d ≥ 1 . Compared to finite difference methods, our methods can achieve a desired accuracy with much fewer points. This fact makes our method much attractive for problems involving variable-order fractional Laplacian where the number of points required is a critical cost. We then apply our method to study solution behaviors of variable-order fractional PDEs arising in different fields, including transition of waves between classical and fractional media, and coexistence of anomalous and normal diffusion in both diffusion equation and the Allen–Cahn equation. These results would provide insights for further understanding and applications of variable-order fractional derivatives. [ABSTRACT FROM AUTHOR]

In this manuscript, we present a new definition of $ (k,\tau) $ (k , τ) -Gauss hypergeometric matrix function and study its analytical properties, like derivative formulas and integral representations. Furthermore, as an application we establish $ {\rm k} $ k -fractional calculus operators for the novel matrix function. We also give some new and known results as special cases of our proposed generalization of the Wright hypergeometric matrix function. [ABSTRACT FROM AUTHOR]

This paper discusses the approximate distributions of eigenvalues of a singular Wishart matrix. We give the approximate joint density of eigenvalues by Laplace approximation for the hypergeometric functions of matrix arguments. Furthermore, we show that the distribution of each eigenvalue can be approximated by the chi-square distribution with varying degrees of freedom when the population eigenvalues are infinitely dispersed. The derived result is applied to testing the equality of eigenvalues in two populations. [ABSTRACT FROM AUTHOR]

We apply known special functions from the literature (and these include the Fox H – function, the exponential function, the Mittag-Leffler function, the Gauss Hypergeometric function, the Wright function, the G – function, the Fox–Wright function and the Meijer G – function) and fuzzy sets and distributions to construct a new class of control functions to consider a novel notion of stability to a fractional-order system and the qualified approximation of its solution. This new concept of stability facilitates the obtention of diverse approximations based on the various special functions that are initially chosen and also allows us to investigate maximal stability, so, as a result, enables us to obtain an optimal solution. In particular, in this paper, we use different tools and methods like the Gronwall inequality, the Laplace transform, the approximations of the Mittag-Leffler functions, delayed trigonometric matrices, the alternative fixed point method, and the variation of constants method to establish our results and theory. [ABSTRACT FROM AUTHOR]

The extended conformable k-hypergeometric function finds various applications in physics due to its ability to describe complex mathematical relationships arising in different physical scenarios. Here are a few instances of its uses in physics, including nuclear physics, fluid dynamics, quantum mechanics, and astronomy. The main objectives of this paper are to introduce the extended conformable k-hypergeometric and confluent hypergeometric functions by utilizing the new definition of the α , k -beta function and studying its important properties, like integral representation, summation formula, derivative formula, transform formula, and generating function. Also, introduce the extension of the Riemann–Liouville fractional derivative and establish some results related to the newly defined fractional operator, such as the Mellin transform and relations to extended α , k -hypergeometric functions. [ABSTRACT FROM AUTHOR]

For a prime p ≡ 3 (mod 4) and a positive integer t, let q = p 2 t . Let g be a primitive element of the finite field F q . The Peisert graph P ∗ (q) is defined as the graph with vertex set F q where ab is an edge if and only if a - b ∈ ⟨ g 4 ⟩ ∪ g ⟨ g 4 ⟩ . We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in P ∗ (q) . We also give a new proof for the number of complete subgraphs of order three contained in P ∗ (q) by evaluating certain character sums. The computations for the number of complete subgraphs of order four are quite tedious, so we further give an asymptotic result for the number of complete subgraphs of any order m in Peisert graphs. [ABSTRACT FROM AUTHOR]

This paper shows that certain   3 F 4 hypergeometric functions can be expanded in sums of pair products of   2 F 3 functions, which reduce in special cases to   2 F 3 functions expanded in sums of pair products of   1 F 2 functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions,   2 F 1 functions, and   3 F 2 functions into the realm of   p F q functions where p < q for both the summand and terms in the series. In addition to its intrinsic value, this result has a specific application in calculating the response of the atoms to laser stimulation in the Strong Field Approximation. [ABSTRACT FROM AUTHOR]

A collection of functions organized according to their indexing based on non-negative integers is grouped by the common factor of fixed integer N. This grouping results in a summation of N series, each consisting of functions partitioned according to this modulo N rule. Notably, when N is equal to two, the functions in the series are divided into two subseries: one containing even-indexed functions and the other containing odd-indexed functions. This partitioning technique is widely utilized in the mathematical literature and finds applications in various contexts, such as in the theory of hypergeometric series. In this paper, we employ this partitioning technique to establish four distinct families of summation formulas for F 3 4 (1) hypergeometric series. Subsequently, we leverage these summation formulas to introduce eight categories of integral formulas. These integrals feature compositions of Beta function-type integrands and F 2 3 (x) hypergeometric functions. Additionally, we highlight that our primary summation formulas can be used to derive some well-known summation results. [ABSTRACT FROM AUTHOR]

A classical problem due to Abel is to determine if a differential equation |$y^{\prime}=\eta y$| admits a non-trivial solution |$y$| algebraic over |$\mathbb C(x)$| when |$\eta $| is a given algebraic function over |$\mathbb C(x)$|⁠. Risch designed an algorithm that, given |$\eta $|⁠ , determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when |$\eta $| admits a Puiseux expansion with rational coefficients at some point in |$\mathbb C\cup \{\infty \}$|⁠ , which can be assumed to be 0 without loss of generality. We prove the following arithmetic characterization: there exists a non-trivial algebraic solution of |$y^{\prime}=\eta y$| if and only if the coefficients of the Puiseux expansion of |$x\eta (x)$| at |$0$| satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations |$y^{\prime}=\eta y$| with an algebraic solution when |$x\eta (x)$| is an algebraic hypergeometric series with rational parameters, and this enables us to prove a prediction Golyshev made using the theory of motives. We also present two other applications, namely to diagonals of rational fractions and to directed two-dimensional walks. [ABSTRACT FROM AUTHOR]

We demonstrate that certain classes of Schlömilch-like infinite series and series that include generalized hypergeometric functions can be calculated in closed form starting from a simple quantum model of a particle trapped inside an infinite potential well and using principles of quantum mechanics. We provide a general framework based on the Hilbert space eigenproblem that can be applied to different exactly solvable quantum models. Obtaining series from normalization conditions in well-defined quantum problems secures their convergence. [ABSTRACT FROM AUTHOR]

Using a flexible form for ladder operators that incorporates confluent hypergeometric functions, we show how one can determine all of the discrete energy eigenvalues and eigenvectors of the time-independent Schrödinger equation via a single factorization step and the satisfaction of boundary (or normalizability) conditions. This approach determines the bound states of all exactly solvable problems whose wavefunctions can be expressed in terms of confluent hypergeometric functions. It is an alternative that shares aspects of the conventional differential equation approach and Schrödinger's factorization method, but is different from both. We also explain how this approach relates to Natanzon's treatment of the same problem and illustrate how to numerically determine nontrivial potentials that can be solved this way. [ABSTRACT FROM AUTHOR]

In this paper, we consider an extended skew CIR processes with sticky points, which is referred to as the sticky skew CIR process. We first calculate the infinitesimal generator and its domain. To explore its trajectory properties, we compute the Laplace transforms and the expectations of first hitting times over a constant boundary. The solutions of Laplace transforms are expressed in terms of Tricomi and Kummer confluent hypergeometric functions. [ABSTRACT FROM AUTHOR]

The main aim this work is to give Hermite–Hadamard inequalities for convex functions via generalized proportional fractional integrals. We also obtained extensions of Hermite–Hadamard type inequalities for generalized proportional fractional integrals. [ABSTRACT FROM AUTHOR]