Search

In this article, we present a new proof of the partial fraction expansion of the cotangent function using the theory of hypergeometric functions. [ABSTRACT FROM AUTHOR]

Let $ F(n,k) $ F (n , k) be a hypergeometric function that may be expressed so that n appears within initial arguments of inverted Pochhammer symbols, as in factors of the form $ \frac {1}{(n)_{k}} $ 1 (n) k . Only in exceptional cases is $ F(n, k) $ F (n , k) such that Zeilberger's algorithm produces a two-term recursion for $ \sum _{k = 0}^{\infty } F(n, k) $ ∑ k = 0 ∞ F (n , k) obtained via the telescoping of the right-hand side of a difference equation of the form $ p_{1}(n) F(n + r, k) + p_{2}(n) F(n, k) = G(n, k+1) - G(n, k) $ p 1 (n) F (n + r , k) + p 2 (n) F (n , k) = G (n , k + 1) − G (n , k) for fixed $ r \in \mathbb {N} $ r ∈ N and polynomials $ p_{1} $ p 1 and $ p_{2} $ p 2 . Building on the work of Wilf, we apply a series acceleration technique based on two-term hypergeometric recursions derived via Zeilberger's algorithm. Fast converging series previously given by Ramanujan, Guillera, Chu and Zhang, Chu, Lupaş, and Amdeberhan are special cases of hypergeometric transforms introduced in our article. [ABSTRACT FROM AUTHOR]

Let Dλd denote the family of monomial deformations of diagonal hypersurface over a finite field 픽q given by Dλd : X 1d + X 2d + ⋯ + X nd = λdX 1h1X2h2⋯Xnhn, where d,n ≥ 2, hi ≥ 1, ∑i=1nh i = d, and gcd(d,h1,h2,…,hn) = 1. The Dwork hypersurface is the case when d = n, that is, h1 = h2 = ⋯ = hn = 1. Formulas for the number of 픽q-points on the Dwork hypersurfaces in terms of McCarthy’s p-adic hypergeometric functions are known. In this paper, we provide a formula for the number of 픽q-points on Dλd in terms of McCarthy’s p-adic hypergeometric function which holds for d ≥ n. [ABSTRACT FROM AUTHOR]

For a general number p\geq 2 of measures, we provide explicit expressions for the Jacobi–Piñeiro and Laguerre of the first kind multiple orthogonal polynomials of type I, presented in terms of generalized hypergeometric functions. [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]

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]

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]

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]

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]

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]

In this article, we first define and then propose to systematically study some new subclasses of the class of analytic and bi-concave functions in the open unit disk. For this purpose, we make use of a combination of the binomial series and the confluent hypergeometric function. Among some other properties and results, we derive the estimates on the initial Taylor-Maclaurin coefficients | a 2 | and | a 3 | for functions in these analytic and bi-concave function classes, which are introduced in this paper. We also derive a number of corollaries and consequences of our main results in this paper. [ABSTRACT FROM AUTHOR]

This paper addresses certain distributional aspects of a scale-free scatter matrix denoted by R that is stemming from a matrix-variate gamma distribution having a positive definite scale parameter matrix B. Under the assumption that B is a diagonal matrix, a structural representation of the determinant of R is derived; the exact density functions of products and ratios of determinants of matrices possessing such a structure are obtained; a closed form expression is given for the density function of R. Moreover, a novel procedure is utilized to establish that certain functions of the determinant of the sample scatter matrix are asymptotically distributed as chi-square or normal random variables. Then, representations of the density function of R that respectively involve multiple integrals, multiple series and Gauss' hypergeometric function are provided for the general case of a positive definite scale parameter matrix, and an illustrative numerical example is presented. Cutting-edge mathematical techniques have been employed to derive the results. Naturally, they also apply to the conventional sample correlation matrix which is encountered in various multivariate inference contexts. [ABSTRACT FROM AUTHOR]

In this paper we construct certain higher Chow cycles in K 1 of the Jacobian of Fermat curves, generalising a construction of Collino. Then we express the regulator of these elements in terms of the special values of hypergeometric functions. [ABSTRACT FROM AUTHOR]

Recently Masjed-Jamei and Koepf obtained generalizations of various classical summation theorems for the 2F1, 3F2, 4F3, 5F4 and 6F5 generalized hypergeometric series. We evaluate a new class of integrals involving generalized hypergeometric function by employing the results given by Masjed-Jamei and Koepf and MacRobert integral, and we give several special cases. [ABSTRACT FROM AUTHOR]

In this paper, we introduce ten new quadruple hypergeometric series. We also obtain their various properties such that integral representations, fractional derivatives, N-fractional connections, operational relations and generating functions. [ABSTRACT FROM AUTHOR]

Two new extensions of the familiar Bernoulli polynomials are considered by using q-sine, q-cosine, q-hypergeometric and q-exponential functions. We call q-sine and q-cosine hypergeometric Bernoulli polynomials. Then, diverse formulas and properties for these polynomials, such as summation formulas, addition formulas, q-derivative properties, q-integral representations and some correlations are derived. Also, q-sine and q-cosine hypergeometric Bernoulli polynomials with two parameters are introduced and some relations and identities are investigated. Furthermore, some computational values are given by tables, and the beautiful zeros representations of the q-sine hypergeometric Bernoulli polynomials and q-cosine hypergeometric Bernoulli polynomials are showed by the figures. [ABSTRACT FROM AUTHOR]

This document discusses recent advances in special functions and their applications across various scientific and engineering disciplines. It highlights the importance of special functions in fields such as mathematics, physics, engineering, statistics, astronomy, computer science, economics, chemistry, biology, medicine, materials science, and environmental science. The document provides summaries of 13 articles that explore topics such as Fibonacci polynomials, Appell and Horn functions, generalized hypergeometric functions, Lagrange-based hypergeometric Bernoulli polynomials, hypergeometric series, Humbert functions, integral formulas, twisted Catalan numbers, incomplete gamma function, existence theorems, finite sum identities, Euler polynomials, and q-generalized tangent-based Apostol-type Frobenius-Euler polynomials. These articles contribute to the understanding and application of special functions in various research realms. [Extracted from the article]

This work aims to introduce κ-Appell hypergeometric matrix functions F1,κ,F2,κ,F3,κ and F4,κ, where κ ∈ N. New relations for these functions, such as the integral representation, important transformation formulas, infinite matrix summation and reduction formulas, are also presented. Finally, new results of the differential formulas associated with κ-Appell hypergeometric matrix functions F1,κ,F2,κ are derived. [ABSTRACT FROM AUTHOR]

We find two series expansions for Legendre's second incomplete elliptic integral E(\lambda, k) in terms of recursively computed elementary functions. Both expansions converge at every point of the unit square in the (\lambda, k) plane. Partial sums of the proposed expansions form a sequence of approximations to E(\lambda,k) which are asymptotic when \lambda and/or k tend to unity, including when both approach the logarithmic singularity \lambda =k=1 from any direction. Explicit two-sided error bounds are given at each approximation order. These bounds yield a sequence of increasingly precise asymptotically correct two-sided inequalities for E(\lambda, k). For the reader's convenience we further present explicit expressions for low-order approximations and numerical examples to illustrate their accuracy. Our derivations are based on series rearrangements, hypergeometric summation algorithms and extensive use of the properties of the generalized hypergeometric functions including some recent inequalities. [ABSTRACT FROM AUTHOR]

The paper is devoted to the proof of infinite linear independence at points that admit high-order approximations by algebraic numbers in non-Archimedean normalized fields. [ABSTRACT FROM AUTHOR]

The paper considers the issue of constructing basises in the solution space of the system of partial differential equations, which is satisfied by the Lauricella hypergeometric function F D (N) (a ; b , c ; z) , depending on N complex variables (z 1 , ... , z N) =: z and having complex parameters (a 1 , ... , a N) =: a , b, c. For an arbitrary number N of variables, we have obtained explicit representations for such basis functions in the vicinity of points (0 , ... , 0) and (∞ , ... , ∞) in terms of the Horn type hypergeometric series in N variables. For some of these functions we have obtained formulas of analytic continuation. The found continuation formulas are important for calculating the solution of the Riemann – Hilbert problem with piecewise constant coefficients and studying its geometrical meaning. Besides, these formulas are effective for solving the parameters problem for the Schwarz – Christoffel integral and calculating conformal mapping of complex-shaped polygons. [ABSTRACT FROM AUTHOR]

This paper deals with Euler-type integrals and the closely related Lauricella function , which is a hypergeometric function of many complex variables . For new analytic continuation formulas are found that represent it in the form of Horn hypergeometric series exponentially converging in corresponding subdomains of , including near hyperplanes of the form , , . The continuation formulas and identities for found in this paper make up an effective apparatus for computing this function and Euler-type integrals expressed in terms of it in the entire complex space , including complicated cases when the variables form one or several groups of closely spaced neighbors. The results are used to compute parameters of the Schwarz–Christoffel integral in the case of crowding and to construct conformal mappings of polygons. [ABSTRACT FROM AUTHOR]

The paper considers the issue of constructing basises in the solution space of the system of partial differential equations, which is satisfied by the Lauricella hypergeometric function F D (N) (a ; b , c ; z) , depending on N complex variables (z 1 , ... , z N) =: z and having complex parameters (a 1 , ... , a N) =: a , b, c. For an arbitrary number N of variables, we have obtained explicit representations for such basis functions in the vicinity of points (0 , ... , 0) and (∞ , ... , ∞) in terms of the Horn type hypergeometric series in N variables. For some of these functions we have obtained formulas of analytic continuation. The found continuation formulas are important for calculating the solution of the Riemann – Hilbert problem with piecewise constant coefficients and studying its geometrical meaning. Besides, these formulas are effective for solving the parameters problem for the Schwarz – Christoffel integral and calculating conformal mapping of complex-shaped polygons. [ABSTRACT FROM AUTHOR]

The resolution of isolated cases of the well-known quadratic transformation due to Kummer was thoroughly examined in two recent publications by Atia as well as Atia and Al-Mohaimeed. The objective of this paper is twofold. We establish generalizations of the quadratic transformation due to Kummer in the most general case in the first section, and in the second section, an effort is made to discuss the resolution of an isolated case of a quadratic transformation contiguous to that of Kummer established by Choi and Rathie. [ABSTRACT FROM AUTHOR]

We provide a novel method to analytically calculate the high-order origin moments of a hypergeometric distribution, that is, the expectation identity method. First, the expectation identity of the hypergeometric distribution is discovered and summarized in a theorem. After that, we analytically calculate the first four origin moments of the hypergeometric distribution by using the expectation identity. Furthermore, we analytically calculate the general kth ( k = 1 , 2 , ... ) origin moment of the hypergeometric distribution by using the expectation identity, and the results are summarized in a theorem. Moreover, we use the general kth origin moment to validate the first four origin moments of the hypergeometric distribution. Next, the coefficients of the first ten origin moments of the hypergeometric distribution are summarized in a table containing Stirling numbers of the second kind. Moreover, the general kth origin moment of the hypergeometric distribution by using the expectation identity is restated by another theorem involving Stirling numbers of the second kind. Finally, we provide some numerical and theoretical results. [ABSTRACT FROM AUTHOR]

Inspired by the work by Brychkov and Saad who gave infinite summation formulas for Appell functions (of two variables) with the help of summation theorems, we, using a new inverse pair of symbolic operators, establish infinite summation formulas for four Lauricella functions of three variables. As an application of the results obtained, we present integral representation formulas. [ABSTRACT FROM AUTHOR]

Recently, Bakhet et al. [9] presented the Wright hypergeometric matrix function 2R(τ) 1 (A,B;C; z) and derived several properties. Abdalla [6] has since applied fractional operators to this function. In this paper, with the help of the generalized Pochhammer matrix symbol (A;B)n and the generalized beta matrix function B(P,Q;X), we introduce and study an extended form of the Wright hypergeometric matrix function, 2R(τ) 1 ((A;A),B;C; z;X). We establish several potentially useful results for this extended form, such as integral representations and fractional derivatives. We also derive some properties of the corresponding incomplete extendedWright hypergeometric matrix function. [ABSTRACT FROM AUTHOR]

We find an explicit formula for an arbitrary -symbol of finite-dimensional irreducible representations of the Lie algebra . It is given by the result of substituting s in a hypergeometric-type series similar to the -series, which is the simplest several-variate hypergeometric series. We present necessary conditions for the -symbol to be nonzero. [ABSTRACT FROM AUTHOR]

We derive the recursion formulas for Horn hypergeometric functions H1 to H7. These recursion formulas help us write these functions as a combination of themselves. [ABSTRACT FROM AUTHOR]

Six families of generalized hypergeometric series in a variable x and an arbitrary number of parameters are considered. Each of them is indexed by an integer n. Linear recurrence relations in n relate these functions and their product by the variable x. We give explicit factorizations of these equations as products of first order recurrence operators. Related recurrences are also derived for the derivative with respect to x. These formulas generalize well-known properties of the classical orthogonal polynomials. [ABSTRACT FROM AUTHOR]

In this paper, using hypergeometric functions, we provide sharp estimates of the remainder of the alternating p-series, ∑n≥1/(−1)n−1 np, where p ≥ 2 is an integer. We show that the largest ρ and the largest σ such that the inequalities 1/2(n+1)p−ρ≤| ∑k=n+1∞ (−1)k−1/kp|≤ 1/2np +σ, hold for any integer n ≥ 1 are ρ (p) = 2p+1− 1/1−(1−2¹−p)ζ (p) and σ (p) = 1/1−(1−21−p)ζ (p) −2, where ζ (p) = ∑k=1∞ 1/kp, the Riemann zeta function. [ABSTRACT FROM AUTHOR]

We present the Gel'fand-Kapranov-Zelevinsky (GKZ) hypergeometric systems of the Feynman integrals of the three-loop vacuum diagrams with arbitrary masses, basing on Mellin-Barnes representations and Miller's transformation. The codimension of derived GKZ hypergeometric systems equals the number of independent dimensionless ratios among the virtual masses squared. Through GKZ hypergeometric systems, the analytical hypergeometric series solutions can be obtained in neighborhoods of origin including infinity. The linear independent hypergeometric series solutions whose convergent regions have non-empty intersection can constitute a fundamental solution system in a proper subset of the whole parameter space. The analytical expression of the vacuum integral can be formulated as a linear combination of the corresponding fundamental solution system in certain convergent region. [ABSTRACT FROM AUTHOR]

Named essentially after their close relationship with the modified Bessel function K ν (z) of the second kind, which is known also as the Macdonald function (or, with a slightly different definition, the Basset function), the so-called Bessel polynomials y n (x) and the generalized Bessel polynomials y n (x ; α , β) stemmed naturally in some systematic investigations of the classical wave equation in spherical polar coordinates. Our main purpose in this invited survey-cum-expository review article is to present an introductory overview of the Bessel polynomials y n (x) and the generalized Bessel polynomials y n (x ; α , β) involving the asymmetric parameters α and β. Each of these polynomial systems, as well as their reversed forms θ n (x) and θ n (x ; α , β) , has been widely and extensively investigated and applied in the existing literature on the subject. We also briefly consider some recent developments based upon the basic (or quantum or q-) extensions of the Bessel polynomials. Several general families of hypergeometric polynomials, which are actually the truncated or terminating forms of the series representing the generalized hypergeometric function r F s with r symmetric numerator parameters and s symmetric denominator parameters, are also investigated, together with the corresponding basic (or quantum or q-) hypergeometric functions and the basic (or quantum or q-) hypergeometric polynomials associated with r Φ s which also involves r symmetric numerator parameters and s symmetric denominator parameters. [ABSTRACT FROM AUTHOR]

Various relations including differentiation formulas, connection with hypergeometric functions, integral representations, formulas of summation and series representations for three types of Neumann polynomials are derived. [ABSTRACT FROM AUTHOR]

We continue a program generalizing classical results from the analysis on symmetric cones to the Dunkl setting for root systems of type A. In particular, we prove a Dunkl-Laplace transform identity for Heckman-Opdam hypergeometric functions of type A and more generally, for the associated Opdam-Cherednik kernel. This is achieved by analytic continuation from a Laplace transform identity for non-symmetric Jack polynomials which was stated, for the symmetric case, as a key conjecture by I.G. Macdonald [arXiv:1309.4568v1]. Our proof for the Jack polynomials is based on Dunkl operator techniques and the raising operator of Knop and Sahi. Moreover, we use these results to establish Laplace transform identities between hypergeometric series in terms of Jack polynomials. Finally, we conclude with a Post-Widder inversion formula for the Dunkl-Laplace transform. [ABSTRACT FROM AUTHOR]

In this paper certain continued fractions of the ratios of 2Ψ2-series have been established. [ABSTRACT FROM AUTHOR]

Over the last two decades, special matrix functions have become a major area of study for mathematicians and physicists. The famous four Appell hypergeometric matrix functions have received considerable attention by many authors from different points of view. The present paper is devoted to provide further investigations on the two variables second Appell hypergeometric matrix functions. Precisely, we derive certain mathematical properties such as limit formulae of Humbert hypergeometric matrix functions, recursion formulae using contiguous matrix relations, some differentiation and summation formulae. We also find general differential operators which induce some differential equations. This enriches the theory of special matrix functions. The obtained results are believed to be newly presented. [ABSTRACT FROM AUTHOR]

The paper deals with the problem of representation of Horn's hypergeometric functions by branched continued fractions. The formal branched continued fraction expansions for three different Horn's hypergeometric function H 4 ratios are constructed. The method employed is a two-dimensional generalization of the classical method of constructing of Gaussian continued fraction. It is proven that the branched continued fraction, which is an expansion of one of the ratios, uniformly converges to a holomorphic function of two variables on every compact subset of some domain H , H ⊂ C 2 , and that this function is an analytic continuation of this ratio in the domain H. The application to the approximation of functions of two variables associated with Horn's double hypergeometric series H 4 is considered, and the expression of solutions of some systems of partial differential equations is indicated. [ABSTRACT FROM AUTHOR]

For a class of even weight functions, generalizations of the classical Jacobi and Laguerre polynomials are defined via fractional Rodrigues' type formulas using the Bessel operators in the form ( d 2 / d x 2) + ((2 β + 1) / x) (d / d x). Their properties, including hypergeometric representation, differential recurrence relations and fractional boundary value problem, are investigated. [ABSTRACT FROM AUTHOR]

In this paper, we derive some branched continued fraction representations for the ratios of the Horn's confluent function H6. The method employed is a two-dimensional generalization of the classical method of constructing of Gaussian continued fraction. We establish the estimates of the rate of convergence for the branched continued fraction expansions in some region Ω (here, region is a domain (open connected set) together with all, part or none of its boundary). It is also proved that the corresponding branched continued fractions uniformly converge to holomorphic functions on every compact subset of some domain Θ, and that these functions are analytic continuations of the ratios of double confluent hypergeometric series in Θ. At the end, several numerical experiments are represented to indicate the power and efficiency of branched continued fractions as an approximation tool compared to double confluent hypergeometric series. [ABSTRACT FROM AUTHOR]

It is well-known that the Kummer quadratic transformation formula is valid provided that its parameters fulfill some specific conditions (see Gradshteyn, Ryzhik, Tables of Integrals, Series and Products, 9.130, 9.134.1). Very recently, one of us established a new identity when one of these conditions is not fulfilled. In this paper, we aim to discuss another isolated case which completely different from the first. Moreover, in the end, we mention two interesting consequences of these two new results. [ABSTRACT FROM AUTHOR]

In this article we obtain new irrationality measures for values of functions which belong to a certain class of hypergeometric functions including shifted logarithmic functions, binomial functions and shifted exponential functions. We explicitly construct Padé approximations by using a formal method and show that the associated sequences satisfy a Poincaré-type recurrence. To study precisely the asymptotic behavior of those sequences, we establish an effective version of the Poincaré-Perron theorem. As a consequence we obtain, among others, effective irrationality measures for values of binomial functions at rational numbers, which might have useful arithmetic applications. A general theorem on simultaneous rational approximations that we need is proven by using new arguments relying on parametric geometry of numbers. [ABSTRACT FROM AUTHOR]

We establish some identities in law for the convolution of a beta prime distribution with itself, involving the square root of beta distributions. The proof of these identities relies on transformations on generalized hypergeometric series obtained via Appell series of the first kind and Thomae's relationships for {}_3F_2(1). Using a self-decomposability argument, the identities are applied to derive complete monotonicity properties for quotients of confluent hypergeometric functions having a doubling character. By means of probability, we also obtain a simple proof of Turán's inequality for the parabolic cylinder function and the confluent hypergeometric function of the second kind. The case of Mill's ratio is discussed in detail. [ABSTRACT FROM AUTHOR]

In the paper, we consider some fractional integral formulas in terms of the Riemann–Liouville, Erdélyi–Kober type, and Weyl fractional integral operators and present the general fractional kinetic model involving the hypergeometric superhyperbolic sine function via the Gauss hypergeometric series. [ABSTRACT FROM AUTHOR]

The Gaussian product-inequality (GPI) conjecture is one of the most famous inequalities associated with Gaussian distributions and has attracted much attention. In this note, we investigate the quantitative versions of the two-dimensional Gaussian product inequalities. For any centered, nondegenerate, and two-dimensional Gaussian random vector (X 1 , X 2) with E [ X 1 2 ] = E [ X 2 2 ] = 1 and the correlation coefficient ρ, we prove that for any real numbers α 1 , α 2 ∈ (− 1 , 0) or α 1 , α 2 ∈ (0 , ∞) , it holds that E [ | X 1 | α 1 | X 2 | α 2 ] − E [ | X 1 | α 1 ] E [ | X 2 | α 2 ] ≥ f (α 1 , α 2 , ρ) ≥ 0 , where the function f (α 1 , α 2 , ρ) will be given explicitly by the Gamma function and is positive when ρ ≠ 0 . When − 1 < α 1 < 0 and α 2 > 0 , Russell and Sun (Statist. Probab. Lett. 191:109656, 2022) proved the "opposite Gaussian product inequality", of which we will also give a quantitative version. These quantitative inequalities are derived by employing the hypergeometric functions and the generalized hypergeometric functions. [ABSTRACT FROM AUTHOR]

We derive two double-series identities involving a bounded sequence employing the Gessel-Stanton and Andrews summation theorems for terminating 3F2 hypergeometric series with arguments 4/3 and 3/4, respectively. Using these double-series identities, we establish two reduction formulas for the Srivastava-Daoust double hypergeometric function with arguments z,-4z/3 and z,-3z=4 expressed in terms of a single generalised hypergeometric function of argument proportional to z³. [ABSTRACT FROM AUTHOR]