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The paper studies discrete structural properties of polynomials that play an important role in the theory of spherical harmonics in any dimensions. These polynomials have their origin in the research on problems of harmonic analysis by means of generalized holomorphic (monogenic) functions of hypercomplex analysis. The Sturm‐Liouville equation that occurs in this context supplements the knowledge about generalized Vietoris number sequences Vn, first encountered as a special sequence (corresponding to n=2) by Vietoris in connection with positivity of trigonometric sums. Using methods of the calculus of holonomic differential equations, we obtain a general recurrence relation for Vn, and we derive an exponential generating function of Vn expressed by Kummer's confluent hypergeometric function. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm that checks if a given second‐order homogeneous linear differential equation can be reduced to the confluent Heun's equation by using the change of variables transformation and the exp‐product transformation. The main purpose of this paper is finding solutions in terms of the confluent Heun's functions for the given differential equation. [ABSTRACT FROM AUTHOR]

In this paper, we state an algorithm that checks whether a given second-order linear differential equation can be reduced to the tri-confluent Heun's equation. The algorithm provides a method for finding solutions of the form exp (∫ r (x) d x) · HeunT (q , α , γ , δ , ϵ , f (x)) , where the parameters α , β , λ ∈ C , the functions r , f ∈ C (x) , and f are not constant. [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]

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]

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]

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]

Indefinite integrals of products of generalized hypergeometric functions satisfying first- order differential equations are considered. Necessary and sufficient conditions for the algebraic independence of the set of these integrals and of their values at algebraic points are studied. All algebraic identities arising in this case are found in closed form. [ABSTRACT FROM AUTHOR]

In this work, the geometric nature of solutions to two second-order differential equations, z y ′ ′ (z) + a (z) y ′ (z) + b (z) y (z) = 0 and z 2 y ′ ′ (z) + a (z) y ′ (z) + b (z) y (z) = d (z) , is studied. Here, a (z) , b (z) , and d (z) are analytic functions defined on the unit disc. Using differential subordination, we established that the normalized solution F (z) (with F(0) = 1) of above differential equations maps the unit disc to the domain bounded by the leminscate curve 1 + z . We construct several examples by the judicious choice of a (z) , b (z) , and d (z) . The examples include Bessel functions, Struve functions, the Bessel–Sturve kernel, confluent hypergeometric functions, and many other special functions. We also established a connection with the nephroid domain. Directly using subordination, we construct functions that are subordinated by a nephroid function. Two open problems are also suggested in the conclusion. [ABSTRACT FROM AUTHOR]

The aim of this work is to establish a connection between Bohr's radius and the analytic and normalized solutions of two differential second-order differential equations, namely y ″ (z) + a (z) y ′ (z) + b (z) y (z) = 0 and z 2 y ″ (z) + a (z) y ′ (z) + b (z) y (z) = d (z) . Using differential subordination, we find the upper bound of the Bohr and Rogosinski radii of the normalized solution F (z) of the above differential equations. We construct several examples by judicious choice of a (z) , b (z) and d (z) . The examples include several special functions like Airy functions, classical and generalized Bessel functions, error functions, confluent hypergeometric functions and associate Laguerre polynomials. [ABSTRACT FROM AUTHOR]

The primary outcome of this study is the construction of partial solutions for a class of multidimensional partial differential equations with multiple singular coefficients of the second order. We consider the generalized multidimensional second-order Euler-Poisson-Darboux equation. Employing a well-known method, we reduce the generalized Euler-Poisson-Darboux equation to a second-order partial differential equation of the hypergeometric type. The solutions to this second order hypergeometric equation comprise 2n functions that contain the first Lauricella hypergeometric function. The Lauricella function, also known as an n-dimensional series, incorporates three distinct parameters - the Pohhammer polynomials. To study the properties of these particular solutions, we require a decomposition formula expressing the first Lauricell function as the product of simpler hypergeometric functions with fewer variables. Through this study of particular solutions and the determination of singularity order at the origin, we establish the uniqueness of these solutions. Thus, having proved the peculiarity of particular solutions at the origin, it can be argued that the constructed particular solutions are fundamental solutions of the generalized multidimensional second-order Euler-Poisson-Darboux equation. [ABSTRACT FROM AUTHOR]

Some well-known results of Grönwall on logarithmic derivative of modified Bessel functions of the first kind concerning exponential bounds are extended to Whittaker functions of the first and second kind M_{\kappa,\mu } and W_{\kappa,\mu }. Moreover, a complete monotonicity result is proved for the logarithmic derivative of the Whittaker function W_{\kappa,\mu }, and some monotonicity results with respect to the parameters and argument are shown for the logarithmic derivative of M_{\kappa,\mu }. The results extend and complement the known results in the literature about modified Bessel functions of the first and second kind. [ABSTRACT FROM AUTHOR]

Two integral transforms involving Gaussian hypergeometric functions in the Kernel, which are generalization of classical Riemann-Liouville and Erdeyi-Kober fractional integral are considered and Applied on the Bessel functions of the first kind and thus two main results are derived, which are given in the form of two theorems. These two main results are obtained in terms of generalized wright functions, which are also expressed in terms of generalized hypergeometric functions, Corollaries & special cases are also considered. [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 discuss the monic polynomials of degree n orthogonal with respect to the perturbed Gaussian weight w (z , t) = | z | α ( z 2 + t) λ e − z 2 , z ∈ R , t > 0 , α > − 1 , λ > 0 , which arises from a symmetrization of a semi-classical Laguerre weight w Lag (z , t) = z γ (z + t) ρ e − z , z ∈ R + , t > 0 , γ > − 1 , ρ > 0. The weight wLag(z) has been widely investigated in multiple-input multi-output antenna wireless communication systems in information theory. Based on the ladder operator method, two auxiliary quantities, Rn(t) and rn(t), which are related to the three-term recurrence coefficients βn(t), are defined, and we show that they satisfy coupled Riccati equations. This turns to be a particular Painlevé V (PV, for short), i.e., P V λ 2 2 , − (1 − (− 1) n α) 2 8 , − 2 n + α + 2 λ + 1 2 , − 1 2 . We also consider the quantity σ n (t) ≔ 2 t d d t ln D n (t) , which is allied to the logarithmic derivative of the Hankel determinant Dn(t). The difference and differential equations satisfied by σn(t), as well as an alternative integral representation of Dn(t), are obtained. The asymptotics of the Hankel determinant under a suitable double scaling, i.e., n → ∞ and t → 0 such that s ≔ 4nt is fixed, are established. Finally, by using the second order difference equation satisfied by the recurrence coefficients, we obtain the large n full asymptotic expansions of βn(t) with the aid of Dyson's Coulomb fluid approach. By employing these results, the second differential equations satisfied by the orthogonal polynomials will be reduced to a confluent Heun equation. [ABSTRACT FROM AUTHOR]

How does the inclusion of a gravitational potential alter an otherwise exact quantum mechanical result? This question motivates this report, with the answer determined from an edited version of a problem with a pair of masses sandwiched between three springs obeying Hooke's law. To elaborate, we begin with the Hamiltonian associated with the vibration of the two masses about their equilibrium positions in one dimension; the Schrödinger equation for the reduced mass is then solved to obtain the parabolic cylinder functions D μ (x) as eigenfunctions, and the eigenvalues of the reduced Hamiltonian are calculated exactly. The introduction of the gravitational potential in the Schrödinger equation alters the eigenfunctions D μ (x) to the bi-confluent H e u n B (q , α , γ , δ , ε , x) function; and the eigenvalues are then determined from a recent series expansion in terms of the Hermite functions for the solution of the differential equation whose exact solution is the aforesaid HeunB function. [ABSTRACT FROM AUTHOR]

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Throughout this paper, we will present a new extension of the Wright hypergeometric matrix function by employing the extended Pochhammer matrix symbol. First, we present the extended hypergeometric matrix function and express certain integral equations and differential formulae concerning it. We also present the Mellin matrix transform of the extended Wright hypergeometric matrix function. After that, we present some fractional calculus findings for these expanded Wright hypergeometric matrix functions. Lastly, we present several theorems of the extended Wright hypergeometric matrix function in fractional Kinetic equations. [ABSTRACT FROM AUTHOR]

We develop the theory of quantization of spectral curves via the topological recursion. We formulate a quantization scheme of spectral curves which is not necessarily admissible in the sense of Bouchard and Eynard. The main result of this paper and the second part (Iwaki et al. in Voros coefficients for the hypergeometric differential equations and Eynard–Orantin's topological recursion, part II: for the confluent family of hypergeometric equations, preprint; arXiv:1810.02946) establishes a relation between the Voros coefficients for the quantum curves and the free energy for spectral curves associated with the confluent family of Gauss hypergeometric differential equations. We focus on the Weber equation in this article and generalize the result for the other members of the confluent family in the second part. We also find explicit formulas of free energy for those spectral curves in terms of the Bernoulli numbers. [ABSTRACT FROM AUTHOR]

The factorization method was introduced by Schrödinger in 1940. Its use in bound-state problems is widely known, including in supersymmetric quantum mechanics; one can create a factorization chain, which simultaneously solves a sequence of auxiliary Hamiltonians that share common eigenvalues with their adjacent Hamiltonians in the chain, except for the lowest eigenvalue. In this work, we generalize the factorization method to continuum energy eigenstates. Here, one does not generically have a factorization chain—instead all energies are solved using a "single-shot factorization", enabled by writing the superpotential in a form that includes the logarithmic derivative of a confluent hypergeometric function. The single-shot factorization approach is an alternative to the conventional method of "deriving a differential equation and looking up its solution", but it does require some working knowledge of confluent hypergeometric functions. This can also be viewed as a method for solving the Ricatti equation needed to construct the superpotential. [ABSTRACT FROM AUTHOR]

We study the reducibility of an isomonodromic family of Fuchsian systems on the Riemann sphere which is determined by some initial Fuchsian system and its monodromy to an isomonodromic family of upper triangular Fuchsian systems. Under some conditions on the monodromy of the initial Fuchsian system, a nonlinear Schlesinger system is reduced to a system of homogeneous and inhomogeneous linear Pfaffian Jordan–Pochhammer differential equations. Such Jordan–Pochhammer systems possess hypergeometric type solutions admitting an explicit description of their integral representations. [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]

Sanford S. Miller and Petru T. Mocanu's theory of second-order differential subordinations was extended for the case of third-order differential subordinations by José A. Antonino and Sanford S. Miller in 2011. In this paper, new results are proved regarding third-order differential subordinations that extend the ones involving the classical second-order differential subordination theory. A method for finding a dominant of a third-order differential subordination is provided when the behavior of the function is not known on the boundary of the unit disc. Additionally, a new method for obtaining the best dominant of a third-order differential subordination is presented. This newly proposed method essentially consists of finding the univalent solution for the differential equation that corresponds to the differential subordination considered in the investigation; previous results involving third-order differential subordinations have been obtained mainly by investigating specific classes of admissible functions. The fractional integral of the Gaussian hypergeometric function, previously associated with the theory of fuzzy differential subordination, is used in this paper to obtain an interesting third-order differential subordination by involving a specific convex function. The best dominant is also provided, and the example presented proves the importance of the theoretical results involving the fractional integral of the Gaussian hypergeometric function. [ABSTRACT FROM AUTHOR]

Polynomial solutions of the confluent Heun differential equation (CHE) are derived by identifying conditions under which the infinite power series expansions around the z = 0 singular point can be terminated. Assuming a specific structure of the expansion coefficients, these conditions lead to four non-trivial polynomials that can be expressed as special cases of the confluent Heun function H c (p , β , γ , δ , σ ; z) . One of these recovers the generalized Laguerre polynomials L N (α) , and another one the rationally extended X 1 type Laguerre polynomials L ^ N (α) . The two remaining solutions represent previously unknown polynomials that do not form an orthogonal set and exhibit features characteristic of semi-classical orthogonal polynomials. A standard method of generating exactly solvable potentials in the one-dimensional Schrödinger equation is applied to the CHE, and all known potentials with solutions expressed in terms of the generalized Laguerre polynomials within, or outside the Natanzon confluent potential class, are recovered. It is also found that the potentials generated from the two new polynomial systems necessarily depend on the N quantum number. General considerations on the application of the Heun type differential differential equations within the present framework are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we construct a (p , k) -hypergeometric function by using the Hadamard product, which we call the generalized (p , k) -hypergeometric function. Several properties, namely, convergence properties, derivative formulas, integral representations and differential equations are indicated of this function. The latter function is a generalization of the usual hypergeometric function, the k-analogue of hypergeometric function and other hypergeometric functions are recently presented. As an application, we obtain the solution of the generalized fractional kinetic equations involving of the generalized (p , k) -hypergeometric function. [ABSTRACT FROM AUTHOR]

Through this article, we will discuss a new extension of the incomplete Wright hypergeometric matrix function by using the extended incomplete Pochhammer matrix symbol. First, we give a generalization of the extended incomplete Wright hypergeometric matrix function and state some integral equations and differential formulas about it. Next, we obtain some results about fractional calculus of these extended incomplete Wright hypergeometric matrix functions. Finally, we discuss an application of the extended incomplete Wright hypergeometric matrix function in the kinetic equations. [ABSTRACT FROM AUTHOR]

For the domain , where , and a function , we consider the Zinger operator and define . In this article, we study a class of periodic functions under the iterations of and show that have interesting properties. A typical element of this class is constructed from the holomorphic solution of a differential equation with maximal unipotent monodromy. For this solution we define a kind of deformation (Zinger deformation) as a member of . This deformation is a natural generalization of what Zinger did for the hypergeometric function Finally for a family of Calabi–Yau manifolds, we consider the associated Picard–Fuchs equation. Then under the mirror symmetry hypothesis, we show that the Yukawa couplings can be interpreted as these new functions . [ABSTRACT FROM AUTHOR]

A special function is a function that is typically entitled after an early scientist who studied its features and has a specific application in mathematical physics or another area of mathematics. There are a few significant examples, including the hypergeometric function and its unique species. These types of special functions are generalized by fractional calculus, fractal, q-calculus, (q , p) -calculus and k-calculus. By engaging the notion of q-fractional calculus (QFC), we investigate the geometric properties of the generalized Prabhakar fractional differential operator in the open unit disk ∇ : = { ξ ∈ C : | ξ | < 1 } . Consequently, we insert the generalized operator in a special class of analytic functions. Our methodology is indicated by the usage of differential subordination and superordination theory. Accordingly, numerous fractional differential inequalities are organized. Additionally, as an application, we study the solution of special kinds of q–fractional differential equation. [ABSTRACT FROM AUTHOR]

In this article, we derived conditions on the coefficient functions a (z) and b (z) of the differential equations y ″ (z) + a (z) y ′ (z) + b (z) y (z) = 0 and z 2 y ″ (z) + a (z) z y ′ (z) + b (z) y (z) = 0 , such their solution f (z) with normalization f (0) = 0 = f ′ (0) − 1 is starlike in the lemniscate domain, equivalently z f ′ (z) / f (z) ≺ 1 + z . We provide several examples with graphical presentations for a clear view of the obtained results. [ABSTRACT FROM AUTHOR]

Agarwal et al. (2021) established the extension of several fundamental contiguous relations for G B . Our aim in this work is to investigate several properties of differentiation formulas, differential equations, recursion relations, differential recursion relations, confluence formulas, series representations, integration formulas, and infinite summations for Horn's hypergeometric function G B of three variables. Some well-known particular cases have additionally been given. [ABSTRACT FROM AUTHOR]

Orthogonal polynomials for the weight x ν exp ⁡ (− x − t / x) , x , t > 0 , ν ∈ R are investigated without the use of the Chen–Ismail ladder operators approach. Differential and differential–difference equations are deduced. [ABSTRACT FROM AUTHOR]

A method related to Wronskians has recently been applied to derive many indefinie integrals involving Bessel functions and associated Legendre functions. Here the same method is applied to derive indefinite integrals of Whittaker functions and Gauss hypergeometric functions. All the integrals presented here appear to be new and are derived either directly from Wronskians or from a differential equation related to Wronskians. All the results presented have been checked numerically using Mathematica. [ABSTRACT FROM AUTHOR]

ferential equations in physics, chemistry, and engineering. Its two power series solutions are the Kummer function, M(a,b,z) often referred to as the confluent hypergeometric function of the first kind, and where a and b are parameters that appear in the differential equation. A third function, the Tricomi function, M(a, b, z), sometimes referred to as the confluent hypergeometric function of the second kind, is also a solution of the confluent hypergeometric equation that is routinely used. Contrary to common procedure, all three of these functions (and more) must be considered in a search for the two linearly independentsolutionsoftheconfluenthypergeometricequation. Therearesituations,when a and a-b are integers,where one of these functions is not defined, or two of the functions are not linearly independent, or one of the linearly independents ol utions of the differentia lequationis different from these three functions. Many of these special cases correspond precisely to cases needed to solve problems in physics. This leads to significant confusion about how to work with confluent hyper geometric equations, inspite of authoritative references such as the NIST Digital Library of Mathematical Functions. Here, we carefully describe all of the different cases one has to consider and what the explicit formulas are for the two linearly independent solutions of the confluent hypergeometric equation. The procedure to properly solve the confluent hypergeometric equation is summarized in a convenient table. As an example, we use these solutions to study the bound states of the hydrogenic atom, correcting the standard treatment in textbooks. We also briefly consider the cutoff Coulomb potential. We hope that this guide will aid physicists to properly solve problems that involve the confluent hypergeometric differential equation. [ABSTRACT FROM AUTHOR]

We answer some questions in a paper by Kaneko and Koike ['On modular forms arising from a differential equation of hypergeometric type', Ramanujan J.7(1–3) (2003), 145–164] about the modularity of the solutions of a certain differential equation. In particular, we provide a number-theoretic explanation of why the modularity of the solutions occurs in some cases and does not occur in others. This also proves their conjecture on the completeness of the list of modular solutions after adding some missing cases. [ABSTRACT FROM AUTHOR]

It turns out that the partial sums g n z = ∑ k = 0 n a 1 k ⋯ a p k b 1 k ⋯ b q k z k k ! of a generalized hypergeometric series pFq(a1,... , ap; b1,... , bq; z) with parameters aj, bl ∈ ℂ\{0,−1,−2,...} are orthogonal Sobolev polynomials. The corresponding monic polynomials Gn(z) are RI-type polynomials. Therefore, they are related to biorthogonal rational functions. The polynomials gn satisfy a differential equation (in z) and a recurrence relation (in n). We study the integral representations for gn and some of their basic properties. It is shown that partial sums of any power series with nonzero coefficients are also related to biorthogonal rational functions. We also establish a relationship between the polynomials gn(z) and the Jacobi-type pencils and their associated polynomials. [ABSTRACT FROM AUTHOR]

Special polynomials play an important role in several subjects of mathematics, engineering, and theoretical physics. Many problems arising in mathematics, engineering, and mathematical physics are framed in terms of differential equations. In this paper, we introduce the family of the Lagrange-based hypergeometric Bernoulli polynomials via the generating function method. We state some algebraic and differential properties for this family of extensions of the Lagrange-based Bernoulli polynomials, as well as a matrix-inversion formula involving these polynomials. Moreover, a generating relation involving the Stirling numbers of the second kind was derived. In fact, future investigations in this subject could be addressed for the potential applications of these polynomials in the aforementioned disciplines. [ABSTRACT FROM AUTHOR]

There have been numerous investigations on the hypergeometric series 2 F 1 and the generalized hypergeometric series p F q such as differential equations, integral representations, analytic continuations, asymptotic expansions, reduction cases, extensions of one and several variables, continued fractions, Riemann's equation, group of the hypergeometric equation, summation, and transformation formulae. Among the various approaches to these functions, the transformation formulae for the hypergeometric series 2 F 1 and the generalized hypergeometric series p F q are significant, both in terms of applications and theory. The purpose of this paper is to establish a number of transformation formulae for p F q , whose particular cases would include Gauss's and Kummer's quadratic transformation formulae for 2 F 1 , as well as their two extensions for 3 F 2 , by making advantageous use of a recently introduced sequence and some techniques commonly used in dealing with p F q theory. The p F q function, which is the most significant function investigated in this study, exhibits natural symmetry. [ABSTRACT FROM AUTHOR]

Many indefinite integrals are derived for Bessel functions and associated Legendre functions from particular transformations of their differential equations which are closely linked to Wronskians. A large portion of the results for Bessel functions is known, but all the results for associated Legendre functions appear to be new. The method can be applied to many other special functions. All results have been checked by differentiation using Mathematica. [ABSTRACT FROM AUTHOR]

In this paper, by applying the differential equation techniques and using integral representations, we obtain the Kummer-type transformations of k-hypergeometric functions 1 F 1 , k ((a , k) ; (b , k) ; x) and 2 F 2 , k ((a , k) , (d , k) ; (b , k) , (c , k) ; x) , which include known results as special cases. [ABSTRACT FROM AUTHOR]

For Lipschitz test functions we propose a new bound of the solution of the Stein equation related to the generalized inverse Gaussian (resp. the Kummer) distribution. This bound is derived using the general approach established in [4] for distributions satisfying a certain differential equation, and thus is optimal for Lipschitz test functions. Themain contribution of this paper is to establish an explicit expression of the bound as a function of the parameters of the distribution in terms of the modified Bessel function of the third kind (resp. the confluent hypergeometric function of the second kind). Under a restriction on the parameters we also obtain an optimal bound for the first derivative of the solution. A recurrence formula is established using the iterative technique developed in [4, 5] in order to bound derivatives of any order, for test functions smooth enough. [ABSTRACT FROM AUTHOR]

We apply the method of [ 7 ] to compute the Stokes matrices of non-resonant confluent hypergeometric differential equations. We discuss the ambiguity of the presentation of the Stokes matrices regarding different choices. The results rely on an explicit description of the perverse sheaf associated to the non-confluent regular singular hypergeometric system arising via Fourier–Laplace transform. We give assumptions on the parameter such that the Stokes matrices have rational or real values. Under some more restrictive conditions, the Stokes matrices had been computed by Duval–Mitschi before. We compare our results with their formulae in the unramified case. [ABSTRACT FROM AUTHOR]

In this paper, we consider the third-order differential equation of the quadratic products of Hermite functions and present the integral representations of products of Hermite functions (Lebedev identities) in terms of the confluent hypergeometric functions. Manipulating the integrands of these identities with various integral representations lead us to get new representations for the products of Hermite functions. [ABSTRACT FROM AUTHOR]

The multiple T-value, which is a variant of the multiple zeta value of level two, was introduced by Kaneko and Tsumura ['Zeta functions connecting multiple zeta values and poly-Bernoulli numbers', in: Various Aspects of Multiple Zeta Functions, Advanced Studies in Pure Mathematics, 84 (Mathematical Society of Japan, Tokyo, 2020), 181–204]. We show that the generating function of a weighted sum of multiple T-values of fixed weight and depth is given in terms of the multiple T-values of depth one by solving a differential equation of Heun type. [ABSTRACT FROM AUTHOR]

The diagrammatic coaction maps any given Feynman graph into pairs of graphs and cut graphs such that, conjecturally, when these graphs are replaced by the corresponding Feynman integrals one obtains a coaction on the respective functions. The coaction on the functions is constructed by pairing a basis of differential forms, corresponding to master integrals, with a basis of integration contours, corresponding to independent cut integrals. At one loop, a general diagrammatic coaction was established using dimensional regularisation, which may be realised in terms of a global coaction on hypergeometric functions, or equivalently, order by order in the ϵ expansion, via a local coaction on multiple polylogarithms. The present paper takes the first steps in generalising the diagrammatic coaction beyond one loop. We first establish general properties that govern the diagrammatic coaction at any loop order. We then focus on examples of two-loop topologies for which all integrals expand into polylogarithms. In each case we determine bases of master integrals and cuts in terms of hypergeometric functions, and then use the global coaction to establish the diagrammatic coaction of all master integrals in the topology. The diagrammatic coaction encodes the complete set of discontinuities of Feynman integrals, as well as the differential equations they satisfy, providing a general tool to understand their physical and mathematical properties. [ABSTRACT FROM AUTHOR]

The paper proposes to introduce incomplete Srivastava's triple hypergeometric matrix functions through application of the incomplete Pochhammer matrix symbols. We also derive certain properties such as matrix differential equation, integral formula, reduction formula, recursion formula, recurrence relation and differentiation formula of the incomplete Srivastava's triple hypergeometric matrix functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the incomplete second Appell hypergeometric matrix functions in terms of the incomplete Pochhammer matrix symbols. We also derive certain properties such as matrix differential equation, integral formula, recurrence relation, recursion formula and differentiation formula of the incomplete second Appell hypergeometric matrix functions. This enriches the theory of special matrix functions. The results listed here are believed to be new. [ABSTRACT FROM AUTHOR]

Reduction formulas for the sum of products of hypergeometric functions can be traced back to Euler. This topic has intimate relations with summation and transformation formulas, contiguous relations, and algebraic properties of the (generalized) hypergeometric differential equation. Over the past several years important discoveries have been made in this topic by Gorelov, Ebisu, Beukers and Jouhet, and Feng, Kuznetsov and Yang. In this paper we give a generalization of the Feng, Kuznetsov, and Yang identity, also covering Ebisu's and Gorelov's formulas as particular cases. [ABSTRACT FROM AUTHOR]