Gauss Hypergeometric Differential Equation Research Articles

Voros coefficients of the Gauss hypergeometric differential equation with a large parameter

The Voros coefficient of the Gauss hypergeometric differential equation with a large parameter is defined for the origin and its explicit form and the details of derivation are given.

The classical β-ensembles with β proportional to 1/N: From loop equations to Dyson’s disordered chain

In the classical β-ensembles of random matrix theory, setting β = 2α/N and taking the N → ∞ limit gives a statistical state depending on α. Using the loop equations for the classical β-ensembles, we study the corresponding eigenvalue density, its moments, covariances of monomial linear statistics, and the moments of the leading 1/N correction to the density. From earlier literature, the limiting eigenvalue density is known to be related to classical functions. Our study gives a unifying mechanism underlying this fact, identifying, in particular, the Gauss hypergeometric differential equation determining the Stieltjes transform of the limiting density in the Jacobi case. Our characterization of the moments and covariances of monomial linear statistics is through recurrence relations. We also extend recent work, which begins with the β-ensembles in the high-temperature limit and constructs a family of tridiagonal matrices referred to as α-ensembles, obtaining a random anti-symmetric tridiagonal matrix with i.i.d. (Independent Identically Distributed) gamma distributed random variables. From this, we can supplement analytic results obtained by Dyson in the study of the so-called type I disordered chain.

Further study on the conformable fractional Gauss hypergeometric function

<abstract> This article presents an exhaustive study on the conformable fractional Gauss hypergeometric function (CFGHF). We start by solving the conformable fractional Gauss hypergeometric differential equation (CFGHDE) about the fractional regular singular points $ x = 1 $ and $ x = \infty $. Next, various generating functions of the CFGHF are established. We also develop some differential forms for the CFGHF. Subsequently, differential operators and contiguous relations are reported. Furthermore, we introduce the conformable fractional integral representation and the fractional Laplace transform of CFGHF. As an application, and after making a suitable change of the independent variable, we provide general solutions of some known conformable fractional differential equations, which could be written by means of the CFGHF. </abstract>

Fractional Gauss hypergeometric differential equation

In this paper, we use the new concept of fractional regular singular point and use the technique of fractional power series to solve the fractional Gauss hypergeometric differential equation. Also, we introduce the forms of the conformable fractional derivative and the integral representation of the fractional Guassian function.

Solutions Modulo p of Gauss–Manin Differential Equations for Multidimensional Hypergeometric Integrals and Associated Bethe Ansatz

We consider the Gauss–Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallel to themselves. We reduce these equations modulo a prime integer p and construct polynomial solutions of the new differential equations as p-analogs of the initial hypergeometric integrals. In some cases, we interpret the p-analogs of the hypergeometric integrals as sums over points of hypersurfaces defined over the finite field Fp. This interpretation is similar to the classical interpretation by Yu. I. Manin of the number of points on an elliptic curve depending on a parameter as a solution of a Gauss hypergeometric differential equation. We discuss the associated Bethe ansatz.

Parametric Stokes phenomena of the Gauss hypergeometric differential equation with a large parameter

Stokes phenomena with respect to parameters are investigated for the Gauss hypergeometric differential equation with a large parameter. For this purpose, the notion of the Voros coefficient is introduced for the equation. The explicit forms of the Voros coefficients are given as well as their Borel sums. By using them, formulas which describe the Stokes phenomena are obtained.

Alien derivatives of the WKB solutions of the Gauss hypergeometric differential equation with a large parameter

We compute alien derivatives of the WKB solutions of the Gauss hypergeometric differential equation with a large parameter and discuss the singularity structures of the Borel transforms of the WKB solution expressed in terms of its alien derivatives.

A new interpretation of the Shimura curve with discriminant 6 in terms of Picard modular forms

In this paper we give a realization of the Shimura curve for the quaternion algebra over Q with discriminant 6 as a quotient space of the complex upper half plane by the triangle group Δ(3, 6, 6). It is given by the Schwarz map for the Gauss hypergeometric differential equation $${E\left(\frac{1}{6},\frac{1}{3},\frac{2}{3}\right)}$$ . The corresponding abelian surfaces are obtained as isogenous components of the Jacobi varieties of the Picard curves $${C(s): w^3=z(z-1)\break \left(z-\frac{1}{2} (1-s)\right)\left(z-\frac{1}{2} (1+s)\right)}$$ .

The formal derivation of an exact series expansion for the principal Schottky–Nordheim barrier function v, using the Gauss hypergeometric differential equation

The standard theory of Fowler–Nordheim tunnelling and cold field electron emission (CFE) employs a mathematical function v, sometimes called the principal field emission elliptic function, but better called the principal Schottky–Nordheim barrier function. This function arises when the simple-JWKB (Jeffreys–Wentzel–Kramers–Brillouin) method is applied to solve the Schrödinger equation approximately, for the image-rounded tunnelling barrier introduced by Schottky and then used by Nordheim in late 1928. An exact series expansion was recently found for v, as a function of a complementary elliptic variable l′ equal to y2, where y is the Nordheim parameter. The expansion was originally found by using an algebraic manipulation package. It was subsequently discovered that v(l′) is a particular solution of the ordinary differential equation (ODE) l′(1 − l′) d2v/dl′2 = (3/16)v. This ODE is now recognized to be a special case of the Gauss hypergeometric differential equation. This paper uses known special-case solutions of the hypergeometric equation to formally derive the series expansion for v(l′). It notes how to derive the defining ODE, and then uses an 1876 result from Cayley to derive the boundary condition that dv/dl′ satisfies as l′ tends to zero. It then establishes the series expansion for v(l′), by applying this and the boundary condition v(0) = 1. This mathematical derivation underpins earlier results, including good approximate expressions for v(l′). Its outcome proves that terms involving ln l′ are part of a mathematically correct solution, but fractional powers of l′ are not. It also implies that simple Taylor-expansion methods cannot easily generate good approximation formulae valid over the whole range 0 ⩽ l′ ⩽ 1; this may also apply to barriers of other shapes. This derivation should bring closure to the particular line of mathematical analysis of CFE theory initiated by Nordheim in 1928. It is hoped that the derivation might also serve as a model for analysing other tunnelling-barrier problems.

Hankel operators that commute with second-order differential operators

Suppose that Γ is a continuous and self-adjoint Hankel operator on L 2 ( 0 , ∞ ) with kernel ϕ ( x + y ) and that L f = − d d x ( a ( x ) d f d x ) + b ( x ) f ( x ) with a ( 0 ) = 0 . If a and b are both quadratic, hyperbolic or trigonometric functions, and ϕ satisfies a suitable form of Gauss's hypergeometric differential equation, or the confluent hypergeometric equation, then Γ L = L Γ . The paper catalogues the commuting pairs Γ and L, including important cases in random matrix theory. There are also results proving rapid decay of the singular numbers of Hankel integral operators with kernels that are analytic and of exponential decay in the right half-plane.

Global Sturm inequalities for the real zeros of the solutions of the Gauss hypergeometric differential equation

Liouville–Green transformations of the Gauss hypergeometric equation with changes of variable z ( x ) = ∫ x t p - 1 ( 1 - t ) q - 1 dt are considered. When p + q = 1 , p = 0 or q = 0 these transformations, together with the application of Sturm theorems, lead to properties satisfied by all the real zeros x i of any of its solutions in the interval ( 0 , 1 ) . Global bounds on the differences z ( x k + 1 ) - z ( x k ) , 0 < x k < x k + 1 < 1 being consecutive zeros, and monotonicity of these distances as a function of k can be obtained. We investigate the parameter ranges for which these two different Sturm-type properties are available. Classical results for Jacobi polynomials (Szegö's bounds, Grosjean's inequality) are particular cases of these more general properties. Similar properties are found for other values of p and q , particularly when | p | = | α | and | q | = | β | , α and β being the usual Jacobi parameters.

On confluences of general hypergeometric systems

We extend the well-known results about the process of confluence for the Gauss hypergeometric differential equation to the case of general hypergeometric systems. We see that the process of confluence comes from the geometry of the set of regular elements of the Lie algebra of complex general linear group. As a consequence, we give a geometric and group-theoretic view on the process of confluence for classical special functions.

A connection between a generalized Pascal matrix and the hypergeometric function

The n × n generalized Pascal matrix P( t) whose elements are related to the hypergeometric function 2F 1(a, b; c; x) is presented and the Cholesky decomposition of P( t) is obtained. As a result, it is shown that 2F 1(−a, −b; 1; x) = ∑ k=0 min(a,b) ( a k )( b k )x k is the solution of the Gauss's hypergeometric differential equation, x(1 − x)y″ + [1 + (a + b − 1)x]y′ − aby = 0 . where a and b are any nonnegative integers. Moreover, a recurrence relation for generating the elements of P( t) is given.

On the Green Function for the Anisotropic Simple Cubic Lattice

The analytical properties of the lattice Green functionG(α, w)=1π3∫π0∫π0∫π0dθ1 dθ2 dθ3w−cosθ1−cosθ2−αcosθ3are investigated, where w=u+iv is a complex variable in the (u, v) plane and α is a real parameter in the interval (0, ∞). In particular, it is shown that the function yG(α, z)≡wG(α, w), where z=1/w2, is a solution of a fourth-order linear differential equation of the type∑j=04fj(α, z)D4−jy=0,where fj(α, z) is a polynomial in the variables α and z and D≡d/dz. It is then proved that the solutions of this differential equation can all be expressed in terms of a product of two functions H1(α, z) and H2(α, z) which satisfy second-order linear differential equations of the normal type[D2+U+(α, z)]y=0,[D2+U−(α, z)]y=0,respectively, where U±(α, z) are complicated algebraic functions of α and z. Next Schwarzian transformation theory is used to reduce both these second-order differential equations to the standard Gauss hypergeometric differential equation. From this result it is deduced thatwG(α, w)=21−(2−α)2z+1−(2+α)2z[2πK(k+)][2πK(k−)],wherek2±≡k2±(α, z)=12−12[1−(2−α)2z+1−(2+α)2z]−3[1+(2−αz1−(2+α)z+1−(2−α)z1+(2+α)z]{±16z+1−α2z[1+(2−α)z1+(2+α)z+1−(2−α)z1−(2+α)z]2}and K(k) denotes the complete elliptic integral of the first kind with a modulus k. This basic formula is valid for all values of w=u+iv which lie in the (u, v) plane, provided that a cut is made along the real axis from w=−2−α to w=2+α. In the remainder of the paper exact series expansions for G(α, w) are derived which are valid in a sufficiently small neighbourhood of the branch-point singularities at w=2+α, w=α, and w=|2−α|. In all cases it is shown that the real and imaginary parts of the coefficients in the analytic part of these expansions can be expressed in terms of complete elliptic integrals of the first and second kinds, while the coefficients in the singular part of the expansions can be expressed in terms of rational functions of α. The behaviour of G(α, w) in the immediate neighbourhood of w=0 is also investigated in a similar manner. Finally, several applications of the results are made in lattice statistics.