We show that many integrals containing products of confluent hypergeometric functions follow directly from one single integral that has a very simple formula in terms of Appell's double series F_2. We present some techniques for computing such series. Applications requiring the matrix elements of singular potentials and the perturbed Kratzer potential are presented.
We show that the Heun confluent equation admits infinitely many solutions in terms of the confluent generalized hypergeometric functions. For each of these solutions a characteristic exponent of a regular singularity of the Heun confluent equation is a non-zero integer and the accessory parameter obeys a polynomial equation. Each of the solutions can be written as a linear combination with constant coefficients of a finite number of either the Kummer confluent hypergeometric functions or the Bessel functions.
In this paper, using the theory of the so-called fractional calculus we show that it is possible to easily obtain the solutions for the confluent hypergeometric equation. Our approach is to be compared with the standard one (Frobenius) which is based on the ordinary calculus of integer order.
We present an explicit difference equation for the Heckman-Opdam hypergeometric function associated with root systems. Via a confluent hypergeometric limit, an analogous difference equation is obtained for the class-one Whittaker function diagonalizing the open quantum Toda chain.
Variants of the q-hypergeometric equation were introduced in our previous paper with Hatano. In this paper, we consider degenerations of the variant of the q-hypergeometric equation, which is a q-analogue of confluence of singularities in the setting of the differential equation. We also consider degenerations of solutions to the q-difference equations.
We obtain "large gap" asymptotics for a Fredholm determinant with a confluent hypergeometric kernel. We also obtain asymptotics for determinants with two types of Bessel kernels which appeared in random matrix theory.
It is shown that analytically soluble bound states of the Schrödinger equation for a large class of systems relevant to atomic and molecular physics can be obtained by means of the Laplace transform of the confluent hypergeometric equation. It is also shown that all closed-form eigenfunctions are expressed in terms of generalized Laguerre polynomials. The generalized Morse potential is used as an illustration.
We use the tridiagonal representation approach to solve the radial Schrödinger equation for the continuum scattering states of the Coulomb problem in a complete basis set of discrete Bessel functions. Consequently, we obtain a new representation of the confluent hypergeometric function as an infinite sum of Bessel functions, which is numerically very stable and more rapidly convergent than another well-known formula.
We show that the each member of the confluent family of the Gauss hypergeometric equations is realized as quantum curves for appropriate spectral curves. As an application, relations between the Voros coefficients of those equations and the free energy of their classical limit computed by the topological recursion are established. We will also find explicit expressions of the free energy and the Voros coefficients in terms of the Bernoulli numbers and Bernoulli polynomials.
Biconfluent Heun (BCH) function, a confluent form of Heun function, is the special case of Grand Confluent Hypergeometric (GCH) function: this has a regular singularity at x=0, and an irregular singularity at infinity of rank 2. In this paper I apply three term recurrence formula (3TRF) [arXiv:1303.0806] to the integral formalism of GCH function including all higher terms of A_n's and the generating function for the GCH polynomial which makes B_n term terminated. I show how to transform power series expansion in closed forms of GCH equation to its integral representation analytically. This pap…
We construct several expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta functions and the Appell generalized hypergeometric functions of two variables of the fist kind. The coefficients of different expansions obey four-, five-, or six-term recurrence relations that are reduced to ones involving less number of terms only in a few exceptional cases. The conditions for deriving finite-sum solutions via termination of the series are discussed.
This paper examines some solutions for confluent and double-confluent Heun equations. In the first place, we review two Leaver's solutions in series of regular and irregular confluent hypergeometric functions for the confluent equation and introduce an additional expansion in series of irregular confluent hypergeometric functions. Then, we find the conditions under which one of these solutions can be written as a linear combination of the others. In the second place, by means of limiting procedures we generate solutions for the double-confluent equation as well as for special limits of both th…
In this paper, we study the zero sets of the confluent hypergeometric function $_{1}F_{1}(α;γ;z):=\sum_{n=0}^{\infty}\frac{(α)_{n}}{n!(γ)_{n}}z^{n}$, where $α, γ, γ-α\not\in \mathbb{Z}_{\leq 0}$, and show that if $\{z_n\}_{n=1}^{\infty}$ is the zero set of $_{1}F_{1}(α;γ;z)$ with multiple zeros repeated and modulus in increasing order, then there exists a constant $M>0$ such that $|z_n|\geq M n$ for all $n\geq 1$.
In this paper we introduce a basic representation for the confluent Cherednik algebras $\mathcal H_{\rm V}$, $\mathcal H_{\rm III}$, $\mathcal H_{\rm III}^{D_7}$ and $\mathcal H_{\rm III}^{D_8}$ defined in arXiv:1307.6140. To prove faithfulness of this basic representation, we introduce the non-symmetric versions of the continuous dual $q$-Hahn, Al-Salam-Chihara, continuous big $q$-Hermite and continuous $q$-Hermite polynomials.
In this paper, we consider the deformed Fredholm determinant of the confluent hypergeometric kernel. This determinant represents the gap probability of the corresponding determinantal point process where each particle is removed independently with probability $1- γ$, $0 \leq γ<1$. We derive asymptotics of the deformed Fredholm determinant when the gap interval tends to infinity, up to and including the constant term. As an application of our results, we establish a central limit theorem for the eigenvalue counting function and a global rigidity upper bound for its maximum deviation.
We construct new solutions in series of confluent hypergeometric functions for the confluent Heun equation (CHE). Some of these solutions are applied to the one-dimensional stationary Schrödinger equation with hyperbolic and trigonometric quasiexactly solvable potentials.
In previous paper I construct an approximative solution of the power series expansion in closed forms of Grand Confluent Hypergeometric (GCH) function only up to one term of A_n's [4]. And I obtain normalized constant and orthogonal relation of GCH function. In this paper I will apply three term recurrence formula [3] to the power series expansion in closed forms of GCH function (infinite series and polynomial) including all higher terms of A_n's. In general most of well-known special function with two recursive coefficients only has one eigenvalue for the polynomial case. However this new fun…
We find sets of solutions to the generalized spheroidal wave equation (GSWE) or, equivalently, to the confluent Heun equation. Each set is constituted by three solutions, one given by a series of ascending powers of the independent variable, and the others by series of regular and irregular confluent hypergeometric functions. For a fixed set, the solutions converge over different regions of the complex plane but present series coefficients proportional to each other. These solutions for the GSWE afford solutions to a double-confluent Heun equation by a taking-limit process due to Leaver. Anoth…
It is known that a large class of integrable hydrodynamic type systems can be constructed through the Lauricella function, a generalization of the classical Gauss hypergeometric function. In this paper, we construct novel class of integrable hydrodynamic type systems which govern the dynamics of critical points of confluent Lauricella type functions defined on finite dimensional Grassmannian Gr(2,n), the set of 2xn matrices of rank two. Those confluent functions satisfy certain degenerate Euler-Poisson-Darboux equations. It is also shown that in general, hydrodynamic type system associated to …
We consider the derivatives of Horn hypergeometric functions of any number variables with respect to their parameters. The derivative of the function in $n$ variables is expressed as a Horn hypergeometric series of $n+1$ infinite summations depending on the same variables and with the same region of convergence as for original Horn function. The derivatives of Appell functions, generalized hypergeometric functions, confluent and non-confluent Lauricella series and generalized Lauricella series are explicitly presented. Applications to the calculation of Feynman diagrams are discussed, especial…
A Palm hierarchy for determinantal point processes with the confluent hypergeometric kernel, the decomposing measures in the problem of harmonic analysis on the infinite-dimensional unitary group
For any complex simple Lie algebra, we generalize primary fileds in the Wess-Zumino-Novikov-Witten conformal field theory with respect to the case of irregular singularities and we construct integral representations of hypergeometric functions of confluent type, as expectation values of products of generalized primary fields. In the case of sl(2), these integral representations coincide with solutions to confluent KZ equations. Computing the operator product expansion of the energy-momentum tensor and the generalized primary field, new differential operators appear in the result. In the case o…
We consider the circular unitary ensemble with a Fisher-Hartwig singularity of both jump type and root type at $z=1$. A rescaling of the ensemble at the Fisher-Hartwig singularity leads to the confluent hypergeometric kernel. By studying the asymptotics of the Toeplitz determinants, we show that the probability of there being no eigenvalues in a symmetric arc about the singularity on the unit circle for a random matrix in the ensemble can be explicitly evaluated via an integral of the Hamiltonian of the coupled Painlevé V system in dimension four. This leads to a Painlevé-type representation o…
In the paper, the well-known quantum mechanical problem of a spin 1/2 particle in external Coulomb potential, reduced to a system of two first-order differential equations, is studied from the point of view of possible applications of the Heun function theory to treat this system. It is shown that in addition to the standard way to solve the problem in terms of the confluent hypergeometric functions (proposed in 1928 by G. Darvin and W. Gordon), there are possible several other possibilities which rely on applying the confluent Heun functions. Namely, in the paper there are elaborated two comb…
In earlier papers [3,4,5,6] Gursey et al. showed development of a bilocal baryon-meson field from two quark-antiquark fields. The Hamiltonian in the case of vanishing quark masses was shown to have a very good agreement with experiments [5]. The theory for vanishing mass was solved using Confluent Hypergeometric functions [6]. In this paper I construct the normalized wave function for the spin-free Hamiltonian with light quark masses (only up to the first order of the mass of quark). I develop the new kind of special function theory in mathematics that generalize all existing theories of Confl…
The cases when the equation for the derivative of the confluent Heun function has only three singularities (in general, the equation has four such points) are examined. It is shown that this occurs only in three specific cases. Further, it is shown that in all these three cases this equation is reduced to some confluent Heun equation with changed parameters. This means that in these cases the derivative of the confluent Heun function is expressed via some other confluent Heun function. The analysis of the obtained relations shows that they can be useful for the construction of some exact solut…
Passive two-plateau relaxation from Tricomi confluent hypergeometric kernels
Instituto de Microelectronica de Barcelona (IMB-CNM), Consejo Superior de Investigaciones Cientificas (CSIC)（巴塞罗那微电子研究所（IMB-CNM），西班牙国家研究委员会（CSIC））；CIBER-BBN, Instituto de Salud Carlos III（CIBER-BBN，卡洛斯三世健康研究所）；Faculty of Engineering, Free University of Bozen-Bolzano（博岑-波尔扎诺自由大学工程学院）
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 [IKoT] 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…
Partial wave expansion of the Coulomb-distorted plane wave is determined and studied. Dominant and sub-dominant asymptotic expansion terms are given and leading order three-dimensional asymptotic form is derived. The generalized hypergeometric function $_2F_2(a,a;a+l+1,a-l;z)$ is expressed with the help of confluent hypergeometric functions and the asymptotic expansion of $_2F_2(a,a;a+l+1,a-l;z)$ is simplified.
We find transformations of variables which preserve the form of the equation for the kernels of integral relations among solutions of the Heun equation. These transformations lead to new kernels for the Heun equation, given by single hypergeometric functions (Lambe-Ward-type kernels) and by products of two hypergeometric functions (Erdélyi-type). Such kernels, by a limiting process, also afford new kernels for the confluent Heun equation.
Several examples of Jacobi matrices with an explicitly solvable spectral problem are worked out in detail. In all discussed cases the spectrum is discrete and coincides with the set of zeros of a special function. Moreover, the components of corresponding eigenvectors are expressible in terms of special functions as well. Our approach is based on a recently developed formalism providing us with explicit expressions for the characteristic function and eigenvectors of Jacobi matrices. This is done under an assumption of a simple convergence condition on matrix entries. Among the treated special …
Identities between Whittaker and modified Bessel functions are derived for particular complex orders. Certain polynomials appear in such identities, which satisfy a fourth order differential equation (not of hypergeometric type), and they themselves can be expressed as particular linear combinations of products of modified Bessel and confluent hypergeometric functions.
We present the exact solution of the stationary Schrödinger equation equation for the potential $V=V_0/{\sqrt{x}}$. Each of the two fundamental solutions that compose the general solution of the problem is given by a combination with non-constant coefficients of two confluent hypergeometric functions of a shifted argument. Alternatively, the solution is written through the first derivative of a tri-confluent Heun function. Apart from the quasi-polynomial solutions provided by the energy specification $E_n=E_1{n^{-2/3}}$, we discuss the bound-state wave functions vanishing both at infinity and …
The established technique of eliminating upper or lower parameters in a general hypergeometric series is profitably exploited to create pathways among confluent hypergeometric functions, binomial functions, Bessel functions, and exponential series. One such pathway, from the mathematical statistics point of view, results in distributions which naturally emerge within nonextensive statistical mechanics and Beck-Cohen superstatistics, as pursued in generalizations of Boltzmann-Gibbs statistics.
An exact solution of the energy shift in each quantum mechanical energy levels in a one dimensional symmetrical linear harmonic oscillator has been investigated. The solution we have used here is firstly derived by manipulating Schrodinger differential equation to be confluent hypergeometric differential equation. The final exact numerical results of the energy shifts are then found by calculating the final analytical solution of the confluent hypergeometric equation with the use of a software (Mathcad Plus 6.0) or a program programmed by using Turbo Pascal 7.0. We find that the results of the…
In this paper we prove some uniform asymptotic estimates for confluent hypergeometric functions making use of the steepest-descent method. As an application, we obtain Strichartz estimates with loss of angular derivatives for the massless Dirac-Coulomb equation in $3D$.
We study the dynamics of a nonlinear two-level crossing model with a cubic modification of the linear Landau-Zener diabatic energies. The solutions are expressed in terms of the bi-confluent Heun functions --- the generalization of the confluent hypergeometric functions. We express the finial transition probability as a convergent series of the parameters of the nonlinear laser detuning, and derive analytical approximations for the state populations in terms of parabolic cylinder and Whittaker functions. Tractable closed-form expressions are derived for a large part of the parameter space. We …
