Power series and integral forms of Lame equation in Weierstrass's form

Choun, Yoon Seok

doi:10.48550/arxiv.1303.0878

Cited by 7 publications

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“…The general expression of the representation in the form of integral of Lame poly-nomial which makes B n term terminated in Weierstrass's form in Ref. [21] is given by 4…”

Section: Definition 1 I Define Thatmentioning

confidence: 99%

“…(Erdelyi 1940) [5] In Ref. [21], I show power series expansion in closed forms of Lame function in Weierstrass's form and its representation of the form of integrals for the cases of infinite series and polynomial in which makes B n term terminated. I show that a 2 F 1 function recurs in each of sub-integral forms of Lame function in Weierstrass's form: the first sub-integral form contains zero term of A ′ n s, the second one contains one term of A n 's, the third one contains two terms of A n 's, etc.…”

Section: Introductionmentioning

confidence: 99%

“…In Ref. [21], we obtain power series expansion and integral form of Lame function of the first type in Weier-…”

Section: Introductionmentioning

confidence: 99%

“…See section 4 for all the papers in the series. Previous paper in series deals with the power series expansion and the integral formalism of Lame equation in Weierstrass's form and its asymptotic behavior [21]. The next paper in the series describes analytic solution for grand confluent hypergeometric function [23].…”

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The generating functions of Lame equation in Weierstrass's form

Choun

2013

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Lame equation arises from deriving Laplace equation in ellipsoidal coordinates; in other words, it's called ellipsoidal harmonic equation. Lame function is applicable to diverse areas such as boundary value problems in ellipsoidal geometry, chaotic Hamiltonian systems, the theory of Bose-Einstein condensates, etc.By applying generating function into modern physics (quantum mechanics, thermodynamics, black hole, supersymmetry, special functions, etc), we are able to obtain the recursion relation, a normalization constant for the wave function and expectation values of any physical quantities. For the case of hydrogen-like atoms, generating function of associated Laguerre polynomial has been used in order to derive expectation values of position and momentum. By applying integral forms of Lame polynomial in Weierstrass's form in which makes B n term terminated[21], I consider generating function of it including all higher terms of A n 's. 1 This paper is 8th out of 10 in series "Special functions and three term recurrence formula (3TRF)". See section 4 for all the papers in the series. Previous paper in series deals with the power series expansion and the integral formalism of Lame equation in Weierstrass's form and its asymptotic behavior [21]. The next paper in the series describes analytic solution for grand confluent hypergeometric function [23].

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“…The general expression of the representation in the form of integral of Lame poly-nomial which makes B n term terminated in Weierstrass's form in Ref. [21] is given by 4…”

Section: Definition 1 I Define Thatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In Ref. [21], we obtain power series expansion and integral form of Lame function of the first type in Weier-…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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The generating functions of Lame equation in Weierstrass's form

Choun

2013

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“…7. "Power series and integral forms of Lame equation in Weierstrass's form" [14]-apply the three term recurrence formula, and derive the power series expansion of Lame function in Weierstrass's form and its integral forms.…”

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confidence: 99%

Asymptotic behavior of Heun function and its integral formalism

Choun

2013

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The Heun function generalizes all well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric 2 F 1 , 1 F 1 and 0 F 1 functions. Heun functions are applicable to diverse areas such as theory of black holes, lattice systems in statistical mechanics, solution of the Schrödinger equation of quantum mechanics, and addition of three quantum spins.In this paper, applying three term recurrence formula [9], we consider asymptotic behaviors of Heun function and its integral formalism including all higher terms of A n 's. 1 We show how the power series expansion of Heun functions can be converted to closed-form integrals for all cases of infinite series and polynomial. One interesting observation resulting from the calculations is the fact that a 2 F 1 function recurs in each of sub-integral forms: the first sub-integral form contains zero term of A n s, the second one contains one term of A n 's, the third one contains two terms of A n 's, etc.Applying three term recurrence formula, we consider asymptotic behaviors of Heun functions and their radius of convergences. And we show why Poincaré-Perron theorem is not always applicable to the Heun equation.In the appendix, I apply the power series expansion and my integral formalism of Heun function to "The 192 solutions of the Heun equation" [34]. Due to space restriction final equations for all 192 Heun functions is not included in the paper, but feel free to contact me for the final solutions. Section 6 contains two additional examples using integral forms of Huen function.This paper is 4th out of 10 in series "Special functions and three term recurrence formula (3TRF)". See section 6 for all the papers in the series. The previous paper in series deals with the power series expansion in closed forms of Heun function. The next paper in the series describes the power series expansion of Mathieu function and its integral formalism analytically.

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Approximative solution of the spin free Hamiltonian involving only scalar potential for the quark-antiquark system

Choun¹

2013

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In earlier papers [5,6,7,8] Gürsey 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 [7]. The theory for vanishing mass was solved using Confluent Hypergeometric functions [8].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 Confluent Hypergeometric types. I call it the Grand Confluent Hypergeometric (GCH) Function. My solution produces previously unknown extra hidden radial quantum numbers relevant for description of supersymmetry and for generating new mass formulas.This paper is 1st out of 10 in series "Special functions and three term recurrence formula (3TRF)". See section 6 for all the papers in the series. The next paper in the series describes generalization of three term recurrence relation in linear ordinary differential equations and its applications [12].

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Power series and integral forms of Lame equation in Weierstrass's form

Cited by 7 publications

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The generating functions of Lame equation in Weierstrass's form

The generating functions of Lame equation in Weierstrass's form

Asymptotic behavior of Heun function and its integral formalism

Approximative solution of the spin free Hamiltonian involving only scalar potential for the quark-antiquark system

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