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Novel relations and new properties of confluent Heun's functions and their derivatives of arbitrary order

Plamen P Fiziev
2009 Journal of Physics A: Mathematical and Theoretical  
The present article reveals important properties of the confluent Heun's functions. We derive a set of novel relations for confluent Heun's functions and their derivatives of arbitrary order.  ...  Specific new subclasses of confluent Heun's functions are introduced and studied. A new alternative derivation of confluent Heun's polynomials is presented.  ...  Then because of the obvious symmetry of the equation (1.1) under the change {α, β, γ, δ, µ, ν, z} → {α, γ, β, δ, ν, µ, z − 1}, the spectral parameter will be −ν.  ... 
doi:10.1088/1751-8113/43/3/035203 fatcat:n7k22uuzsbb5nlirgmrs3bq6mm

Heun's equation, generalized hypergeometric function and exceptional Jacobi polynomial

Kouichi Takemura
2012 Journal of Physics A: Mathematical and Theoretical  
We study Heun's differential equation in the case that one of the singularities is apparent.  ...  In particular we conjecture a relationship with generalized hypergeometric differential equation and establish it in some cases. We apply our results to exceptional Jacobi polynomials.  ...  He is supported by the Grant-in-Aid for Young Scientists (B) (No. 22740107) from the Japan Society for the Promotion of Science.  ... 
doi:10.1088/1751-8113/45/8/085211 fatcat:46pdklk6jzal5m4edctz6iyt3m

Generalized Heun and Lamé's equations: factorization [article]

Mahouton Norbert Hounkonnou, André Ronveaux
2009 arXiv   pre-print
This paper addresses new results on the factorization of the general Heun's operator, extending the investigations performed in previous works [ Applied Mathematics and Computation 141 (2003), 177 - 184  ...  Acknowledgements The authors are thankful to Dr Alain Moussiaux from the Facultés Universitaires Notre Dame de la Paix (FUNDP), Namur, Belgium, for helpful discussions.  ...  Table 5 . 5 Factorization of the Heun's operator H 2 [y] Table 6 . 6 Factorization of the Heun's operator H 2 [y] (continuation).  ... 
arXiv:0902.2991v1 fatcat:3tho3cuvgnek7oteez3i3throi

Towards Finite-Gap Integration of the Inozemtsev Model

Kouichi Takemura
2007 Symmetry, Integrability and Geometry: Methods and Applications  
The Inozemtsev model is considered to be a multivaluable generalization of Heun's equation.  ...  We review results on Heun's equation, the elliptic Calogero-Moser-Sutherland model and the Inozemtsev model, and discuss some approaches to the finite-gap integration for multivariable models.  ...  Acknowledgements The author would like to thank the referees for valuable comments.  ... 
doi:10.3842/sigma.2007.038 fatcat:5zv2xzvjgzb4zdao3rzcwehvwm

Elliptic solitons and Heun's equation [article]

A.O. Smirnov
2001 arXiv   pre-print
We find a new class of algebraic geometric solutions of Heun's equation with the accessory parameter belonging to a hyperelliptic curve.  ...  Dependence of these solutions from the accessory parameter as well as their relation to Heun's polynomials is studied.  ...  of periodic solution of a differential equation for the product of eigenfunctions of Schrödinger's operator with periodic potential [1, 2, 18] .  ... 
arXiv:math/0109149v2 fatcat:lgzpy5lmdfd6zfzd5ivcz7wxu4

Teukolsky-Starobinsky identities: A novel derivation and generalizations

Plamen P. Fiziev
2009 Physical Review D  
Symmetry of parameters of confluent Heun's functions is shown to stay behind the behavior of the known solutions under the change of the sign of their spin weights.  ...  We present a novel derivation of the Teukolsky-Starobinsky identities, based on properties of the confluent Heun functions.  ...  (IV.2) Here an ordered operator product is used. The arrow indicates the operator ordering and points in the direction of the increase of the integer k. 2.  ... 
doi:10.1103/physrevd.80.124001 fatcat:nbmwugbjifdwnl7btcjcyas3wq

Page 7365 of Mathematical Reviews Vol. , Issue 2004i [page]

2004 Mathematical Reviews  
By matching the solutions across the boundaries we obtain the ellipsoidal particle polarizability, which is written in terms of the standard depolarization factors and logarithmic derivatives of the Heun  ...  In addition, there is a treatment of the nonrelativis- tic interpretation of the covariant operator of real synchronous differentiation. Some Heaviside-Feynman formulae are also men- tioned. LL. G.  ... 

Toric Sasaki–Einstein manifolds and Heun equations

Takeshi Oota, Yukinori Yasui
2006 Nuclear Physics B  
The eigenvalue problem leads to two Heun's differential equations and the exponents at regular singularities are directly related to toric data.  ...  The scaling dimensions of the holomorphic functions are simply given by scalar products of the Reeb vector and the integral vectors, which are consistent with R-charges of BPS states in the dual quiver  ...  We study the polynomial solutions for the two Heun's differential equations.  ... 
doi:10.1016/j.nuclphysb.2006.03.003 fatcat:lsjqrul6v5h4xiy3o34xtgeiz4

HEUN FUNCTIONS VERSUS ELLIPTIC FUNCTIONS

GALLIANO VALENT
2007 Difference Equations, Special Functions and Orthogonal Polynomials  
We describe Picard's generalization of Floquet's theory for differential equations with doubly periodic coefficients and give the detailed forms of the level one Heun functions in terms of Jacobi theta  ...  The finite-gap solutions give an interesting alternative integral representation which, at level one, is shown to be equivalent to their elliptic form.  ...  For all these solutions Heun's differential operator remains obviously factorized and, as shown in [22] , this happens only for these cases.  ... 
doi:10.1142/9789812770752_0057 fatcat:vlmgyp6xo5hqzgsonmolj7r5ly

Heun functions versus elliptic functions [article]

Galliano Valent
2005 arXiv   pre-print
We describe Picard's generalization of Floquet's theory for differential equations with doubly periodic coefficients and give the detailed forms of the level one Heun functions in terms of Jacobi theta  ...  The finite-gap solutions give an interesting alternative integral representation which, at level one, is shown to be equivalent to their elliptic form.  ...  For all these solutions Heun's differential operator remains obviously factorized and, as shown in [22] , this happens only for these cases.  ... 
arXiv:math-ph/0512006v1 fatcat:e7lrhgiiwzdx7ihy74ilsz67ma

Nonlinear Equations [chapter]

2005 Applied Numerical Methods Using MATLAB®  
Factorization): Cholesky, QR, and SVD / 97 2.5 Iterative Methods to Solve Equations / 98 2.5.1 Jacobi Iteration / 98 2.5.2 Gauss-Seidel Iteration / 100 2.5.3 The Convergence of Jacobi and Gauss-Seidel  ...  Linear Equations 2.1 Solution for a System of Linear Equations / 72 2.1.1 The Nonsingular Case (M = N) I 72 2.1.2 The Underdetermined Case (M < N): Minimum-Norm Solution / 72 2.1.3 The Overdetermined  ... 
doi:10.1002/0471705195.ch4 fatcat:oksrzi2lvrd53egmpj36dikqqu

Klein-Gordon Equation with Coulomb Potential in the Presence of a Minimal Length [article]

Djamil Bouaziz
2013 arXiv   pre-print
The zero energy solution is obtained analytically in momentum space in terms of Heun's functions.  ...  The equation with nonzero energy is established in a particular case in the first order of the deformation parameter; it is a generalized Heun's equation.  ...  [12] that, in the particular case β ′ = 2β, this operator can be factorized in the first order of the deformation parameter, and thus the KG equation can be obtained in this special case.  ... 
arXiv:1311.7405v1 fatcat:cxwi6avfbvfznnwj3erv2k3qre

Analytic Solutions of the Teukolsky Equation in Kerr-de Sitter and Kerr-Newman-de Sitter Geometries

Hisao Suzuki, Eiichi Takasugi, Hiroshi Umetsu
1999 Progress of theoretical physics  
The analytic solution of Teukolsky equation in Kerr-de Sitter and Kerr-Newman-de Sitter geometries is presented and the properties of the solution are examined.  ...  In particular, we show that our solution satisfies the Teukolsky-Starobinsky identities explicitly and fix the relative normalization between solutions with the spin weight s and -s.  ...  (A) Differential operators Since the solutions are expressed in terms of x or z, we rewrite the differential operator D † Q as D † Q = − r + − r − (r + − r − )(r − − r − ) (−x) −a 1 (1 − x) −a 2 x − x  ... 
doi:10.1143/ptp.102.253 fatcat:xxoxgjc4xjeizb2bmc7p4v7jsy

Solving Heun's equation using conformal blocks [article]

Marcin Piatek, Artur R. Pietrykowski
2018 arXiv   pre-print
Moreover, it is shown that the factorization property yields a practical method of computation of the Floquet type Heun's solutions.  ...  In the present work the semi-classical heavy-light factorization of the 5-point degenerate conformal blocks is studied.  ...  we will look once again into depths of eq. ( 2 .22) in order to explicitly compute the limit (2.25). 3 Heavy-light factorization and Floquet type Heun's solutions Path-multiplicative solutions from  ... 
arXiv:1708.06135v3 fatcat:4by5cm5u7jevlekpvxoe6tmnou

Higher length-twist coordinates, generalized Heun's opers, and twisted superpotentials [article]

Lotte Hollands, Omar Kidwai
2017 arXiv   pre-print
Second, we give an explicit parametrization of the locus of opers and determine the generating functions of this Lagrangian subvariety in terms of the higher rank Darboux coordinates in some specific examples  ...  Last, we relate the approach of Nekrasov, Rosly and Shatashvili to the approach using quantum periods via the exact WKB method.  ...  We define the rescaled Heun's differential equation by substituting z = qt in Heun's differential equation itself.  ... 
arXiv:1710.04438v1 fatcat:m7joi5lh65apxg55y4o2t2f2l4
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