The Heun Biconfluent function - Maple Help

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HeunB - The Heun Biconfluent function

HeunBPrime - The derivative of the Heun Biconfluent function

Calling Sequence

HeunB(α, β, γ, δ, z)

HeunBPrime(α, β, γ, δ, z)

Parameters

α

-

algebraic expression

β

-

algebraic expression

γ

-

algebraic expression

δ

-

algebraic expression

z

-

algebraic expression

Description

• 

The HeunB function is the solution of the Heun Biconfluent equation. Following the first reference (at the end), the equation and the conditions at the origin satisfied by HeunB are

FunctionAdvisor(definition, HeunB);

HeunBα,β,γ,δ,z=DESolⅆ2ⅆz2_Yzβz+2z2α1ⅆⅆz_Yzz122α2γ+4z+βα+β+δ_Yzz,_Yz,_Y0=1,D_Y0=αβ+β+δ2α+2

(1)
• 

The HeunB(α, β, γ, δ, z) function is a local (Frobenius) solution to Heun's Biconfluent equation, computed as a power series expansion around the origin, a regular singular point. Because the next singularity is located at , this series converges in the whole complex plane.

• 

The Biconfluent Heun Equation (BHE) above is obtained from the Confluent Heun Equation (CHE) through a confluence process, that is, a process where two singularities coalesce, performed by redefining parameters and taking limits. In this case one regular singularity of the CHE is coalesced with its irregular singularity at . The resulting Heun Biconfluent equation, thus, has one regular singularity at the origin, one irregular one at , and includes as a particular case the 1F1 hypergeometric confluent equation

DEtools[hyperode]( hypergeom([a],[c],z), y(z) ) = 0;

ayz+c+zⅆⅆzyzzⅆ2ⅆz2yz=0

(2)
  

So besides the standard hypergeometric solution of this equation, a solution expressed in terms of HeunB functions can also be constructed, and in this way HeunB contains as particular cases all the hypergeometric functions of the 1F1 class. Some of these specializations are listed at the end of the Examples section.

• 

A special case happens when in HeunB(α, β, γ, δ, z) the third parameter satisfies γ=21+n+α, where n is a positive integer. In this case the n+1th coefficient in the series expansion is a polynomial of degree n in δ. When δ is a root of this polynomial, the n+1th and subsequent coefficients cancel and the series truncates, resulting in a polynomial form of degree n for HeunB.

Examples

Heun's Biconfluent equation,

BHE:=ⅆ2ⅆz2yz=2z21α+βzⅆⅆzyzz+12γ+2α+4z+δ+β+βαyz2z

BHE:=ⅆ2ⅆz2yz=βz+2z2α1ⅆⅆzyzz+122γ+2α+4z+δ+β+βαyzz

(3)

can be transformed into another version of itself, that is, an equation with one regular and one irregular singularity respectively located at 0 and  through transformations of the form

z=κt,yz=zε1α2ⅇz+β1κ2z2uz

z=κt,yz=z12ε1αⅇ12z+βκ2+1zuz

(4)

where t,ut are new variables, ϵ2=1 and κ2=1. Under this transformation, the HeunB parameters transform according to α -> ϵα, β -> κ3β,  γ -> κ2γ and δ -> κδ. These transformations form a group and imply on a number of identities, among which you have

FunctionAdvisoridentities,HeunB

HeunBα,β,γ,δ,z=zαHeunBα,β,γ,δ,z,Andα::negint,z0,HeunBα,β,γ,δ,z=Iαⅇβz+z2HeunBα,Iβ,γ,Iδ,Iz,Andα::negint,z0,HeunBα,β,γ,δ,z=1αHeunBα,β,γ,δ,z,Andα::negint,z0,HeunBα,β,γ,δ,z=ⅇβz+z2HeunBα,Iβ,γ,Iδ,Iz,Andα::Notnegint,HeunBα,β,γ,δ,z=HeunBα,β,γ,δ,z,Andα::Notnegint

(5)

A relation between HeunB and the confluent 1F1 hypergeometric function is

FunctionAdvisorspecialize,HeunB,hypergeom

HeunBα,β,γ,δ,z=hypergeom1214α14γ,112α,z2zα,Andα::negint,β=0,δ=0,HeunBα,β,γ,δ,z=hypergeom12+14α14γ,1+12α,z2,Andα::Notnegint,β=0,δ=0

(6)

When, in HeunB(α,β,γ,δ,z), γ=21+n+α, with n a positive integer, the n+1th coefficient in the series expansion is a polynomial in δ of order n. If δ is a root of that polynomial, that n+1th coefficient and the subsequent ones are zero. The series then truncates and HeunB reduces to a polynomial. For example, this is the necessary condition for a polynomial form

HeunBα,β,2n+2+α,δ,z

HeunBα,β,2n+2+α,δ,z

(7)

Considering the first non-trivial case, for n=1, the function is

HB:=subsn=1,

HB:=HeunBα,β,4+α,δ,z

(8)

So the coefficient of z2 in the series expansion is

Q:=simplifyseriesHB,z,3,size

Q:=1+αβ+β+δ2α+2z+18α2β2+4β2+2βδ8α+3β2+4βδ+δ28α+1α+2z2+Oz3

(9)

c2:=coeffQ,z,2

c2:=18α2β2+4β2+2βδ8α+3β2+4βδ+δ28α+1α+2

(10)

solving for δ, requesting from solve to return using RootOf, you have

_EnvExplicit:=false

_EnvExplicit:=false

(11)

δ=solvec2,δ

δ=RootOf_Z2+2αβ+4β_Z+α2β2+4αβ2+3β28α8

(12)

substituting in HB we have

HB_polynomial:=subs,HB

HB_polynomial:=HeunBα,β,4+α,RootOf_Z2+2αβ+4β_Z+α2β2+4αβ2+3β28α8,z

(13)

When the function admits a polynomial form, as is the case of HB_polynomial by construction, to obtain the actual polynomial of degree n (in this case n=1) use

HeunB=HeunB:-SpecialValues:-Polynomial|HeunB=HeunB:-SpecialValues:-Polynomial

1+RootOf_Z2+2αβ+4β_Z+α2β2+4αβ2+3β28α8+βα+βz2α+2

(14)

_EnvExplicit:='_EnvExplicit'

_EnvExplicit:=_EnvExplicit

(15)

See Also

FunctionAdvisor, Heun, HeunC, HeunD, HeunG, HeunT, hypergeom

References

  

Decarreau, A.; Dumont-Lepage, M.C.; Maroni, P.; Robert, A.; and Ronveaux, A. "Formes Canoniques de Equations confluentes de l'equation de Heun." Annales de la Societe Scientifique de Bruxelles. Vol. 92 I-II, (1978): 53-78.

  

Ronveaux, A. ed. Heun's Differential Equations. Oxford University Press, 1995.

  

Slavyanov, S.Y., and Lay, W. Special Functions, A Unified Theory Based on Singularities. Oxford Mathematical Monographs, 2000.


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