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HeunB

The Heun Biconfluent function

HeunBPrime

The derivative of the Heun Biconfluent function

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

HeunB(, , , , )

HeunBPrime(, , , , )

Parameters

-

algebraic expression

-

algebraic expression

-

algebraic expression

-

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);

(1)
• 

The HeunB(, , , , 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;

(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(, , , , z) the third parameter satisfies , where  is a positive integer. In this case the th coefficient in the series expansion is a polynomial of degree  in . When  is a root of this polynomial, the th and subsequent coefficients cancel and the series truncates, resulting in a polynomial form of degree  for HeunB.

Examples

Heun's Biconfluent equation,

(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

(4)

where  are new variables,  and . Under this transformation, the HeunB parameters transform according to  -> ,  -> ,   ->  and  -> . These transformations form a group and imply on a number of identities, among which you have

(5)

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

(6)

When, in HeunB(,,,,z), , with  a positive integer, the th coefficient in the series expansion is a polynomial in  of order . If  is a root of that polynomial, that th 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

(7)

Considering the first non-trivial case, for , the function is

(8)

So the coefficient of  in the series expansion is

(9)

(10)

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

(11)

(12)

substituting in  we have

(13)

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

(14)

(15)

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.

See Also

FunctionAdvisor

Heun

HeunC

HeunD

HeunG

HeunT

hypergeom

 


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