The Heun Biconfluent function
The derivative of the Heun Biconfluent function
HeunB(α, β, γ, δ, z)
HeunBPrime(α, β, γ, δ, z)
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
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;
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=2⁡n+1+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.
Heun's Biconfluent equation,
BHE ≔ ⅆ2ⅆz2⁢y⁡z=2⁢z2−1−α+β⁢z⁢ⅆⅆz⁢y⁡zz+1⁢−2⁢gamma+2⁢α+4⁢z+δ+β+β⁢α⁢y⁡z2⁢z
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
where t,u⁡t are new variables, ε2=1 and κ2=1. Under this transformation, the HeunB parameters transform according to α -> ε⁢α, β -> κ3⁢β, γ -> κ2⁢γ and δ -> κ⁢δ. These transformations form a group and imply on a number of identities, among which you have
A relation between HeunB and the confluent 1F1 hypergeometric function is
When, in HeunB(α,β,γ,δ,z), γ=2⁡n+1+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
Considering the first non-trivial case, for n=1, the function is
HB ≔ subs⁡n=1,
So the coefficient of z2 in the series expansion is
Q ≔ simplify⁡series⁡HB,z,3,size
c2 ≔ coeff⁡Q,z,2
solving for δ, requesting from solve to return using RootOf, you have
_EnvExplicit ≔ false
substituting in HB we have
HB_polynomial ≔ subs⁡,HB
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
_EnvExplicit ≔ '_EnvExplicit'
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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