The Heun Triconfluent function
The derivative of the Heun Triconfluent function
HeunT(α, β, γ, z)
HeunTPrime(α, β, γ, z)
The HeunT function is the solution of the Heun Triconfluent equation. Following the first reference (at the end), the equation and the conditions at the origin satisfied by HeunT are
The HeunT(α,β,γ,z) function is a local solution to Heun's Triconfluent equation, computed as a standard power series expansion around the origin, a regular point. Because the single singularity is located at ∞, this series converges in the whole complex plane.
The Triconfluent Heun Equation (THE) above is obtained from the Doubleconfluent Heun Equation (DHE) through a confluence process, that is, a process where two singularities coalesce, performed by redefining parameters and taking limits. In this case the two irregular singularities of the DHE are coalesced into one irregular singularity at ∞. The resulting Heun Triconfluent equation, thus, has the structure of singularities f of the 0F1 hypergeometric equation and so can be related to the Airy functions.
A special case happens when in HeunT(α,β,γ,z), the second parameter satisfies β=3⁡n+1, where n is a positive integer. In this case the nth+1, nth+2 and nth+3 coefficients form a polynomial system for the remaining parameters α and γ; when this system is identically satisfied all the subsequent coefficients cancel too and the series truncates, resulting in a polynomial form of degree n for HeunT. Remark: for n=0 this situation leads to a constant, for n=1 HeunT will also be a constant since its series expansion satisfies HeunT⁢' at 0 = 0 and for n=2 the polynomial system for α and γ is inconsistent. So the non-trivial polynomial forms of HeunT are of degree 3≤n.
Heun's Triconfluent equation,
THE ≔ ⅆ2ⅆz2⁢y⁡z=3⁢z2+gamma⁢ⅆⅆz⁢y⁡z+3−β⁢z−α⁢y⁡z
can be transformed into another version of itself, that is, an equation with one regular and one irregular singularities respectively located at 0 and ∞ through transformations of the form
where t,u⁡t are new variables and κ6=1. Under this transformation, the HeunT parameters transform according to α -> ακ2, β -> βκ3, γ -> γκ4. These transformations form a group of six elements and imply on identities, among which you have
When, in HeunT(α,β,γ,z), β=3⁡n+1, where n is a positive integer, the nth+1, nth+2 and nth+3 coefficients form a polynomial system for the remaining parameters α and γ. When this system is identically satisfied all the subsequent coefficients cancel too and the series truncates, resulting in a polynomial form of degree n for HeunT. For example, this is the necessary condition for a polynomial form
Considering the first non-trivial case, for n=3, the function is
HT ≔ subs⁡n=3,
So the coefficients of zm for m equal to 4, 5, and 6 in the series expansion are
Q ≔ simplify⁡series⁡HT,z,7,size
c4,c5,c6 ≔ coeff⁡Q,z,4,coeff⁡Q,z,5,coeff⁡Q,z,6
solving for α and γ, requesting from solve to return using RootOf, you have
_EnvExplicit ≔ false
substituting for instance the first of these two solutions in HT we have
HT_polynomial ≔ subs⁡1,HT
When the function admits a polynomial form, as is the case of HT_polynomial by construction, to obtain the actual polynomial of degree n (in this case n=3) use
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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