The Heun Doubleconfluent function
The derivative of the Heun Doubleconfluent function
HeunD(α, β, γ, δ, z)
HeunDPrime(α, β, γ, δ, z)
The HeunD function is the solution of the Heun Doubleconfluent equation. Following the first reference (at the end), the equation and the conditions at the origin satisfied by HeunD are
The HeunD(α,β,γ,z) function is a local solution to Heun's Doubleconfluent equation, computed as a standard power series expansion around the origin, a regular point. Because of the presence of two irregular singularities located at -1 and 1, the radius of convergence of this series is z<1. An analytic continuation of HeunD outside the unit circle is obtained through the identity
The Doubleconfluent Heun Equation (DHE) above is obtained from the Confluent Heun Equation (CHE) through an additional confluence process, with the two regular singularities of the CHE coalescing into one irregular singularity at the origin. The resulting Heun equation, with two irregular singularities at 0 and ∞, is further transformed using x -> x+1x−1, relocating these singularities symmetrically at -1 and 1, leaving the origin as a regular point. The Doubleconfluent equation, thus, has a structure of singularities that can be transformed into that of the 0F1 hypergeometric equation and particular cases of HeunD are related to the Bessel functions.
Heun's Doubleconfluent equation,
DHE ≔ ⅆ2ⅆz2⁢y⁡z=−2⁢z5+4⁢z3+4⁢α⁢z4−2⁢z−4⁢α⁢ⅆⅆz⁢y⁡zz−13⁢z+13+−4⁢β−2⁢α2−4⁢δ−4⁢gamma+1⁢z2+8⁢β−α−gamma⁢z−4⁢gamma+4⁢δ−4⁢β+2⁢α2−1⁢y⁡zz−13⁢z+13
can be transformed into another version of itself, that is, an equation with two irregular singularities located at -1 and 1 through transformations of the form
where t,u⁡t are new variables, and σ4=1, κ2=σ⁢ε⁢α⁢κ4. Under this transformation, the HeunD parameters transform according to α = 2⁢κ−σ⁢ε⁢α4, β = ε⁢γ+β+δ16⁢σ, γ = −ε⁢γ+β+δ⁢σ16 and δ = β−δ−ε2⁢α242+14, where ε2=1.
These transformations form a group of 32 elements and imply on identities, among which you have
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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