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

FunctionAdvisor(definition, HeunC);

This Heun (singly) Confluent equation is obtained from the Heun General equation through a confluence process, that is, a process where two singularities coalesce, performed by redefining parameters and taking limits, resulting in a single (typically irregular) singularity. The Heun Confluent equation thus has two regular singularities and one irregular singularity, and includes as particular cases both the 2F1 and 1F1 hypergeometric equations. The solution to the 2F1 equation,

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

can then be expressed in terms of HeunC functions

dsolve((2), [HeunC]);

and the same for the 1F1 hypergeometric confluent equation

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

dsolve((4), [HeunC]);

HeunC, thus, contains as particular cases all the hypergeometric functions of the 2F1 and 1F1 classes - some of these specializations are listed at the end of the Examples section.

Two other important non-hypergeometric case of Heun's Confluent equation, are the "spheroidal wave function" equation

diff(y(z),z,z) + 2*(gamma+1)*z*diff(y(z),z)/(z^2-1) + (4*delta*z^2-c)/(z^2-1)*y(z) = 0;

obtained from Heun's Confluent equation taking  and changing  -> ;

and the rational form of Mathieu's equation,

diff(y(z),z,z) + z/(z^2-1)*diff(y(z),z) + (2*delta*(2*z^2-1)-a)/(z^2-1)*y(z) = 0;

obtained from the spheroidal wave function equation above by taking  and further specializing .

The HeunC(,,,,, z) function is a local (Frobenius) solution to Heun's Confluent equation, computed as a power series expansion around the origin, a regular singular point. The series converges for , where the second regular singularity is located. An analytic continuation of HeunC is obtained through identities, relating the values of the function in different regions of the  plane, for given values of the other parameters, or by expanding the solution around 1, the other regular singularity, and overlapping the series. General formulas relating these series expansions at different singularities and for arbitrary values of the other parameters, however, are not known at present.

A special case happens when the parameters entering HeunC are such that the function is, simultaneously, a Frobenius solution around the two regular singularities and hence analytic in a domain containing both of them. In such a case the series expansion for HeunC truncates and the function becomes a polynomial. A necessary (not sufficient) condition for this case is that , with  a positive integer, and  has one of a finite number of characteristic values, so that the function is a polynomial of degree .

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.