# §31.12 Confluent Forms of Heun’s Equation

Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity. This is analogous to the derivation of the confluent hypergeometric equation from the hypergeometric equation in §13.2(i). There are four standard forms, as follows:

## Confluent Heun Equation

 31.12.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\gamma}{z}+\frac{% \delta}{z-1}+\epsilon\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{% z(z-1)}w=0.$

This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\infty$.

Mathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions (§30.12) are special cases of solutions of the confluent Heun equation.

## Doubly-Confluent Heun Equation

 31.12.2 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\delta}{z^{2}}+\frac{% \gamma}{z}+1\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{z^{2}}w=0.$

This has irregular singularities at $z=0$ and $\infty$, each of rank $1$.

## Biconfluent Heun Equation

 31.12.3 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-\left(\frac{\gamma}{z}+\delta+z% \right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{z}w=0.$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter Referenced by: Erratum (V1.0.7) for Equation (31.12.3) Permalink: http://dlmf.nist.gov/31.12.E3 Encodings: TeX, pMML, png Errata (effective with 1.0.7): Originally the sign in front of the second term in this equation was $+$. The correct sign is $-$. Reported 2013-10-31 by Henryk Witek See also: Annotations for §31.12, §31.12 and Ch.31

This has a regular singularity at $z=0$, and an irregular singularity at $\infty$ of rank $2$.

## Triconfluent Heun Equation

 31.12.4 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\gamma+z\right)z\frac{% \mathrm{d}w}{\mathrm{d}z}+\left(\alpha z-q\right)w=0.$

This has one singularity, an irregular singularity of rank $3$ at $z=\infty$.

For properties of the solutions of (31.12.1)–(31.12.4), including connection formulas, see Bühring (1994), Ronveaux (1995, Parts B,C,D,E), Wolf (1998), Lay and Slavyanov (1998), and Slavyanov and Lay (2000).