§31.4 Solutions Analytic at Two Singularities: Heun Functions

Notes:: See Erdélyi et al. (1955, Chapter XV) and Arscott (1964b, Chapter IX).
Defines:: $\mathop{(s_{1},s_{2})\mathit{Hf}_{{m}}\/}\nolimits\!\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right)$ : Heun functions
Keywords:: Heun functions, Heun’s equation
Referenced by:: §31.11(iii), §31.9(i)
Permalink:: http://dlmf.nist.gov/31.4

For an infinite set of discrete values $q_{m}$ , $m=0,1,2,\dots$ , of the accessory parameter $q$ , the function $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ is analytic at $z=1$ , and hence also throughout the disk $|z|<a$ . To emphasize this property this set of functions is denoted by

31.4.1 $\mathop{(0,1)\mathit{Hf}_{{m}}\/}\nolimits\!\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),$ $m=0,1,2,\dots$ .

Symbols:: $\mathop{(s_{1},s_{2})\mathit{Hf}_{{m}}\/}\nolimits\!\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Referenced by:: §31.6
Permalink:: http://dlmf.nist.gov/31.4.E1
Encodings:: TeX, pMML, png

The eigenvalues $q_{m}$ satisfy the continued-fraction equation

31.4.2 $q=\cfrac{a\gamma P_{1}}{Q_{1}+q-\cfrac{R_{1}P_{2}}{Q_{2}+q-\cfrac{R_{2}P_{3}}{Q_{3}+q-\cdots}}},$

Symbols:: $\gamma$ : real or complex parameter, $a$ : complex parameter, $q$ : real or complex parameter, $P_{j}$ : coefficient, $Q_{j}$ : coefficient and $R_{j}$ : coefficient
Referenced by:: §31.18
Permalink:: http://dlmf.nist.gov/31.4.E2
Encodings:: TeX, pMML, png

in which $P_{j},Q_{j},R_{j}$ are as in §31.3(i).

More generally,

31.4.3 $\mathop{(s_{1},s_{2})\mathit{Hf}_{{m}}\/}\nolimits\!\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),$ $m=0,1,2,\dots$ ,

Symbols:: $\mathop{(s_{1},s_{2})\mathit{Hf}_{{m}}\/}\nolimits\!\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Referenced by:: §31.4, §31.6
Permalink:: http://dlmf.nist.gov/31.4.E3
Encodings:: TeX, pMML, png

with $(s_{1},s_{2})\in\{ 0,1,a,\infty\}$ , denotes a set of solutions of (31.2.1), each of which is analytic at $s_{1}$ and $s_{2}$ . The set $q_{m}$ depends on the choice of $s_{1}$ and $s_{2}$ .

The solutions (31.4.3) are called the Heun functions. See Ronveaux (1995, pp. 39–41).