31 Heun Functions Properties 31.3 Basic Solutions 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

§31.4 Solutions Analytic at Two Singularities: Heun Functions

ⓘ

Defines:: $(\NVar{s_{1}},\NVar{s_{2}})\mathit{Hf}_{\NVar{m}}\left(\NVar{a},\NVar{q_{m}};% \NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions
Keywords:: Heun functions, Heun’s equation, definition, eigenvalues of accessory parameter
Notes:: See Erdélyi et al. (1955, Chapter XV) and Arscott (1964b, Chapter IX).
Referenced by:: §31.11(iii), §31.11(iv), §31.9(i)
Permalink:: http://dlmf.nist.gov/31.4
See also:: Annotations for Ch.31

For an infinite set of discrete values $q_{m}$ , $m=0,1,2,\dots$ , of the accessory parameter $q$ , the function $\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ 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		$(0,1)\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),$
$m=0,1,2,\dots$ .
ⓘ Symbols: $(\NVar{s_{1}},\NVar{s_{2}})\mathit{Hf}_{\NVar{m}}\left(\NVar{a},\NVar{q_{m}};% \NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : 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 See also: Annotations for §31.4 and Ch.31

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 See also: Annotations for §31.4 and Ch.31

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

More generally,

31.4.3		$(s_{1},s_{2})\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),$
$m=0,1,2,\dots$ ,
ⓘ Symbols: $(\NVar{s_{1}},\NVar{s_{2}})\mathit{Hf}_{\NVar{m}}\left(\NVar{a},\NVar{q_{m}};% \NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : 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 See also: Annotations for §31.4 and Ch.31

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).

31.3 Basic Solutions 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

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