31 Heun Functions Notation 31 Heun Functions 31.2 Differential Equations

§31.1 Special Notation

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Keywords:: Heun functions, Heun’s equation, differential equation, notation, solutions analytic at three singularities, solutions analytic at two singularities
Permalink:: http://dlmf.nist.gov/31.1
See also:: Annotations for Ch.31

(For other notation see Notation for the Special Functions.)

$x$ , $y$	real variables.
$z$ , $\zeta$ , $w$ , $W$	complex variables.
$j$ , $k$ , $\ell$ , $m$ , $n$	nonnegative integers.
$a$	complex parameter, $\|a\|\geq 1,a\neq 1$ .
$q,\alpha,\beta,\gamma,\delta,\epsilon,\nu$	complex parameters.

The main functions treated in this chapter are $\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ , $(s_{1},s_{2})\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right)$ , $(s_{1},s_{2})\mathit{Hf}_{m}^{\nu}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right)$ , and the polynomial $\mathit{Hp}_{n,m}\left(a,q_{n,m};-n,\beta,\gamma,\delta;z\right)$ . These notations were introduced by Arscott in Ronveaux (1995, pp. 34–44). Sometimes the parameters are suppressed.

31 Heun Functions 31.2 Differential Equations

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