# Heun functions

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##### 1: 31.1 Special Notation
 $x$, $y$ real variables. …
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 $\mathit{Hp}_{n,m}\left(a,q_{n,m};-n,\beta,\gamma,\delta;z\right)$. …Sometimes the parameters are suppressed.
##### 2: 31.4 Solutions Analytic at Two Singularities: Heun Functions
###### §31.4 Solutions Analytic at Two Singularities: HeunFunctions
To emphasize this property this set of functions is denoted by
31.4.1 $(0,1)\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),$ $m=0,1,2,\dots$.
31.4.3 $(s_{1},s_{2})\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),$ $m=0,1,2,\dots$,
The solutions (31.4.3) are called the Heun functions. …
##### 3: 31.6 Path-Multiplicative Solutions
A further extension of the notation (31.4.1) and (31.4.3) is given by
31.6.1 $(s_{1},s_{2})\mathit{Hf}_{m}^{\nu}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z% \right),$ $m=0,1,2,\dots$,
##### 5: 31.9 Orthogonality
###### §31.9(i) Single Orthogonality
31.9.1 $w_{m}(z)=(0,1)\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),$
31.9.2 $\int_{\zeta}^{(1+,0+,1-,0-)}t^{\gamma-1}(1-t)^{\delta-1}(t-a)^{\epsilon-1}\*w_% {m}(t)w_{k}(t)\,\mathrm{d}t=\delta_{m,k}\theta_{m}.$
For corresponding orthogonality relations for Heun functions31.4) and Heun polynomials (§31.5), see Lambe and Ward (1934), Erdélyi (1944), Sleeman (1966a), and Ronveaux (1995, Part A, pp. 59–64).
##### 6: 31.18 Methods of Computation
###### §31.18 Methods of Computation
The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (31.4.2) and (31.11.13) in the same way as for the Mathieu, Lamé, and spheroidal wave functions in Chapters 2830.
##### 7: 31.17 Physical Applications
###### §31.17(ii) Other Applications
Heun functions appear in the theory of black holes (Kerr (1963), Teukolsky (1972), Chandrasekhar (1984), Suzuki et al. (1998), Kalnins et al. (2000)), lattice systems in statistical mechanics (Joyce (1973, 1994)), dislocation theory (Lay and Slavyanov (1999)), and solution of the Schrödinger equation of quantum mechanics (Bay et al. (1997), Tolstikhin and Matsuzawa (2001), and Hall et al. (2010)). … More applications—including those of generalized spheroidal wave functions and confluent Heun functions in mathematical physics, astrophysics, and the two-center problem in molecular quantum mechanics—can be found in Leaver (1986) and Slavyanov and Lay (2000, Chapter 4). …
##### 8: 31.10 Integral Equations and Representations
###### Kernel Functions
Then the integral equation (31.10.1) is satisfied by $w(z)=w_{m}(z)$ and $W(z)=\kappa_{m}w_{m}(z)$, where $w_{m}(z)=(0,1)\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right)$ and $\kappa_{m}$ is the corresponding eigenvalue. … Fuchs–Frobenius solutions $W_{m}(z)=\tilde{\kappa}_{m}z^{-\alpha}\mathit{H\!\ell}\left(1/a,q_{m};\alpha,% \alpha-\gamma+1,\alpha-\beta+1,\delta;1/z\right)$ are represented in terms of Heun functions $w_{m}(z)=(0,1)\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right)$ by (31.10.1) with $W(z)=W_{m}(z)$, $w(z)=w_{m}(z)$, and with kernel chosen from …
###### Kernel Functions
For integral equations for special confluent Heun functions31.12) see Kazakov and Slavyanov (1996). …
##### 10: 31.7 Relations to Other Functions
###### §31.7(i) Reductions to the Gauss Hypergeometric Function
31.7.2 $\mathit{H\!\ell}\left(2,\alpha\beta;\alpha,\beta,\gamma,\alpha+\beta-2\gamma+1% ;z\right)={{}_{2}F_{1}}\left(\tfrac{1}{2}\alpha,\tfrac{1}{2}\beta;\gamma;1-(1-% z)^{2}\right),$
31.7.3 $\mathit{H\!\ell}\left(4,\alpha\beta;\alpha,\beta,\tfrac{1}{2},\tfrac{2}{3}(% \alpha+\beta);z\right)={{}_{2}F_{1}}\left(\tfrac{1}{3}\alpha,\tfrac{1}{3}\beta% ;\tfrac{1}{2};1-(1-z)^{2}(1-\tfrac{1}{4}z)\right),$
31.7.4 $\mathit{H\!\ell}\left(\tfrac{1}{2}+i\tfrac{\sqrt{3}}{2},\alpha\beta(\tfrac{1}{% 2}+i\tfrac{\sqrt{3}}{6});\alpha,\beta,\tfrac{1}{3}(\alpha+\beta+1),\tfrac{1}{3% }(\alpha+\beta+1);z\right)={{}_{2}F_{1}}\left(\tfrac{1}{3}\alpha,\tfrac{1}{3}% \beta;\tfrac{1}{3}(\alpha+\beta+1);1-\left(1-\left(\tfrac{3}{2}-i\tfrac{\sqrt{% 3}}{2}\right)z\right)^{3}\right).$