# relation to Heun equation

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##### 2: 29.2 Differential Equations
For the Weierstrass function $\wp$ see §23.2(ii). …
##### 3: 31.8 Solutions via Quadratures
For $\mathbf{m}=(m_{0},0,0,0)$, these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form. …
##### 4: Bibliography T
• O. I. Tolstikhin and M. Matsuzawa (2001) Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.
• ##### 5: Bibliography S
• S. Yu. Slavyanov and N. A. Veshev (1997) Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.
• ##### 6: 31.12 Confluent Forms of Heun’s Equation
This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\infty$. …
##### 7: 31.3 Basic Solutions
###### §31.3(i) Fuchs–Frobenius Solutions at $z=0$
$\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ denotes the solution of (31.2.1) that corresponds to the exponent $0$ at $z=0$ and assumes the value $1$ there. …
###### §31.3(iii) Equivalent Expressions
For example, $\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ is equal to
##### 8: 31.7 Relations to Other Functions
###### §31.7(i) Reductions to the Gauss Hypergeometric Function
Other reductions of $\mathit{H\!\ell}$ to a ${{}_{2}F_{1}}$, with at least one free parameter, exist iff the pair $(a,p)$ takes one of a finite number of values, where $q=\alpha\beta p$. …
###### §31.7(ii) Relationsto Lamé Functions
Similar specializations of formulas in §31.3(ii) yield solutions in the neighborhoods of the singularities $\zeta=K$, $K+i{K^{\prime}}$, and $i{K^{\prime}}$, where $K$ and ${K^{\prime}}$ are related to $k$ as in §19.2(ii).
##### 9: 31.17 Physical Applications
###### §31.17(ii) Other Applications
For applications of Heun’s equation and functions in astrophysics see Debosscher (1998) where different spectral problems for Heun’s equation are also considered. …
##### 10: Bibliography M
• R. S. Maier (2005) On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.
• R. S. Maier (2007) The 192 solutions of the Heun equation. Math. Comp. 76 (258), pp. 811–843.
• I. Marquette and C. Quesne (2016) Connection between quantum systems involving the fourth Painlevé transcendent and $k$-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. J. Math. Phys. 57 (5), pp. Paper 052101, 15 pp..
• M. Mazzocco (2001b) Picard and Chazy solutions to the Painlevé VI equation. Math. Ann. 321 (1), pp. 157–195.
• S. C. Milne (1985c) A new symmetry related to $\mathit{SU}(n)$ for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.