# Heun polynomials

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##### 1: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
###### §31.5 Solutions Analytic at Three Singularities: HeunPolynomials
These solutions are the Heun polynomials. …
##### 2: 31.16 Mathematical Applications
###### §31.16(ii) HeunPolynomial Products
Expansions of Heun polynomial products in terms of Jacobi polynomial18.3) products are derived in Kalnins and Miller (1991a, b, 1993) from the viewpoint of interrelation between two bases in a Hilbert space:
31.16.1 $\mathit{Hp}_{n,m}\left(x\right)\mathit{Hp}_{n,m}\left(y\right)=\sum_{j=0}^{n}A% _{j}{\sin}^{2j}\theta\*P^{(\gamma+\delta+2j-1,\epsilon-1)}_{n-j}\left(\cos% \left(2\theta\right)\right)P^{(\delta-1,\gamma-1)}_{j}\left(\cos\left(2\phi% \right)\right),$
31.16.3 $A_{0}=\frac{n!}{{\left(\gamma+\delta\right)_{n}}}\mathit{Hp}_{n,m}\left(1% \right),\quad Q_{0}A_{0}+R_{0}A_{1}=0,$
##### 3: 31.1 Special Notation
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.
##### 4: 31.9 Orthogonality
The right-hand side may be evaluated at any convenient value, or limiting value, of $\zeta$ in $(0,1)$ since it is independent of $\zeta$. For corresponding orthogonality relations for Heun functions (§31.4) and Heun polynomials31.5), see Lambe and Ward (1934), Erdélyi (1944), Sleeman (1966a), and Ronveaux (1995, Part A, pp. 59–64).
###### §31.9(ii) Double Orthogonality
Heun polynomials $w_{j}=\mathit{Hp}_{n_{j},m_{j}}$, $j=1,2$, satisfy …
##### 5: 31.11 Expansions in Series of Hypergeometric Functions
Series of Type II (§31.11(iv)) are expansions in orthogonal polynomials, which are useful in calculations of normalization integrals for Heun functions; see Erdélyi (1944) and §31.9(i). … The case $\alpha=-n$ for nonnegative integer $n$ corresponds to the Heun polynomial $\mathit{Hp}_{n,m}\left(z\right)$. …
$\mu=\gamma+\delta-2.$
##### 6: 31.10 Integral Equations and Representations
For integral equations satisfied by the Heun polynomial $\mathit{Hp}_{n,m}\left(z\right)$ we have $\sigma=\frac{1}{2}-\delta-j$, $j=0,1,\dots,n$. For suitable choices of the branches of the $P$-symbols in (31.10.9) and the contour $C$, we can obtain both integral equations satisfied by Heun functions, as well as the integral representations of a distinct solution of Heun’s equation in terms of a Heun function (polynomial, path-multiplicative solution). …
##### 7: Gerhard Wolf
Wolf has published papers on Mathieu functions, orthogonal polynomials, and Heun functions. …
##### 8: 31.8 Solutions via Quadratures
Here $\Psi_{g,N}(\lambda,z)$ is a polynomial of degree $g$ in $\lambda$ and of degree $N=m_{0}+m_{1}+m_{2}+m_{3}$ in $z$, that is a solution of the third-order differential equation satisfied by a product of any two solutions of Heun’s equation. … When $\lambda=-4q$ approaches the ends of the gaps, the solution (31.8.2) becomes the corresponding Heun polynomial. …
##### 9: Bibliography K
• E. G. Kalnins and W. Miller (1991a) Hypergeometric expansions of Heun polynomials. SIAM J. Math. Anal. 22 (5), pp. 1450–1459.
• E. G. Kalnins and W. Miller (1991b) Addendum: “Hypergeometric expansions of Heun polynomials. SIAM J. Math. Anal. 22 (6), pp. 1803.
• E. G. Kalnins and W. Miller (1993) Orthogonal Polynomials on $n$-spheres: Gegenbauer, Jacobi and Heun. In Topics in Polynomials of One and Several Variables and their Applications, pp. 299–322.
• ##### 10: Bibliography P
• J. Patera and P. Winternitz (1973) A new basis for the representation of the rotation group. Lamé and Heun polynomials. J. Mathematical Phys. 14 (8), pp. 1130–1139.