# §31.5 Solutions Analytic at Three Singularities: Heun Polynomials

Let $\alpha=-n$, $n=0,1,2,\dots$, and $q_{n,m}$, $m=0,1,\dots,n$, be the eigenvalues of the tridiagonal matrix

 31.5.1 $\begin{bmatrix}0&a\gamma&0&\dots&0\\ P_{1}&-Q_{1}&R_{1}&\dots&0\\ 0&P_{2}&-Q_{2}&&\vdots\\ \vdots&\vdots&&\ddots&R_{n-1}\\ 0&0&\dots&P_{n}&-Q_{n}\end{bmatrix},$

where $P_{j},Q_{j},R_{j}$ are again defined as in §31.3(i). Then

 31.5.2 $\mathop{\mathit{Hp}_{n,m}\/}\nolimits\!\left(a,q_{n,m};-n,\beta,\gamma,\delta;% z\right)=\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q_{n,m};-n,\beta,\gamma% ,\delta;z\right)$ Defines: $\mathop{\mathit{Hp}_{\NVar{n},\NVar{m}}\/}\nolimits\!\left(\NVar{a},\NVar{q_{n% ,m}};\NVar{-n},\NVar{\beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$: Heun polynomials Symbols: $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(\NVar{a},\NVar{q};\NVar{\alpha},% \NVar{\beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $\beta$: real or complex parameter Permalink: http://dlmf.nist.gov/31.5.E2 Encodings: TeX, pMML, png See also: Annotations for 31.5

is a polynomial of degree $n$, and hence a solution of (31.2.1) that is analytic at all three finite singularities $0,1,a$. These solutions are the Heun polynomials. Some properties are included as special cases of properties given in §31.15 below.