§31.5 Solutions Analytic at Three Singularities: Heun Polynomials

Notes:: See e.g. Erdélyi et al. (1955, Chapter XV) and Arscott (1964b, Chapter IX).
Keywords:: Heun polynomials
Referenced by:: §31.9(i)
Permalink:: http://dlmf.nist.gov/31.5

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

Symbols:: $\gamma$ : real or complex parameter, $n$ : nonnegative integer, $a$ : complex parameter, $P_{j}$ : coefficient, $Q_{j}$ : coefficient and $R_{j}$ : coefficient
Permalink:: http://dlmf.nist.gov/31.5.E1
Encodings:: TeX, pMML, png

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:: $\mathop{\mathit{Hp}_{{n,m}}\/}\nolimits\!\left(a,q_{{n,m}};-n,\beta,\gamma,\delta;z\right)$ : Heun polynomials
Symbols:: $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ : 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

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.