31 Heun Functions Properties 31.4 Solutions Analytic at Two Singularities: Heun Functions 31.6 Path-Multiplicative Solutions

§31.5 Solutions Analytic at Three Singularities: Heun Polynomials

ⓘ

Keywords:: Heun polynomials, definitions, notation
Notes:: See e.g. Erdélyi et al. (1955, Chapter XV) and Arscott (1964b, Chapter IX).
Referenced by:: §31.9(i)
Permalink:: http://dlmf.nist.gov/31.5
See also:: Annotations for Ch.31

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 See also: Annotations for §31.5 and Ch.31

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

31.5.2		$\mathit{Hp}_{n,m}\left(a,q_{n,m};-n,\beta,\gamma,\delta;z\right)=\mathit{H\!% \ell}\left(a,q_{n,m};-n,\beta,\gamma,\delta;z\right)$
ⓘ Defines: $\mathit{Hp}_{\NVar{n},\NVar{m}}\left(\NVar{a},\NVar{q_{n,m}};\NVar{-n},\NVar{% \beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun polynomials Symbols: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : 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 and Ch.31

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.

31.4 Solutions Analytic at Two Singularities: Heun Functions 31.6 Path-Multiplicative Solutions

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