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

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§31.5 Solutions Analytic at Three Singularities: Heun Polynomials

pFad - (p)hone/(F)rame/(a)nonymizer/(d)eclutterfier! Saves Data!