§31.16 Mathematical Applications

§31.16(i) Uniformization Problem for Heun’s Equation

The main part of Smirnov (1996) consists of V. I. Smirnov’s 1918 M. Sc. thesis “Inversion problem for a second-order linear differential equation with four singular points”. It describes the monodromy group of Heun’s equation for specific values of the accessory parameter.

§31.16(ii) Heun Polynomial Products

Expansions of Heun polynomial products in terms of Jacobi polynomial (§18.3) 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 $\mathit{Hp}_{n,m}\left(x\right)\mathit{Hp}_{n,m}\left(y\right)=\sum_{j=0}^{n}A% _{j}{\sin^{2j}}\theta\*P^{(\gamma+\delta+2j-1,\epsilon-1)}_{n-j}\left(\cos% \left(2\theta\right)\right)P^{(\delta-1,\gamma-1)}_{j}\left(\cos\left(2\phi% \right)\right),$

where $n=0,1,\dots$, $m=0,1,\dots,n$, and $x,y$ are implicitly defined by

 31.16.2 $\displaystyle xy$ $\displaystyle=a{\sin^{2}}\theta{\cos^{2}}\phi,$ $\displaystyle(x-1)(y-1)$ $\displaystyle=(1-a){\sin^{2}}\theta{\sin^{2}}\phi,$ $\displaystyle(x-a)(y-a)$ $\displaystyle=a(a-1){\cos^{2}}\theta.$ ⓘ Defines: $\theta$: angle (locally), $\phi$: angle (locally), $x$: variable (locally) and $y$: variable (locally) Symbols: $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function and $a$: complex parameter Referenced by: §31.16(ii), §31.16(ii), Erratum Equations (31.16.2) and (31.16.3), Erratum Equations (31.16.2) and (31.16.3) Permalink: http://dlmf.nist.gov/31.16.E2 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png Correction (effective with 1.0.21): Originally $x,y$ were incorrectly defined by the set of equations given previously as “$x={\sin^{2}}\theta{\cos^{2}}\phi,$ $y={\sin^{2}}\theta{\sin^{2}}\phi$”. In fact, $x,y$ are implicitly defined by the corrected set of equations. See also: Annotations for §31.16(ii), §31.16 and Ch.31

The coefficients $A_{j}$ satisfy the relations:

 31.16.3 $A_{0}=\frac{n!}{{\left(\gamma+\delta\right)_{n}}}\mathit{Hp}_{n,m}\left(1% \right),\quad Q_{0}A_{0}+R_{0}A_{1}=0,$
 31.16.4 $P_{j}A_{j-1}+Q_{j}A_{j}+R_{j}A_{j+1}=0,$ $j=1,2,\dots,n$,

where

 31.16.5 $\displaystyle P_{j}$ $\displaystyle=\frac{(\epsilon-j+n)j(\beta+j-1)(\gamma+\delta+j-2)}{(\gamma+% \delta+2j-3)(\gamma+\delta+2j-2)},$ ⓘ Defines: $P_{j}$ (locally) Symbols: $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $\epsilon$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer and $\beta$: real or complex parameter Permalink: http://dlmf.nist.gov/31.16.E5 Encodings: TeX, pMML, png See also: Annotations for §31.16(ii), §31.16 and Ch.31 31.16.6 $\displaystyle Q_{j}$ $\displaystyle=-aj(j+\gamma+\delta-1)-q+\frac{(j-n)(j+\beta)(j+\gamma)(j+\gamma% +\delta-1)}{(2j+\gamma+\delta)(2j+\gamma+\delta-1)}+\frac{(j+n+\gamma+\delta-1% )j(j+\delta-1)(j-\beta+\gamma+\delta-1)}{(2j+\gamma+\delta-1)(2j+\gamma+\delta% -2)},$ ⓘ Defines: $Q_{j}$ (locally) Symbols: $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $\beta$: real or complex parameter Permalink: http://dlmf.nist.gov/31.16.E6 Encodings: TeX, pMML, png See also: Annotations for §31.16(ii), §31.16 and Ch.31 31.16.7 $\displaystyle R_{j}$ $\displaystyle=\frac{(n-j)(j+n+\gamma+\delta)(j+\gamma)(j+\delta)}{(\gamma+% \delta+2j)(\gamma+\delta+2j+1)}.$ ⓘ Defines: $R_{j}$ (locally) Symbols: $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $j$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/31.16.E7 Encodings: TeX, pMML, png See also: Annotations for §31.16(ii), §31.16 and Ch.31

By specifying either $\theta$ or $\phi$ in (31.16.1) and (31.16.2) we obtain expansions in terms of one variable.