§31.16 Mathematical Applications

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.

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		$${\mathit{Hp}}_{n,m}\left(x\right){\mathit{Hp}}_{n,m}\left(y\right)=\sum _{j=0}^n\begin{array}{l}{A}_j{\sin }^{2j}\theta P_{n-j}^{(\gamma +\delta +2j-1,ϵ-1)}\left(\cos \left(2\theta \right)\right)\\ \times P_j^{(\delta -1,\gamma -1)}\left(\cos \left(2\varphi \right)\right),\end{array}$$
ⓘ Symbols: ${\mathit{Hp}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\cos z$: cosine function, $\sin z$: sine function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $j$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\beta$: real or complex parameter, $\theta$: angle, $\varphi$: angle, $x$: variable, $y$: variable and $A_j$: coefficients Referenced by: §31.16(ii), §31.16(ii) Permalink: http://dlmf.nist.gov/31.16.E1 Encodings: TeX, pMML, png See also: Annotations for §31.16(ii), §31.16 and Ch.31

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

31.16.2		$xy$	$=a{\sin }^2\theta {\cos }^2\varphi ,$
		$(x-1)(y-1)$	$=(1-a){\sin }^2\theta {\sin }^2\varphi ,$
		$(x-a)(y-a)$	$=a(a-1){\cos }^2\theta .$
ⓘ Defines: $\theta$: angle (locally), $\varphi$: angle (locally), $x$: variable (locally) and $y$: variable (locally) Symbols: $\cos z$: cosine function, $\sin z$: sine function and $a$: complex parameter Referenced by: §31.16(ii), §31.16(ii), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3), Erratum (V1.0.21) for 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 “$x={\sin }^2\theta {\cos }^2\varphi ,$ $y={\sin }^2\theta {\sin }^2\varphi$”. 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		$$A_0=\frac{n!}{{\left(\gamma +\delta \right)}_n}{\mathit{Hp}}_{n,m}\left(1\right),Q_0{A}_0+R_0{A}_1=0,$$
ⓘ Symbols: ${\mathit{Hp}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\beta$: real or complex parameter, $A_j$: coefficients, $Q_j$ and $R_j$ Referenced by: §31.16(ii), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3) Permalink: http://dlmf.nist.gov/31.16.E3 Encodings: TeX, pMML, png Correction (effective with 1.0.21): The initial data $A_0$, previously missing, has now been included. See also: Annotations for §31.16(ii), §31.16 and Ch.31

31.16.4		$$P_j{A}_{j-1}+Q_j{A}_j+R_j{A}_{j+1}=0,$$
$j=1,2,\mathrm{\dots },n$,
ⓘ Symbols: $j$: nonnegative integer, $n$: nonnegative integer, $A_j$: coefficients, $P_j$, $Q_j$ and $R_j$ Permalink: http://dlmf.nist.gov/31.16.E4 Encodings: TeX, pMML, png See also: Annotations for §31.16(ii), §31.16 and Ch.31

where

31.16.5		$P_j$	$={\displaystyle \frac{(ϵ-j+n)j(\beta +j-1)(\gamma +\delta +j-2)}{(\gamma +\delta +2j-3)(\gamma +\delta +2j-2)}},$
ⓘ Defines: $P_j$ (locally) Symbols: $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: 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		$Q_j$	$=\begin{array}{l}-aj(j+\gamma +\delta -1)-q+{\displaystyle \frac{(j-n)(j+\beta )(j+\gamma )(j+\gamma +\delta -1)}{(2j+\gamma +\delta )(2j+\gamma +\delta -1)}}\\ +{\displaystyle \frac{(j+n+\gamma +\delta -1)j(j+\delta -1)(j-\beta +\gamma +\delta -1)}{(2j+\gamma +\delta -1)(2j+\gamma +\delta -2)}},\end{array}$
ⓘ 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		$R_j$	$={\displaystyle \frac{(n-j)(j+n+\gamma +\delta )(j+\gamma )(j+\delta )}{(\gamma +\delta +2j)(\gamma +\delta +2j+1)}}.$
ⓘ 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 $\varphi$ in (31.16.1) and (31.16.2) we obtain expansions in terms of one variable.