# §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 $\mathop{\mathit{Hp}_{n,m}\/}\nolimits\!\left(x\right)\mathop{\mathit{Hp}_{n,m}% \/}\nolimits\!\left(y\right)=\sum_{j=0}^{n}A_{j}{\mathop{\sin\/}\nolimits^{2j}% }\theta\*P_{n-j}^{(\gamma+\delta+2j-1,\epsilon-1)}(\mathop{\cos\/}\nolimits 2% \theta)P_{j}^{(\delta-1,\gamma-1)}(\mathop{\cos\/}\nolimits 2\phi),$

where $n=0,1,\dots$, $m=0,1,\dots,n$, and

 31.16.2 $\displaystyle x$ $\displaystyle={\mathop{\sin\/}\nolimits^{2}}\theta{\mathop{\cos\/}\nolimits^{2% }}\phi,$ $\displaystyle y$ $\displaystyle={\mathop{\sin\/}\nolimits^{2}}\theta{\mathop{\sin\/}\nolimits^{2% }}\phi.$ Defines: $\theta$: angle (locally), $\phi$: angle (locally), $x$: variable (locally) and $y$: variable (locally) Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function and $\mathop{\sin\/}\nolimits\NVar{z}$: sine function Referenced by: §31.16(ii) Permalink: http://dlmf.nist.gov/31.16.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 31.16(ii)

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

 31.16.3 $Q_{0}A_{0}+R_{0}A_{1}=0,$ Symbols: $A_{j}$: coefficients, $Q_{j}$ and $R_{j}$ Permalink: http://dlmf.nist.gov/31.16.E3 Encodings: TeX, pMML, png See also: Annotations for 31.16(ii)
 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) 31.16.6 $\displaystyle Q_{j}$ $\displaystyle=-aj(j+\gamma+\delta-1)-q+\frac{(j-n)(j+\beta)(j+\gamma)(j+\gamma% +\delta-1)}{(2j+\gamma+\delta)(2j+\gamma+\delta-1)}+\frac{(j+n+\gamma+\delta-1% )j(j+\delta-1)(j-\beta+\gamma+\delta-1)}{(2j+\gamma+\delta-1)(2j+\gamma+\delta% -2)},$ 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) 31.16.7 $\displaystyle R_{j}$ $\displaystyle=\frac{(n-j)(j+n+\gamma+\delta)(j+\gamma)(j+\delta)}{(\gamma+% \delta+2j)(\gamma+\delta+2j+1)}.$ 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)

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