# §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

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
 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 31.16.6 $\displaystyle Q_{j}$ $\displaystyle=-aj(j+\gamma+\delta-1)-q+\frac{(j-n)(j+\beta)(j+\gamma)(j+\gamma% +\delta-1)}{(2j+\gamma+\delta)(2j+\gamma+\delta-1)}+\frac{(j+n+\gamma+\delta-1% )j(j+\delta-1)(j-\beta+\gamma+\delta-1)}{(2j+\gamma+\delta-1)(2j+\gamma+\delta% -2)},$ 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

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