31 Heun Functions Applications 31.15 Stieltjes Polynomials 31.17 Physical Applications

§31.16 Mathematical Applications

Keywords:: Heun functions, Heun’s equation
Permalink:: http://dlmf.nist.gov/31.16

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

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

Keywords:: Heun functions, Heun’s equation, monodromy groups
Permalink:: http://dlmf.nist.gov/31.16.i

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

Keywords:: Heun functions, Heun polynomials, Heun’s equation, Hilbert space
Referenced by:: §31.11(i)
Permalink:: http://dlmf.nist.gov/31.16.ii

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

$x={\mathop{\sin\/}\nolimits^{{2}}}\theta{\mathop{\cos\/}\nolimits^{{2}}}\phi,$

$y={\mathop{\sin\/}\nolimits^{{2}}}\theta{\mathop{\sin\/}\nolimits^{{2}}}\phi.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\theta$ : angle, $\phi$ : angle, : variable and : variable
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 $P_{j}A_{{j-1}}+Q_{j}A_{j}+R_{j}A_{{j+1}}=0,$ $j=1,2,\dots,n$ ,

Symbols:: : nonnegative integer, : nonnegative integer, $A_{j}$ : coefficients, $P_{j}$ , $Q_{j}$ and $R_{j}$
Permalink:: http://dlmf.nist.gov/31.16.E4
Encodings:: TeX, pMML, png

where

31.16.5 $P_{j}=\frac{(\epsilon-j+n)j(\beta+j-1)(\gamma+\delta+j-2)}{(\gamma+\delta+2j-3% )(\gamma+\delta+2j-2)},$

Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, : nonnegative integer, : nonnegative integer, $\beta$ : real or complex parameter and $P_{j}$
Permalink:: http://dlmf.nist.gov/31.16.E5
Encodings:: TeX, pMML, png

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

Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, : nonnegative integer, : nonnegative integer, : complex parameter, : real or complex parameter, $\beta$ : real or complex parameter and $Q_{j}$
Permalink:: http://dlmf.nist.gov/31.16.E6
Encodings:: TeX, pMML, png

31.16.7 $R_{j}=\frac{(n-j)(j+n+\gamma+\delta)(j+\gamma)(j+\delta)}{(\gamma+\delta+2j)(% \gamma+\delta+2j+1)}.$

Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, : nonnegative integer, : nonnegative integer and $R_{j}$
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.

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§31.16 Mathematical Applications

Contents

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

§31.16(ii) Heun Polynomial Products