31 Heun Functions Applications 31.15 Stieltjes Polynomials 31.17 Physical Applications

§31.16 Mathematical Applications

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Keywords:: Heun functions, Heun’s equation, applications, mathematical
Permalink:: http://dlmf.nist.gov/31.16
See also:: Annotations for Ch.31

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

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Keywords:: Heun functions, Heun’s equation, inversion problem, monodromy group, monodromy groups, uniformization problem
Permalink:: http://dlmf.nist.gov/31.16.i
See also:: Annotations for §31.16 and Ch.31

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

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Keywords:: Heun functions, Heun polynomial products, Heun polynomials, Heun’s equation, Hilbert space, applications, interrelation between bases, mathematical, products
Referenced by:: §31.11(i)
Permalink:: http://dlmf.nist.gov/31.16.ii
Correction (effective with 1.0.21):: The Jacobi polynomials in (31.16.1) which previously linked incorrectly to the constants $P_{j}$ , now link correctly. Originally $x, y$ were incorrectly defined by the set of equations (31.16.2), given previously as “ $x={\sin}^{2}\theta{\cos}^{2}\phi,$ $y={\sin}^{2}\theta{\sin}^{2}\phi$ .” It now states that $x, y$ are implicitly defined by the corrected set of equations. In (31.16.3), the initial data $A_{0}$ , previously missing, has now been included.
See also:: Annotations for §31.16 and Ch.31

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		$\mathit{Hp}_{n,m}\left(x\right)\mathit{Hp}_{n,m}\left(y\right)=\sum_{j=0}^{n}A% _{j}{\sin}^{2j}\theta\*P^{(\gamma+\delta+2j-1,\epsilon-1)}_{n-j}\left(\cos% \left(2\theta\right)\right)P^{(\delta-1,\gamma-1)}_{j}\left(\cos\left(2\phi% \right)\right),$
ⓘ Symbols: $\mathit{Hp}_{\NVar{n},\NVar{m}}\left(\NVar{a},\NVar{q_{n,m}};\NVar{-n},\NVar{% \beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun polynomials, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : 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, $\phi$ : 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 $n=0,1,\dots$ , $m=0,1,\dots,n$ , 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 (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 “ $x={\sin}^{2}\theta{\cos}^{2}\phi,$ $y={\sin}^{2}\theta{\sin}^{2}\phi$ ”. 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		$A_{0}=\frac{n!}{{\left(\gamma+\delta\right)_{n}}}\mathit{Hp}_{n,m}\left(1% \right),\quad Q_{0}A_{0}+R_{0}A_{1}=0,$
ⓘ Symbols: $\mathit{Hp}_{\NVar{n},\NVar{m}}\left(\NVar{a},\NVar{q_{n,m}};\NVar{-n},\NVar{% \beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun polynomials, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : 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		$P_{j}A_{j-1}+Q_{j}A_{j}+R_{j}A_{j+1}=0,$
$j=1,2,\dots,n$ ,
ⓘ 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	$\displaystyle P_{j}$	$\displaystyle=\frac{(\epsilon-j+n)j(\beta+j-1)(\gamma+\delta+j-2)}{(\gamma+% \delta+2j-3)(\gamma+\delta+2j-2)},$
ⓘ 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.

31.15 Stieltjes Polynomials 31.17 Physical Applications

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§31.16 Mathematical Applications

Contents

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

§31.16(ii) Heun Polynomial Products

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