31 Heun Functions Properties 31.8 Solutions via Quadratures 31.10 Integral Equations and Representations

§31.9 Orthogonality

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Permalink:: http://dlmf.nist.gov/31.9
See also:: Annotations for Ch.31

§31.9(i) Single Orthogonality

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Keywords:: Heun functions, Heun polynomials, Pochhammer double-loop contour, Pochhammer’s integral, orthogonality, single
Notes:: See Becker (1997).
Referenced by:: §31.10(i), §31.11(i)
Permalink:: http://dlmf.nist.gov/31.9.i
See also:: Annotations for §31.9 and Ch.31

With

31.9.1		$w_{m}(z)=(0,1)\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),$
ⓘ Symbols: $(\NVar{s_{1}},\NVar{s_{2}})\mathit{Hf}_{\NVar{m}}\left(\NVar{a},\NVar{q_{m}};% \NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter Permalink: http://dlmf.nist.gov/31.9.E1 Encodings: TeX, pMML, png See also: Annotations for §31.9(i), §31.9 and Ch.31

we have

31.9.2		$\int_{\zeta}^{(1+,0+,1-,0-)}t^{\gamma-1}(1-t)^{\delta-1}(t-a)^{\epsilon-1}\*w_% {m}(t)w_{k}(t)\,\mathrm{d}t=\delta_{m,k}\theta_{m}.$
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $\zeta$ : point and $\theta_{m}$ : normalization constant Permalink: http://dlmf.nist.gov/31.9.E2 Encodings: TeX, pMML, png See also: Annotations for §31.9(i), §31.9 and Ch.31

Here $\zeta$ is an arbitrary point in the interval $(0,1)$ . The integration path begins at $z=\zeta$ , encircles $z=1$ once in the positive sense, followed by $z=0$ once in the positive sense, and so on, returning finally to $z=\zeta$ . The integration path is called a Pochhammer double-loop contour (compare Figure 5.12.3). The branches of the many-valued functions are continuous on the path, and assume their principal values at the beginning.

The normalization constant $\theta_{m}$ is given by

31.9.3		$\theta_{m}=(1-{\mathrm{e}}^{2\pi i\gamma})(1-{\mathrm{e}}^{2\pi i\delta})\zeta% ^{\gamma}(1-\zeta)^{\delta}(\zeta-a)^{\epsilon}\*\frac{f_{0}(q,\zeta)}{f_{1}(q% ,\zeta)}\left.\frac{\partial}{\partial q}\mathscr{W}\left\{f_{0}(q,\zeta),f_{1% }(q,\zeta)\right\}\right\|_{q=q_{m}},$
ⓘ Defines: $\theta_{m}$ : normalization constant (locally) Symbols: $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter, $\zeta$ : point and $f_{0}(q_{m},z)$ , $f_{1}(q_{m},z)$ : coefficients Permalink: http://dlmf.nist.gov/31.9.E3 Encodings: TeX, pMML, png See also: Annotations for §31.9(i), §31.9 and Ch.31

where

31.9.4	$\displaystyle f_{0}(q_{m},z)$	$\displaystyle=\mathit{H\!\ell}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),$
31.9.4	$\displaystyle f_{1}(q_{m},z)$	$\displaystyle=\mathit{H\!\ell}\left(1-a,\alpha\beta-q_{m};\alpha,\beta,\delta,% \gamma;1-z\right),$
ⓘ Defines: $f_{0}(q_{m},z)$ , $f_{1}(q_{m},z)$ : coefficients (locally) Symbols: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter Permalink: http://dlmf.nist.gov/31.9.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §31.9(i), §31.9 and Ch.31

and $\mathscr{W}$ denotes the Wronskian (§1.13(i)). The right-hand side may be evaluated at any convenient value, or limiting value, of $\zeta$ in $(0,1)$ since it is independent of $\zeta$ .

For corresponding orthogonality relations for Heun functions (§31.4) and Heun polynomials (§31.5), see Lambe and Ward (1934), Erdélyi (1944), Sleeman (1966a), and Ronveaux (1995, Part A, pp. 59–64).

§31.9(ii) Double Orthogonality

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Keywords:: Heun functions, Heun polynomials, Heun’s equation, biorthogonality, double, orthogonality, path-multiplicative solutions
Permalink:: http://dlmf.nist.gov/31.9.ii
See also:: Annotations for §31.9 and Ch.31

Heun polynomials $w_{j}=\mathit{Hp}_{n_{j},m_{j}}$ , $j=1,2$ , satisfy

31.9.5		$\int_{\mathcal{L}_{1}}\int_{\mathcal{L}_{2}}\rho(s,t)w_{1}(s)w_{1}(t)w_{2}(s)w% _{2}(t)\,\mathrm{d}s\,\mathrm{d}t=0,$
$\|n_{1}-n_{2}\|+\|m_{1}-m_{2}\|\neq 0$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $m$ : nonnegative integer, $n$ : nonnegative integer, $\rho(s,t)$ : weight function and $\mathcal{L}_{1}$ , $\mathcal{L}_{2}$ : integration paths Permalink: http://dlmf.nist.gov/31.9.E5 Encodings: TeX, pMML, png See also: Annotations for §31.9(ii), §31.9 and Ch.31

where

31.9.6		$\rho(s,t)=(s-t)(st)^{\gamma-1}\left((s-1)(t-1)\right)^{\delta-1}\*\left((s-a)(% t-a)\right)^{\epsilon-1},$
ⓘ Defines: $\rho(s,t)$ : weight function (locally) Symbols: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter and $a$ : complex parameter Permalink: http://dlmf.nist.gov/31.9.E6 Encodings: TeX, pMML, png See also: Annotations for §31.9(ii), §31.9 and Ch.31

and the integration paths $\mathcal{L}_{1}$ , $\mathcal{L}_{2}$ are Pochhammer double-loop contours encircling distinct pairs of singularities $\{0,1\}$ , $\{0,a\}$ , $\{1,a\}$ .

For further information, including normalization constants, see Sleeman (1966a). For bi-orthogonal relations for path-multiplicative solutions see Schmidt (1979, §2.2). For other generalizations see Arscott (1964b, pp. 206–207 and 241).

31.8 Solutions via Quadratures 31.10 Integral Equations and Representations

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§31.9 Orthogonality

Contents

§31.9(i) Single Orthogonality

§31.9(ii) Double Orthogonality