§31.9 Orthogonality

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§31.9(i) Single Orthogonality
§31.9(ii) Double Orthogonality

§31.9(i) Single Orthogonality

Notes:: See Becker (1997).
Keywords:: Heun functions, Heun polynomials, Pochhammer double-loop contour, Pochhammer’s integral
Referenced by:: ¶ ‣ §31.10(i), §31.11(i)
Permalink:: http://dlmf.nist.gov/31.9.i

With

31.9.1 $w_{m}(z)=\mathop{(0,1)\mathit{Hf}_{{m}}\/}\nolimits\!\left(a,q_{{m}};\alpha,\beta,\gamma,\delta;z\right),$

Symbols:: $\mathop{(s_{1},s_{2})\mathit{Hf}_{{m}}\/}\nolimits\!\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right)$ : 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
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we have

31.9.2 $\int _{{\zeta}}^{{(1+,0+,1-,0-)}}t^{{\gamma-1}}(1-t)^{{\delta-1}}(t-a)^{{{\textstyle\epsilon}-1}}\* w_{m}(t)w_{k}(t)dt=\delta _{{m,k}}\theta _{m}.$

Symbols:: $\delta _{{j,k}}$ : Kronecker delta, $dx$ : 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
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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 $\theta _{m}=(1-e^{{2\pi i\gamma}})(1-e^{{2\pi i\delta}})\zeta^{\gamma}(1-\zeta)^{\delta}(\zeta-a)^{{\textstyle\epsilon}}\*\frac{f_{0}(q,\zeta)}{f_{1}(q,\zeta)}\left.\frac{\partial}{\partial q}\mathop{\mathscr{W}\/}\nolimits\left\{ f_{0}(q,\zeta),f_{1}(q,\zeta)\right\}\right|_{{q=q_{m}}},$

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $e$ : base of exponential function, $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $\gamma$ : 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, $\theta _{m}$ : normalization constant and $f_{0}(q_{m},z)$ , $f_{1}(q_{m},z)$ : coefficients
Permalink:: http://dlmf.nist.gov/31.9.E3
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where

31.9.4

$f_{0}(q_{m},z)=\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),$

$f_{1}(q_{m},z)=\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(1-a,\alpha\beta-q_{m};\alpha,\beta,\delta,\gamma;1-z\right),$

Symbols:: $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ : 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, $\beta$ : real or complex parameter and $f_{0}(q_{m},z)$ , $f_{1}(q_{m},z)$ : coefficients
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and $\mathop{\mathscr{W}\/}\nolimits$ 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

Keywords:: Heun functions, Heun polynomials, Heun’s equation
Permalink:: http://dlmf.nist.gov/31.9.ii

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

31.9.5 $\int _{{\mathcal{L}_{1}}}\int _{{\mathcal{L}_{2}}}\rho(s,t)w_{1}(s)w_{1}(t)w_{2}(s)w_{2}(t)dsdt=0,$ $|n_{1}-n_{2}|+|m_{1}-m_{2}|\neq 0$ ,

Symbols:: $dx$ : 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
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where

31.9.6 $\rho(s,t)=(s-t)(st)^{{\gamma-1}}\left((s-1)(t-1)\right)^{{\delta-1}}\*\left((s-a)(t-a)\right)^{{{\textstyle\epsilon}-1}},$

Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter and $\rho(s,t)$ : weight function
Permalink:: http://dlmf.nist.gov/31.9.E6
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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).

§31.9 Orthogonality

Contents

§31.9(i) Single Orthogonality

§31.9(ii) Double Orthogonality