# §31.9 Orthogonality

## §31.9(i) Single Orthogonality

With

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

we have

 31.9.2 $\int_{\zeta}^{(1+,0+,1-,0-)}t^{\gamma-1}(1-t)^{\delta-1}(t-a)^{\epsilon-1}\*w_% {m}(t)w_{k}(t)\,\mathrm{d}t=\delta_{m,k}\theta_{m}.$

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 $\theta_{m}=(1-{\mathrm{e}}^{2\pi i\gamma})(1-{\mathrm{e}}^{2\pi i\delta})\zeta% ^{\gamma}(1-\zeta)^{\delta}(\zeta-a)^{\epsilon}\*\frac{f_{0}(q,\zeta)}{f_{1}(q% ,\zeta)}\left.\frac{\partial}{\partial q}\mathscr{W}\left\{f_{0}(q,\zeta),f_{1% }(q,\zeta)\right\}\right|_{q=q_{m}},$

where

 31.9.4 $\displaystyle f_{0}(q_{m},z)$ $\displaystyle=\mathit{H\!\ell}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),$ $\displaystyle f_{1}(q_{m},z)$ $\displaystyle=\mathit{H\!\ell}\left(1-a,\alpha\beta-q_{m};\alpha,\beta,\delta,% \gamma;1-z\right),$ ⓘ 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

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

 31.9.5 $\int_{\mathcal{L}_{1}}\int_{\mathcal{L}_{2}}\rho(s,t)w_{1}(s)w_{1}(t)w_{2}(s)w% _{2}(t)\,\mathrm{d}s\,\mathrm{d}t=0,$ $|n_{1}-n_{2}|+|m_{1}-m_{2}|\neq 0$,

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).