§31.9 Orthogonality

With

31.9.1		$$w_m(z)=(0,1){\mathit{Hf}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z),$$
ⓘ Symbols: $(s_1,s_2){\mathit{Hf}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$: 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		$${\int }_{\zeta }^{(1+,0+,1-,0-)}{t}^{\gamma -1}{(1-t)}^{\delta -1}{(t-a)}^{ϵ-1}{w}_m(t)w_k(t)dt={\delta }_{m,k}{\theta }_m.$$
ⓘ Symbols: ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$, $\int$: integral, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: 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		$${\theta }_m={(1-{\mathrm{e}}^{2\pi \mathrm{i}\gamma })(1-{\mathrm{e}}^{2\pi \mathrm{i}\delta }){\zeta }^{\gamma }{(1-\zeta )}^{\delta }{(\zeta -a)}^{ϵ}\frac{f_0(q,\zeta )}{f_1(q,\zeta )}\frac{\partial }{\partial q}𝒲\left\{f_0(q,\zeta ),f_1(q,\zeta )\right\}|}_{q=q_m},$$
ⓘ Defines: ${\theta }_m$: normalization constant (locally) Symbols: $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: 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		$f_0(q_m,z)$	$=H\mathrm{\ell }(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z),$
		$f_1(q_m,z)$	$=H\mathrm{\ell }(1-a,\alpha \beta -q_m;\alpha ,\beta ,\delta ,\gamma ;1-z),$
ⓘ Defines: $f_0(q_m,z)$, $f_1(q_m,z)$: coefficients (locally) Symbols: $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: 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 $𝒲$ 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).

Heun polynomials $w_j={\mathit{Hp}}_{n_j,m_j}$, $j=1,2$, satisfy

31.9.5		$${\int }_{{ℒ}_1}{\int }_{{ℒ}_2}\rho (s,t)w_1(s)w_1(t)w_2(s)w_2(t)dsdt=0,$$
$|n_1-n_2|+|m_1-m_2|\ne 0$,
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral, $m$: nonnegative integer, $n$: nonnegative integer, $\rho (s,t)$: weight function and ${ℒ}_1$, ${ℒ}_2$: 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		$$\rho (s,t)=(s-t){(st)}^{\gamma -1}{\left((s-1)(t-1)\right)}^{\delta -1}{\left((s-a)(t-a)\right)}^{ϵ-1},$$
ⓘ Defines: $\rho (s,t)$: weight function (locally) Symbols: $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: 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 ${ℒ}_1$, ${ℒ}_2$ 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).