# eigenvalues of accessory parameters

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##### 1: 31.13 Asymptotic Approximations
###### §31.13 Asymptotic Approximations
For asymptotic approximations for the accessory parameter eigenvalues $q_{m}$, see Fedoryuk (1991) and Slavyanov (1996). …
##### 2: 31.4 Solutions Analytic at Two Singularities: Heun Functions
For an infinite set of discrete values $q_{m}$, $m=0,1,2,\dots$, of the accessory parameter $q$, the function $\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ is analytic at $z=1$, and hence also throughout the disk $|z|. …
31.4.1 $(0,1)\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),$ $m=0,1,2,\dots$.
The eigenvalues $q_{m}$ satisfy the continued-fraction equation
31.4.3 $(s_{1},s_{2})\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),$ $m=0,1,2,\dots$,
##### 3: 28.7 Analytic Continuation of Eigenvalues
###### §28.7 Analytic Continuation of Eigenvalues
As functions of $q$, $a_{n}\left(q\right)$ and $b_{n}\left(q\right)$ can be continued analytically in the complex $q$-plane. The only singularities are algebraic branch points, with $a_{n}\left(q\right)$ and $b_{n}\left(q\right)$ finite at these points. …The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). …
##### 5: 28.6 Expansions for Small $q$
###### §28.6(i) Eigenvalues
Leading terms of the power series for $a_{m}\left(q\right)$ and $b_{m}\left(q\right)$ for $m\leq 6$ are: … The coefficients of the power series of $a_{2n}\left(q\right)$, $b_{2n}\left(q\right)$ and also $a_{2n+1}\left(q\right)$, $b_{2n+1}\left(q\right)$ are the same until the terms in $q^{2n-2}$ and $q^{2n}$, respectively. … Higher coefficients in the foregoing series can be found by equating coefficients in the following continued-fraction equations: … Here $j=1$ for $a_{2n}\left(q\right)$, $j=2$ for $b_{2n+2}\left(q\right)$, and $j=3$ for $a_{2n+1}\left(q\right)$ and $b_{2n+1}\left(q\right)$. …
##### 6: 28.34 Methods of Computation
###### §28.34(ii) Eigenvalues
Methods for computing the eigenvalues $a_{n}\left(q\right)$, $b_{n}\left(q\right)$, and $\lambda_{\nu}\left(q\right)$, defined in §§28.2(v) and 28.12(i), include: …
• (d)

Solution of the matrix eigenvalue problem for each of the five infinite matrices that correspond to the linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4). See Zhang and Jin (1996, pp. 479–482) and §3.2(iv).

• (f)

Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

• Also, once the eigenvalues $a_{n}\left(q\right)$, $b_{n}\left(q\right)$, and $\lambda_{\nu}\left(q\right)$ have been computed the following methods are applicable: …
##### 7: 29.3 Definitions and Basic Properties
###### §29.3(ii) Distribution
The eigenvalues interlace according to …The eigenvalues coalesce according to …
##### 8: 28.15 Expansions for Small $q$
###### §28.15(i) Eigenvalues$\lambda_{\nu}\left(q\right)$
28.15.1 $\lambda_{\nu}\left(q\right)=\nu^{2}+\frac{1}{2(\nu^{2}-1)}q^{2}+\frac{5\nu^{2}% +7}{32(\nu^{2}-1)^{3}(\nu^{2}-4)}q^{4}+\frac{9\nu^{4}+58\nu^{2}+29}{64(\nu^{2}% -1)^{5}(\nu^{2}-4)(\nu^{2}-9)}q^{6}+\cdots.$
Higher coefficients can be found by equating powers of $q$ in the following continued-fraction equation, with $a=\lambda_{\nu}\left(q\right)$:
28.15.2 $a-\nu^{2}-\cfrac{q^{2}}{a-(\nu+2)^{2}-\cfrac{q^{2}}{a-(\nu+4)^{2}-\cdots}}=% \cfrac{q^{2}}{a-(\nu-2)^{2}-\cfrac{q^{2}}{a-(\nu-4)^{2}-\cdots}}.$
28.15.3 $\operatorname{me}_{\nu}\left(z,q\right)=e^{\mathrm{i}\nu z}-\frac{q}{4}\left(% \frac{1}{\nu+1}e^{\mathrm{i}(\nu+2)z}-\frac{1}{\nu-1}e^{\mathrm{i}(\nu-2)z}% \right)+\frac{q^{2}}{32}\left(\frac{1}{(\nu+1)(\nu+2)}e^{\mathrm{i}(\nu+4)z}+% \frac{1}{(\nu-1)(\nu-2)}e^{\mathrm{i}(\nu-4)z}-\frac{2(\nu^{2}+1)}{(\nu^{2}-1)% ^{2}}e^{\mathrm{i}\nu z}\right)+\cdots;$
##### 10: 29.7 Asymptotic Expansions
###### §29.7(i) Eigenvalues
29.7.1 $a^{m}_{\nu}\left(k^{2}\right)\sim p\kappa-\tau_{0}-\tau_{1}\kappa^{-1}-\tau_{2% }\kappa^{-2}-\cdots,$
29.7.3 $\tau_{0}=\frac{1}{2^{3}}(1+k^{2})(1+p^{2}),$
The same Poincaré expansion holds for $b^{m+1}_{\nu}\left(k^{2}\right)$, since
29.7.5 $b^{m+1}_{\nu}\left(k^{2}\right)-a^{m}_{\nu}\left(k^{2}\right)=O\left(\nu^{m+% \frac{3}{2}}\left(\frac{1-k}{1+k}\right)^{\nu}\right),$ $\nu\to\infty$.