# §28.7 Analytic Continuation of Eigenvalues

As functions of $q$, $\mathop{a_{n}\/}\nolimits\!\left(q\right)$ and $\mathop{b_{n}\/}\nolimits\!\left(q\right)$ can be continued analytically in the complex $q$-plane. The only singularities are algebraic branch points, with $\mathop{a_{n}\/}\nolimits\!\left(q\right)$ and $\mathop{b_{n}\/}\nolimits\!\left(q\right)$ finite at these points. The number of branch points is infinite, but countable, and there are no finite limit points. In consequence, the functions can be defined uniquely by introducing suitable cuts in the $q$-plane. See Meixner and Schäfke (1954, §2.22). The branch points are called the exceptional values, and the other points normal values. The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). All real values of $q$ are normal values. To 4D the first branch points between $\mathop{a_{0}\/}\nolimits\!\left(q\right)$ and $\mathop{a_{2}\/}\nolimits\!\left(q\right)$ are at $q_{0}=\pm i1.4688$ with $\mathop{a_{0}\/}\nolimits\!\left(q_{0}\right)=\mathop{a_{2}\/}\nolimits\!\left% (q_{0}\right)=2.0886$, and between $\mathop{b_{2}\/}\nolimits\!\left(q\right)$ and $\mathop{b_{4}\/}\nolimits\!\left(q\right)$ they are at $q_{1}=\pm i6.9289$ with $\mathop{b_{2}\/}\nolimits\!\left(q_{1}\right)=\mathop{b_{4}\/}\nolimits\!\left% (q_{1}\right)=11.1904$. For real $q$ with $|q|<|q_{0}|$, $\mathop{a_{0}\/}\nolimits\!\left(iq\right)$ and $\mathop{a_{2}\/}\nolimits\!\left(iq\right)$ are real-valued, whereas for real $q$ with $|q|>|q_{0}|$, $\mathop{a_{0}\/}\nolimits\!\left(iq\right)$ and $\mathop{a_{2}\/}\nolimits\!\left(iq\right)$ are complex conjugates. See also Mulholland and Goldstein (1929), Bouwkamp (1948), Meixner et al. (1980), Hunter and Guerrieri (1981), Hunter (1981), and Shivakumar and Xue (1999).

For a visualization of the first branch point of $\mathop{a_{0}\/}\nolimits\!\left(i\hat{q}\right)$ and $\mathop{a_{2}\/}\nolimits\!\left(i\hat{q}\right)$ see Figure 28.7.1.

All the $\mathop{a_{2n}\/}\nolimits\!\left(q\right)$, $n=0,1,2,\dots$, can be regarded as belonging to a complete analytic function (in the large). Therefore $w^{\prime}_{\mbox{\tiny I}}(\frac{1}{2}\pi;a,q)$ is irreducible, in the sense that it cannot be decomposed into a product of entire functions that contain its zeros; see Meixner et al. (1980, p. 88). Analogous statements hold for $\mathop{a_{2n+1}\/}\nolimits\!\left(q\right)$, $\mathop{b_{2n+1}\/}\nolimits\!\left(q\right)$, and $\mathop{b_{2n+2}\/}\nolimits\!\left(q\right)$, also for $n=0,1,2,\dots$. Closely connected with the preceding statements, we have

 28.7.1 $\displaystyle\sum_{n=0}^{\infty}\left(\mathop{a_{2n}\/}\nolimits\!\left(q% \right)-(2n)^{2}\right)$ $\displaystyle=0,$ 28.7.2 $\displaystyle\sum_{n=0}^{\infty}\left(\mathop{a_{2n+1}\/}\nolimits\!\left(q% \right)-(2n+1)^{2}\right)$ $\displaystyle=q,$ 28.7.3 $\displaystyle\sum_{n=0}^{\infty}\left(\mathop{b_{2n+1}\/}\nolimits\!\left(q% \right)-(2n+1)^{2}\right)$ $\displaystyle=-q,$ 28.7.4 $\displaystyle\sum_{n=0}^{\infty}\left(\mathop{b_{2n+2}\/}\nolimits\!\left(q% \right)-(2n+2)^{2}\right)$ $\displaystyle=0.$