§28.6 Expansions for Small $q$

Leading terms of the power series for $a_m\left(q\right)$ and $b_m\left(q\right)$ for $m\le 6$ are:

Leading terms of the of the power series for $m=7,8,9,\mathrm{\dots }$ are:

For more details on these expansions and recurrence relations for the coefficients see Frenkel and Portugal (2001, §2).

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. Then

Higher coefficients in the foregoing series can be found by equating coefficients in the following continued-fraction equations:

Numerical values of the radii of convergence ${\rho }_n^{(j)}$ of the power series (28.6.1)–(28.6.14) for $n=0,1,\mathrm{\dots },9$ are given in Table 28.6.1. 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 $b_{2n+1}\left(q\right)$. (Table 28.6.1 is reproduced from Meixner et al. (1980, §2.4).)

It is conjectured that for large $n$, the radii increase in proportion to the square of the eigenvalue number $n$; see Meixner et al. (1980, §2.4). It is known that

where $k$ is the unique root of the equation $2E\left(k\right)=K\left(k\right)$ in the interval $(0,1)$, and $k^{\prime }=\sqrt{1-k^2}$. For $E\left(k\right)$ and $K\left(k\right)$ see §19.2(ii).

Leading terms of the power series for the normalized functions are:

For $m=3,4,5,\mathrm{\dots }$,

For the corresponding expansions of ${\mathrm{se}}_m(z,q)$ for $m=3,4,5,\mathrm{\dots }$ change $\cos$ to $\sin$ everywhere in (28.6.26). For more details on these expansions and recurrence relations for the coefficients see Frenkel and Portugal (2001, §2).’

The radii of convergence of the series (28.6.21)–(28.6.26) are the same as the radii of the corresponding series for $a_n\left(q\right)$ and $b_n\left(q\right)$; compare Table 28.6.1 and (28.6.20).