§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:

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).

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).