28 Mathieu Functions and Hill’s Equation Mathieu Functions of Integer Order 28.5 Second Solutions

\operatorname{fe}_{n}

\operatorname{ge}_{n}

28.7 Analytic Continuation of Eigenvalues

§28.6 Expansions for Small $q$

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Permalink:: http://dlmf.nist.gov/28.6
See also:: Annotations for Ch.28

§28.6(i) Eigenvalues

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Keywords:: Mathieu’s equation, continued-fraction equations, eigenvalues (or characteristic values), power-series expansions in $q$
Notes:: See Meixner and Schäfke (1954, pp. 118, 120, and 122) and Meixner et al. (1980, p. 101 and §2.4). For (28.6.20) see Volkmer (1998).
Referenced by:: §28.2(v), item (a), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii)
Permalink:: http://dlmf.nist.gov/28.6.i
Clarification (effective with 1.1.10):: Just below (28.6.14), a sentence was added referring the reader to Frenkel and Portugal (2001, §2).
See also:: Annotations for §28.6 and Ch.28

Leading terms of the power series for $a_{m}\left(q\right)$ and $b_{m}\left(q\right)$ for $m\leq 6$ are:

28.6.1	$\displaystyle a_{0}\left(q\right)$	$\displaystyle=-\tfrac{1}{2}q^{2}+\tfrac{7}{128}q^{4}-\tfrac{29}{2304}q^{6}+% \tfrac{68687}{188\;74368}q^{8}+\cdots,$
ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter A&S Ref: 20.2.25 Referenced by: §28.6(i) Permalink: http://dlmf.nist.gov/28.6.E1 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28
28.6.2	$\displaystyle a_{1}\left(q\right)$	$\displaystyle=1+q-\tfrac{1}{8}q^{2}-\tfrac{1}{64}q^{3}-\tfrac{1}{1536}q^{4}+% \tfrac{11}{36864}q^{5}+\tfrac{49}{5\;89824}q^{6}+\tfrac{55}{94\;37184}q^{7}-% \tfrac{83}{353\;89440}q^{8}+\cdots,$
ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter Permalink: http://dlmf.nist.gov/28.6.E2 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28
28.6.3	$\displaystyle b_{1}\left(q\right)$	$\displaystyle=1-q-\tfrac{1}{8}q^{2}+\tfrac{1}{64}q^{3}-\tfrac{1}{1536}q^{4}-% \tfrac{11}{36864}q^{5}+\tfrac{49}{5\;89824}q^{6}-\tfrac{55}{94\;37184}q^{7}-% \tfrac{83}{353\;89440}q^{8}+\cdots,$
ⓘ Symbols: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter Permalink: http://dlmf.nist.gov/28.6.E3 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28
28.6.4	$\displaystyle a_{2}\left(q\right)$	$\displaystyle=4+\tfrac{5}{12}q^{2}-\tfrac{763}{13824}q^{4}+\tfrac{10\;02401}{7% 96\;26240}q^{6}-\tfrac{16690\;68401}{45\;86471\;42400}q^{8}+\cdots,$
ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter Permalink: http://dlmf.nist.gov/28.6.E4 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28
28.6.5	$\displaystyle b_{2}\left(q\right)$	$\displaystyle=4-\tfrac{1}{12}q^{2}+\tfrac{5}{13824}q^{4}-\tfrac{289}{796\;2624% 0}q^{6}+\tfrac{21391}{45\;86471\;42400}q^{8}+\cdots,$
ⓘ Symbols: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter Permalink: http://dlmf.nist.gov/28.6.E5 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28
28.6.6	$\displaystyle a_{3}\left(q\right)$	$\displaystyle=9+\tfrac{1}{16}q^{2}+\tfrac{1}{64}q^{3}+\tfrac{13}{20480}q^{4}-% \tfrac{5}{16384}q^{5}-\tfrac{1961}{235\;92960}q^{6}-\tfrac{609}{1048\;57600}q^% {7}+\cdots,$
ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter Permalink: http://dlmf.nist.gov/28.6.E6 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28
28.6.7	$\displaystyle b_{3}\left(q\right)$	$\displaystyle=9+\tfrac{1}{16}q^{2}-\tfrac{1}{64}q^{3}+\tfrac{13}{20480}q^{4}+% \tfrac{5}{16384}q^{5}-\tfrac{1961}{235\;92960}q^{6}+\tfrac{609}{1048\;57600}q^% {7}+\cdots,$
ⓘ Symbols: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter Permalink: http://dlmf.nist.gov/28.6.E7 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28
28.6.8	$\displaystyle a_{4}\left(q\right)$	$\displaystyle=16+\tfrac{1}{30}q^{2}+\tfrac{433}{8\;64000}q^{4}-\tfrac{5701}{27% 216\;00000}q^{6}+\cdots,$
ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter Permalink: http://dlmf.nist.gov/28.6.E8 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28
28.6.9	$\displaystyle b_{4}\left(q\right)$	$\displaystyle=16+\tfrac{1}{30}q^{2}-\tfrac{317}{8\;64000}q^{4}+\tfrac{10049}{2% 7216\;00000}q^{6}+\cdots,$
ⓘ Symbols: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter Permalink: http://dlmf.nist.gov/28.6.E9 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28
28.6.10	$\displaystyle a_{5}\left(q\right)$	$\displaystyle=25+\tfrac{1}{48}q^{2}+\tfrac{11}{7\;74144}q^{4}+\tfrac{1}{1\;474% 56}q^{5}+\tfrac{37}{8918\;13888}q^{6}+\cdots,$
ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter Permalink: http://dlmf.nist.gov/28.6.E10 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28
28.6.11	$\displaystyle b_{5}\left(q\right)$	$\displaystyle=25+\tfrac{1}{48}q^{2}+\tfrac{11}{7\;74144}q^{4}-\tfrac{1}{1\;474% 56}q^{5}+\tfrac{37}{8918\;13888}q^{6}+\cdots,$
ⓘ Symbols: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter Permalink: http://dlmf.nist.gov/28.6.E11 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28
28.6.12	$\displaystyle a_{6}\left(q\right)$	$\displaystyle=36+\tfrac{1}{70}q^{2}+\tfrac{187}{439\;04000}q^{4}+\tfrac{67\;43% 617}{9293\;59872\;00000}q^{6}+\cdots,$
ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter Permalink: http://dlmf.nist.gov/28.6.E12 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28
28.6.13	$\displaystyle b_{6}\left(q\right)$	$\displaystyle=36+\tfrac{1}{70}q^{2}+\tfrac{187}{439\;04000}q^{4}-\tfrac{58\;61% 633}{9293\;59872\;00000}q^{6}+\cdots.$
ⓘ Symbols: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter Permalink: http://dlmf.nist.gov/28.6.E13 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28

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

28.6.14		$\rselection{a_{m}\left(q\right)\\ b_{m}\left(q\right)}=m^{2}+\frac{1}{2(m^{2}-1)}q^{2}+\frac{5m^{2}+7}{32(m^{2}-% 1)^{3}(m^{2}-4)}q^{4}+\frac{9m^{4}+58m^{2}+29}{64(m^{2}-1)^{5}(m^{2}-4)(m^{2}-% 9)}q^{6}+\cdots.$
ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $m$ : integer and $q=h^{2}$ : parameter A&S Ref: 20.2.26 Referenced by: §28.6(i), §28.6(i), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii) Permalink: http://dlmf.nist.gov/28.6.E14 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28

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

28.6.15		$a_{m}\left(q\right)-b_{m}\left(q\right)=\frac{2q^{m}}{\left(2^{m-1}(m-1)!% \right)^{2}}\left(1+O\left(q^{2}\right)\right).$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $!$ : factorial (as in $n!$ ), $m$ : integer and $q=h^{2}$ : parameter Permalink: http://dlmf.nist.gov/28.6.E15 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28

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

28.6.16		$a-(2n)^{2}-\cfrac{q^{2}}{a-(2n-2)^{2}-\cfrac{q^{2}}{a-(2n-4)^{2}-}}\cdots% \cfrac{q^{2}}{a-2^{2}-\cfrac{2q^{2}}{a}}=-\cfrac{q^{2}}{(2n+2)^{2}-a-\cfrac{q^% {2}}{(2n+4)^{2}-a-\cdots}},$
$a=a_{2n}\left(q\right)$ ,
ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter, $n$ : integer and $a$ : parameter Referenced by: item (e) Permalink: http://dlmf.nist.gov/28.6.E16 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28

28.6.17		$a-(2n+1)^{2}-\cfrac{q^{2}}{a-(2n-1)^{2}-}\cdots\cfrac{q^{2}}{a-3^{2}-\cfrac{q^% {2}}{a-1^{2}-q}}=-\cfrac{q^{2}}{(2n+3)^{2}-a-\cfrac{q^{2}}{(2n+5)^{2}-a-\cdots% }},$
$a=a_{2n+1}\left(q\right)$ ,
ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter, $n$ : integer and $a$ : parameter Permalink: http://dlmf.nist.gov/28.6.E17 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28

28.6.18		$a-(2n+1)^{2}-\cfrac{q^{2}}{a-(2n-1)^{2}-}\cdots\cfrac{q^{2}}{a-3^{2}-\cfrac{q^% {2}}{a-1^{2}+q}}=-\cfrac{q^{2}}{(2n+3)^{2}-a-\cfrac{q^{2}}{(2n+5)^{2}-a-\cdots% }},$
$a=b_{2n+1}\left(q\right)$ ,
ⓘ Symbols: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter, $n$ : integer and $a$ : parameter Permalink: http://dlmf.nist.gov/28.6.E18 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28

28.6.19		$a-(2n+2)^{2}-\cfrac{q^{2}}{a-(2n)^{2}-\cfrac{q^{2}}{a-(2n-2)^{2}-}}\cdots% \cfrac{q^{2}}{a-2^{2}}=-\cfrac{q^{2}}{(2n+4)^{2}-a-\cfrac{q^{2}}{(2n+6)^{2}-a-% \cdots}},$
$a=b_{2n+2}\left(q\right)$ .
ⓘ Symbols: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter, $n$ : integer and $a$ : parameter Referenced by: item (e) Permalink: http://dlmf.nist.gov/28.6.E19 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28

Numerical values of the radii of convergence $\rho_{n}^{(j)}$ of the power series (28.6.1)–(28.6.14) for $n=0,1,\dots,9$ 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).)

Table 28.6.1: Radii of convergence for power-series expansions of eigenvalues of Mathieu’s equation.

$n$	$\rho^{(1)}_{n}$	$\rho^{(2)}_{n}$	$\rho^{(3)}_{n}$
0 or 1	$1.46876\;86138$	$6.92895\;47588$	$3.76995\;74940$
2	$7.26814\;68935$	$16.80308\;98254$	$11.27098\;52655$
3	$16.47116\;58923$	$30.09677\;28376$	$22.85524\;71216$
4	$30.42738\;20960$	$48.13638\;18593$	$38.52292\;50099$
5	$47.80596\;57026$	$69.59879\;32769$	$58.27413\;84472$
6	$69.92930\;51764$	$95.80595\;67052$	$82.10894\;36067$
7	$95.47527\;27072$	$125.43541\;1314$	$110.02736\;9210$
8	$125.76627\;89677$	$159.81025\;4642$	$142.02943\;1279$
9	$159.47921\;26694$	$197.60667\;8692$	$178.11513\;940$

ⓘ

Symbols:: $n$ : integer and $\rho_{n}^{(j)}$ : radii of convergence
Referenced by:: §28.6(i), §28.6(ii)
Permalink:: http://dlmf.nist.gov/28.6.T1
See also:: Annotations for §28.6(i), §28.6 and Ch.28

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

28.6.20		$\liminf_{n\to\infty}\frac{\rho_{n}^{(j)}}{n^{2}}\geq kk^{\prime}(K\left(k% \right))^{2}=2.04183\;4\dots,$
ⓘ Symbols: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\liminf$ : least limit point, $n$ : integer, $j$ : integer, $\rho_{n}^{(j)}$ : radii of convergence and $k$ : root Referenced by: §28.6(i), §28.6(ii) Permalink: http://dlmf.nist.gov/28.6.E20 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28

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).

§28.6(ii) Functions $\operatorname{ce}_{n}$ and $\operatorname{se}_{n}$

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Keywords:: Mathieu functions, power series in $q$
Notes:: See Meixner and Schäfke (1954, pp. 122–123).
Referenced by:: §28.15(ii), item (a), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii)
Permalink:: http://dlmf.nist.gov/28.6.ii
Clarification (effective with 1.1.10):: Just below (28.6.26), a sentence was added referring the reader to Frenkel and Portugal (2001, §2).
See also:: Annotations for §28.6 and Ch.28

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

28.6.21	$\displaystyle 2^{\ifrac{1}{2}}\operatorname{ce}_{0}\left(z,q\right)$	$\displaystyle=1-\tfrac{1}{2}q\cos 2z+\tfrac{1}{32}q^{2}\left(\cos 4z-2\right)-% \tfrac{1}{128}q^{3}\left(\tfrac{1}{9}\cos 6z-11\cos 2z\right)+\cdots,$
ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cos\NVar{z}$ : cosine function, $q=h^{2}$ : parameter and $z$ : complex variable A&S Ref: 20.2.27 Referenced by: §28.6(ii) Permalink: http://dlmf.nist.gov/28.6.E21 Encodings: TeX, pMML, png See also: Annotations for §28.6(ii), §28.6 and Ch.28
28.6.22	$\displaystyle\operatorname{ce}_{1}\left(z,q\right)$	$\displaystyle=\cos z-\tfrac{1}{8}q\cos 3z+\tfrac{1}{128}q^{2}\left(\tfrac{2}{3% }\cos 5z-2\cos 3z-\cos z\right)-\tfrac{1}{1024}q^{3}\left(\tfrac{1}{9}\cos 7z-% \tfrac{8}{9}\cos 5z-\tfrac{1}{3}\cos 3z+2\cos z\right)+\cdots,$
ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cos\NVar{z}$ : cosine function, $q=h^{2}$ : parameter and $z$ : complex variable Permalink: http://dlmf.nist.gov/28.6.E22 Encodings: TeX, pMML, png See also: Annotations for §28.6(ii), §28.6 and Ch.28
28.6.23	$\displaystyle\operatorname{se}_{1}\left(z,q\right)$	$\displaystyle=\sin z-\tfrac{1}{8}q\sin 3z+\tfrac{1}{128}q^{2}\left(\tfrac{2}{3% }\sin 5z+2\sin 3z-\sin z\right)-\tfrac{1}{1024}q^{3}\left(\tfrac{1}{9}\sin 7z+% \tfrac{8}{9}\sin 5z-\tfrac{1}{3}\sin 3z-2\sin z\right)+\cdots,$
ⓘ Symbols: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\sin\NVar{z}$ : sine function, $q=h^{2}$ : parameter and $z$ : complex variable Permalink: http://dlmf.nist.gov/28.6.E23 Encodings: TeX, pMML, png See also: Annotations for §28.6(ii), §28.6 and Ch.28
28.6.24	$\displaystyle\operatorname{ce}_{2}\left(z,q\right)$	$\displaystyle=\cos 2z-\tfrac{1}{4}q\left(\tfrac{1}{3}\cos 4z-1\right)+\tfrac{1% }{128}q^{2}\left(\tfrac{1}{3}\cos 6z-\tfrac{76}{9}\cos 2z\right)+\cdots,$
ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cos\NVar{z}$ : cosine function, $q=h^{2}$ : parameter and $z$ : complex variable Permalink: http://dlmf.nist.gov/28.6.E24 Encodings: TeX, pMML, png See also: Annotations for §28.6(ii), §28.6 and Ch.28
28.6.25	$\displaystyle\operatorname{se}_{2}\left(z,q\right)$	$\displaystyle=\sin 2z-\tfrac{1}{12}q\sin 4z+\tfrac{1}{128}q^{2}\left(\tfrac{1}% {3}\sin 6z-\tfrac{4}{9}\sin 2z\right)+\cdots.$
ⓘ Symbols: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\sin\NVar{z}$ : sine function, $q=h^{2}$ : parameter and $z$ : complex variable Permalink: http://dlmf.nist.gov/28.6.E25 Encodings: TeX, pMML, png See also: Annotations for §28.6(ii), §28.6 and Ch.28

For $m=3,4,5,\dots$ ,

28.6.26		$\operatorname{ce}_{m}\left(z,q\right)=\cos mz-\frac{q}{4}\left(\frac{1}{m+1}% \cos(m+2)z-\frac{1}{m-1}\cos(m-2)z\right)+\frac{q^{2}}{32}\left(\frac{1}{(m+1)% (m+2)}\cos(m+4)z+\frac{1}{(m-1)(m-2)}\cos(m-4)z-\frac{2(m^{2}+1)}{(m^{2}-1)^{2% }}\cos mz\right)+\cdots.$
ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cos\NVar{z}$ : cosine function, $m$ : integer, $q=h^{2}$ : parameter and $z$ : complex variable A&S Ref: 20.2.28 (in slightly different form) Referenced by: §28.6(ii), §28.6(ii), §28.6(ii), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii) Permalink: http://dlmf.nist.gov/28.6.E26 Encodings: TeX, pMML, png See also: Annotations for §28.6(ii), §28.6 and Ch.28

For the corresponding expansions of $\operatorname{se}_{m}\left(z,q\right)$ for $m=3,4,5,\dots$ 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).