# power-series expansions in q

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## 1—10 of 16 matching pages

##### 1: 28.15 Expansions for Small $q$

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###### §28.15(i) Eigenvalues ${\lambda}_{\nu}\left(q\right)$

…##### 2: 28.6 Expansions for Small $q$

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###### §28.6(i) Eigenvalues

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28.6.19
$$a-{(2n+2)}^{2}-\frac{{q}^{2}}{a-{(2n)}^{2}-}\frac{{q}^{2}}{a-{(2n-2)}^{2}-}\mathrm{\cdots}\frac{{q}^{2}}{a-{2}^{2}}=-\frac{{q}^{2}}{{(2n+4)}^{2}-a-}\frac{{q}^{2}}{{(2n+6)}^{2}-a-}\mathrm{\cdots},$$
$a={b}_{2n+2}\left(q\right)$.

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##### 3: 28.34 Methods of Computation

##### 4: 23.17 Elementary Properties

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###### §23.17(ii) Power and Laurent Series

►When $$ ►
23.17.4
$$\lambda \left(\tau \right)=16q(1-8q+44{q}^{2}+\mathrm{\cdots}),$$

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►In (23.17.5) for terms up to ${q}^{48}$ see Zuckerman (1939), and for terms up to ${q}^{100}$ see van Wijngaarden (1953).
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►with ${q}^{1/12}={\mathrm{e}}^{\mathrm{i}\pi \tau /12}$.
##### 5: 16.5 Integral Representations and Integrals

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►In this event, the formal power-series expansion of the left-hand side (obtained from (16.2.1)) is the asymptotic expansion of the right-hand side as $z\to 0$
in the sector $|\mathrm{ph}\left(-z\right)|\le (p+1-q-\delta )\pi /2$, where $\delta $ is an arbitrary small positive constant.
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##### 6: 3.10 Continued Fractions

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###### §3.10(ii) Relations to Power Series

… ►We say that it*corresponds*to the formal power series …if the expansion of its $n$th convergent ${C}_{n}$ in ascending powers of $z$ agrees with (3.10.7) up to and including the term in ${z}^{n-1}$, $n=1,2,3,\mathrm{\dots}$. … ►For several special functions the $S$-fractions are known explicitly, but in any case the coefficients ${a}_{n}$ can always be calculated from the power-series coefficients by means of the*quotient-difference algorithm*; see Table 3.10.1. … ►We say that it is*associated*with the formal power series $f(z)$ in (3.10.7) if the expansion of its $n$th convergent ${C}_{n}$ in ascending powers of $z$, agrees with (3.10.7) up to and including the term in ${z}^{2n-1}$, $n=1,2,3,\mathrm{\dots}$. …##### 7: 19.5 Maclaurin and Related Expansions

###### §19.5 Maclaurin and Related Expansions

… ► … ►Series expansions of $F(\varphi ,k)$ and $E(\varphi ,k)$ are surveyed and improved in Van de Vel (1969), and the case of $F(\varphi ,k)$ is summarized in Gautschi (1975, §1.3.2). For series expansions of $\mathrm{\Pi}(\varphi ,{\alpha}^{2},k)$ when $$ see Erdélyi et al. (1953b, §13.6(9)). …##### 8: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

###### §28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

… ►With ${\mathcal{C}}_{\mu}^{(j)}$, ${c}_{n}^{\nu}(q)$, ${A}_{n}^{m}(q)$, and ${B}_{n}^{m}(q)$ as in §28.23, … ►In the case when $\nu $ is an integer, … ►The expansions (28.24.1)–(28.24.13) converge absolutely and uniformly on compact sets of the $z$-plane. ►For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).##### 9: 3.11 Approximation Techniques

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###### §3.11(ii) Chebyshev-Series Expansions

… ►In fact, (3.11.11) is the Fourier-series expansion of $f(\mathrm{cos}\theta )$; compare (3.11.6) and §1.8(i). … ►For further details on Chebyshev-series expansions in the complex plane, see Mason and Handscomb (2003, §5.10). … ►be a formal power series. … ►When $F$ has an explicit power-series expansion a possible choice of $R$ is a Padé approximation to $F$. …##### 10: 20.11 Generalizations and Analogs

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###### §20.11(ii) Ramanujan’s Theta Function and $q$-Series

… ►In the case $z=0$ identities for theta functions become identities in the complex variable $q$, with $$, that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … ►As in §20.11(ii), the modulus $k$ of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in $q$-series via (20.9.1). However, in this case $q$ is no longer regarded as an independent complex variable within the unit circle, because $k$ is related to the variable $\tau =\tau (k)$ of the theta functions via (20.9.2). … ►For applications to rapidly convergent expansions for $\pi $ see Chudnovsky and Chudnovsky (1988), and for applications in the construction of*elliptic-hypergeometric series*see Rosengren (2004). …