# power series in q

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## 1—10 of 26 matching pages

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

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

… ►###### §28.15(ii) Solutions ${\mathrm{me}}_{\nu}(z,q)$

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

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###### §28.6(i) Eigenvalues

… ►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. … ►
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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###### §28.6(ii) Functions ${\mathrm{ce}}_{n}$ and ${\mathrm{se}}_{n}$

…##### 3: 28.2 Definitions and Basic Properties

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►Near $q=0$, ${a}_{n}\left(q\right)$ and ${b}_{n}\left(q\right)$ can be expanded in power series in
$q$ (see §28.6(i)); elsewhere they are determined by analytic continuation (see §28.7).
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##### 4: 28.34 Methods of Computation

##### 5: 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 ${q}^{1/12}={\mathrm{e}}^{\mathrm{i}\pi \tau /12}$.
##### 6: 20.11 Generalizations and Analogs

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►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).
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##### 7: 17.18 Methods of Computation

###### §17.18 Methods of Computation

… ►The two main methods for computing basic hypergeometric functions are: (1) numerical summation of the defining series given in §§17.4(i) and 17.4(ii); (2) modular transformations. Method (1) is applicable within the circles of convergence of the defining series, although it is often cumbersome owing to slowness of convergence and/or severe cancellation. Method (2) is very powerful when applicable (Andrews (1976, Chapter 5)); however, it is applicable only rarely. … ►Shanks (1955) applies such methods in several $q$-series problems; see Andrews et al. (1986).##### 8: 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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##### 9: 16.8 Differential Equations

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►For other values of the ${b}_{j}$, series solutions in powers of $z$ (possibly involving also $\mathrm{ln}z$) can be constructed via a limiting process; compare §2.7(i) in the case of second-order differential equations.
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►In this reference it is also explained that in general when $q>1$ no simple representations in terms of generalized hypergeometric functions are available for the fundamental solutions near $z=1$.
Analytical continuation formulas for ${}_{q+1}{}^{}F_{q}^{}(\mathbf{a};\mathbf{b};z)$ near $z=1$ are given in Bühring (1987b) for the case $q=2$, and in Bühring (1992) for the general case.
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►If $p\le q$, then
…Thus in the case $p=q$ the regular singularities of the function on the left-hand side at $\alpha $ and $\mathrm{\infty}$ coalesce into an irregular singularity at $\mathrm{\infty}$.
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##### 10: 3.10 Continued Fractions

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

… ►###### Stieltjes Fractions

… ►We say that it*corresponds*to the formal power series … ►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 ${z}^{2n-1}$, $n=1,2,3,\mathrm{\dots}$. …