# power series in q

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## 1—10 of 26 matching pages

##### 2: 28.6 Expansions for Small $q$
###### §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 $a-(2n+2)^{2}-\cfrac{q^{2}}{a-(2n)^{2}-\cfrac{q^{2}}{a-(2n-2)^{2}-}}\cdots% \cfrac{q^{2}}{a-2^{2}}=-\cfrac{q^{2}}{(2n+4)^{2}-a-\cfrac{q^{2}}{(2n+6)^{2}-a-% \cdots}},$ $a=b_{2n+2}\left(q\right)$.
##### 3: 28.2 Definitions and Basic Properties
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). …
##### 4: 28.34 Methods of Computation
• (a)

Summation of the power series in §§28.6(i) and 28.15(i) when $\left|q\right|$ is small.

• (a)

Summation of the power series in §§28.6(ii) and 28.15(ii) when $\left|q\right|$ is small.

• ##### 5: 23.17 Elementary Properties
###### §23.17(ii) Power and Laurent Series
When $|q|<1$ In (23.17.5) for terms up to $q^{48}$ see Zuckerman (1939), and for terms up to $q^{100}$ see van Wijngaarden (1953). … with $q^{1/12}=e^{i\pi\tau/12}$.
##### 6: 20.11 Generalizations and Analogs
In the case $z=0$ identities for theta functions become identities in the complex variable $q$, with $\left|q\right|<1$, 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). …
##### 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
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 $|\operatorname{ph}\left(-z\right)|\leq(p+1-q-\delta)\pi/2$, where $\delta$ is an arbitrary small positive constant. …
##### 9: 16.8 Differential Equations
For other values of the $b_{j}$, series solutions in powers of $z$ (possibly involving also $\ln z$) can be constructed via a limiting process; compare §2.7(i) in the case of second-order differential equations. … 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 ${{}_{q+1}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)$ near $z=1$ are given in Bühring (1987b) for the case $q=2$, and in Bühring (1992) for the general case. … If $p\leq q$, then …Thus in the case $p=q$ the regular singularities of the function on the left-hand side at $\alpha$ and $\infty$ coalesce into an irregular singularity at $\infty$. …
##### 10: 3.10 Continued Fractions
###### Stieltjes Fractions
We say that it corresponds to the formal power seriesFor 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 $z^{2n-1}$, $n=1,2,3,\dots$. …