# power-series expansions in q

(0.006 seconds)

## 1—10 of 16 matching pages

##### 2: 28.6 Expansions for Small $q$
###### §28.6(i) Eigenvalues
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: 28.34 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.

• (b)

Use of asymptotic expansions and approximations for large $q$ (§§28.8(ii)28.8(iv)).

• (a)

Numerical summation of the expansions in series of Bessel functions (28.24.1)–(28.24.13). These series converge quite rapidly for a wide range of values of $q$ and $z$.

• (c)

Use of asymptotic expansions for large $z$ or large $q$. See §§28.25 and 28.26.

• ##### 4: 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 $q^{1/12}=e^{i\pi\tau/12}$.
##### 5: 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. …
##### 6: 3.10 Continued Fractions
###### §3.10(ii) Relations to PowerSeries
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 $z^{n-1}$, $n=1,2,3,\dots$. … 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 $z^{2n-1}$, $n=1,2,3,\dots$. …
##### 7: 19.5 Maclaurin and Related Expansions
###### §19.5 Maclaurin and Related Expansions
Series expansions of $F\left(\phi,k\right)$ and $E\left(\phi,k\right)$ are surveyed and improved in Van de Vel (1969), and the case of $F\left(\phi,k\right)$ is summarized in Gautschi (1975, §1.3.2). For series expansions of $\Pi\left(\phi,\alpha^{2},k\right)$ when $|\alpha^{2}|<1$ 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 ExpansionsinSeries of Cross-Products of Bessel Functions or Modified Bessel Functions
With ${\cal C}_{\mu}^{(j)}$, $c^{\nu}_{n}(q)$, $A_{n}^{m}(q)$, 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
###### §3.11(ii) Chebyshev-SeriesExpansions
In fact, (3.11.11) is the Fourier-series expansion of $f(\cos\theta)$; 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
###### §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 $\left|q\right|<1$, 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). … 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). …