power series in q

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1: 28.15 Expansions for Small $q$

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

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

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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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Symbols:

$b_n\left(q\right)$: 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:

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See also:

Annotations for §28.6(i), §28.6 and Ch.28

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§28.6(ii) Functions ${\mathrm{ce}}_n$ and ${\mathrm{se}}_n$

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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

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

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

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5: 23.17 Elementary Properties

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§23.17(ii) Power and Laurent Series

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

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Symbols:

$\lambda \left(\tau \right)$: elliptic modular function, $q$: nome and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §23.17(ii), §23.17 and Ch.23

… ►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 $q^{1/12}={\mathrm{e}}^{\mathrm{i}\pi \tau /12}$.

6: 20.11 Generalizations and Analogs

… ►In the case $z=0$ identities for theta functions become identities in the complex variable $q$, with , 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 $|\mathrm{ph}\left(-z\right)|\le (p+1-q-\delta )\pi /2$, 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 ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;z)$ near $z=1$ are given in Bühring (1987b) for the case $q=2$, and in Bühring (1992) for the general case. … ►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 $\mathrm{\infty }$. …

10: 3.10 Continued Fractions

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§3.10(ii) Relations to Power Series

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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 $z^{2n-1}$, $n=1,2,3,\mathrm{\dots }$. …