power-series expansions 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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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}-\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)

ⓘ

Symbols:: $b_{\NVar{n}}\left(\NVar{q}\right)$ : 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:: pMML, png, TeX
See also:: Annotations for §28.6(i), §28.6 and Ch.28

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3: 28.34 Methods of Computation

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(a)

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

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(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)).

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(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 $z$ .

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(c)

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

4: 23.17 Elementary Properties

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§23.17(ii) Power and Laurent Series

►When

|q|<1

►

23.17.4

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

ⓘ

Symbols:: $\lambda\left(\NVar{\tau}\right)$ : 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}=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

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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,\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 Expansions in Series 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

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§3.11(ii) Chebyshev-Series Expansions

… ►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

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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

\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). …