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1: 16.11 Asymptotic Expansions

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§16.11(i) Formal Series

►For subsequent use we define two formal infinite series, $E_{p,q}(z)$ and $H_{p,q}(z)$, as follows: ►

16.11.1 $$E_{p,q}(z)={(2\pi )}^{(p-q)/2}{\kappa }^{-\nu -(1/2)}{\mathrm{e}}^{\kappa z^{1/\kappa }}\sum _{k=0}^{\mathrm{\infty }}{c}_k{\left(\kappa z^{1/\kappa }\right)}^{\nu -k},$$ ,

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Defines:

$E_{p,q}(z)$: formal infinite series (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $\kappa$ and $\nu$

Referenced by:

§16.11(i), §16.11(ii)

Permalink:

http://dlmf.nist.gov/16.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(i), §16.11 and Ch.16

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16.11.2 $$H_{p,q}(z)=\sum _{m=1}^p\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k}{k!}\mathrm{\Gamma }\left(a_m+k\right)\left(\prod _{\begin{array}{c}\mathrm{\ell }=1\\ \mathrm{\ell }\ne m\end{array}}^p\mathrm{\Gamma }\left(a_{\mathrm{\ell }}-a_m-k\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-a_m-k\right)\right)z^{-a_m-k}.$$

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Defines:

$H_{p,q}(z)$: formal infinite series (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.11(ii), §16.11(ii)

Permalink:

http://dlmf.nist.gov/16.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(i), §16.11 and Ch.16

… ►The formal series (16.11.2) for $H_{q+1,q}(z)$ converges if $\left|z\right|>1$, and …

2: 10.70 Zeros

… ►Let $\mu =4{\nu }^2$ and $f(t)$ denote the formal series …

3: 3.10 Continued Fractions

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

4: 1.17 Integral and Series Representations of the Dirac Delta

… ►By analogy with §1.17(ii) we have the formal series representation …

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 $|\mathrm{ph}\left(-z\right)|\le (p+1-q-\delta )\pi /2$, where $\delta$ is an arbitrary small positive constant. …

6: 1.8 Fourier Series

§1.8 Fourier Series

… ►Formally, if $f(x)$ is a real- or complex-valued $2\pi$-periodic function, … … ►

Uniqueness of Fourier Series

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7: 2.1 Definitions and Elementary Properties

… ►Let $\sum a_s{x}^{-s}$ be a formal power series (convergent or divergent) and for each positive integer $n$, …

8: Bibliography B

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Formal Power Series and Algebraic Combinatorics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 24, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-0324-7, MathReview, ZentralBlatt

Referenced by:

§26.19.

Encodings:

BibTeX

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9: 18.2 General Orthogonal Polynomials

… ►where $f(t)$ and $u(t)$ are formal power series in $t$, with $f(0)=1$, $u(0)=0$ and $u^{\prime }(0)=1$. …If $v(s)$ is the formal power series such that $v(u(t))=t$ then a property equivalent to (18.2.45) with $c_n=1$ is that …

10: 16.4 Argument Unity

… ►Contiguous balanced series have parameters shifted by an integer but still balanced. … … ►when the series on the right terminates and the series on the left converges. … ►

§16.4(v) Bilateral Series

►Denote, formally, the bilateral hypergeometric function …