power series

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41: 22.17 Moduli Outside the Interval [0,1]

… ►In terms of the coefficients of the power series of §22.10(i), the above equations are polynomial identities in $k$. …

42: 22.15 Inverse Functions

… ►For power-series expansions see Carlson (2008).

43: 29.3 Definitions and Basic Properties

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§29.3(vii) Power Series

►For power-series expansions of the eigenvalues see Volkmer (2004b).

44: 18.3 Definitions

… ►For finite power series of the Jacobi, ultraspherical, Laguerre, and Hermite polynomials, see §18.5(iii) (in powers of $x-1$ for Jacobi polynomials, in powers of $x$ for the other cases). Explicit power series for Chebyshev, Legendre, Laguerre, and Hermite polynomials for $n=0,1,\mathrm{\dots },6$ are given in §18.5(iv). For explicit power series coefficients up to $n=12$ for these polynomials and for coefficients up to $n=6$ for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). …

45: 13.14 Definitions and Basic Properties

… ►The series … ►

13.14.9 $$W_{\kappa ,\pm \frac{1}{2}n}\left(z\right)={(-1)}^{\kappa -\frac{1}{2}n-\frac{1}{2}}{\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}n+\frac{1}{2}}\sum _{k=0}^{\kappa -\frac{1}{2}n-\frac{1}{2}}\left(\genfrac{}{}{0.0pt}{}{\kappa -\frac{1}{2}n-\frac{1}{2}}{k}\right){\left(n+1+k\right)}_{\kappa -k-\frac{1}{2}n-\frac{1}{2}}{(-z)}^k,$$ $\kappa -\frac{1}{2}n-\frac{1}{2}=0,1,2,\mathrm{\dots }$.

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.14(i)

Permalink:

http://dlmf.nist.gov/13.14.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

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46: 7.18 Repeated Integrals of the Complementary Error Function

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47: 27.13 Functions

… ►Mordell (1917) notes that $r_k\left(n\right)$ is the coefficient of $x^n$ in the power-series expansion of the $k$th power of the series for $\vartheta \left(x\right)$. …

48: 8.21 Generalized Sine and Cosine Integrals

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Power-Series Expansions

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49: 12.14 The Function $W(a,x)$

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§12.14(v) Power-Series Expansions

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50: 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). …