series expansions

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11: 35.10 Methods of Computation

… ►Koev and Edelman (2006) utilizes combinatorial identities for the zonal polynomials to develop computational algorithms for approximating the series expansion (35.8.1). …

12: 8.7 Series Expansions

§8.7 Series Expansions

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8.7.6 $$\mathrm{\Gamma }(a,x)=x^a{\mathrm{e}}^{-x}\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(a)}\left(x\right)}{n+1},$$ $x>0$, .

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{\Re }$: real part, $x$: real variable, $z$: complex variable, $a$: parameter and $n$: nonnegative integer

Proof sketch:

Integrate (18.12.13) from $z=0$ to $z=1$ and for the integral on the left-hand side use the substitution $z=\frac{t}{1+t}$. The resulting integral is (8.6.5). The constraint follows from (18.15.14).

Referenced by:

Erratum (V1.1.8) for Equation (8.7.6)

Permalink:

http://dlmf.nist.gov/8.7.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.1.8):

The constraint was updated to include .

Suggested 2022-10-14 by Walter Gautschi

See also:

Annotations for §8.7 and Ch.8

►For an expansion for $\gamma (a,\mathrm{i}x)$ in series of Bessel functions $J_n\left(x\right)$ that converges rapidly when $a>0$ and $x$ ($\ge 0$) is small or moderate in magnitude see Barakat (1961).

13: 13.31 Approximations

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§13.31(i) Chebyshev-Series Expansions

►Luke (1969b, pp. 35 and 25) provides Chebyshev-series expansions of $M(a,b,x)$ and $U(a,b,x)$ that include the intervals $0\le x\le \alpha$ and , respectively, where $\alpha$ is an arbitrary positive constant. …

14: 7.6 Series Expansions

§7.6 Series Expansions

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§7.6(i) Power Series

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§7.6(ii) Expansions in Series of Spherical Bessel Functions

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15: 11.13 Methods of Computation

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§11.13(ii) Series Expansions

►Although the power-series expansions (11.2.1) and (11.2.2), and the Bessel-function expansions of §11.4(iv) converge for all finite values of $z$, they are cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. …

16: 25.20 Approximations

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Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of $s\zeta \left(s+1\right)$ and $\zeta \left(s+k\right)$, $k=2,3,4,5,8$, for $0\le s\le 1$ (23D).

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Luke (1969b, p. 306) gives coefficients in Chebyshev-series expansions that cover $\zeta \left(s\right)$ for $0\le s\le 1$ (15D), $\zeta \left(s+1\right)$ for $0\le s\le 1$ (20D), and $\ln \xi \left(\frac{1}{2}+\mathrm{i}x\right)$ (§25.4) for $-1\le x\le 1$ (20D). For errata see Piessens and Branders (1972).

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17: 14.32 Methods of Computation

… ►In particular, for small or moderate values of the parameters $\mu$ and $\nu$ the power-series expansions of the various hypergeometric function representations given in §§14.3(i)–14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real. …

18: 19.38 Approximations

… ►Cody (1965b) gives Chebyshev-series expansions (§3.11(ii)) with maximum precision 25D. …

19: 6.16 Mathematical Applications

… ►Compare Figure 6.16.1. … ►It occurs with Fourier-series expansions of all piecewise continuous functions. … …

20: 30.10 Series and Integrals

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