expansions in series of

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11: 31.11 Expansions in Series of Hypergeometric Functions

§31.11 Expansions in Series of Hypergeometric Functions

… ►Series of Type II (§31.11(iv)) are expansions in orthogonal polynomials, which are useful in calculations of normalization integrals for Heun functions; see Erdélyi (1944) and §31.9(i). … ►

$\mu =\gamma +\delta -2.$

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§31.11(v) Doubly-Infinite Series

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12: 18.18 Sums

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Legendre

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Laguerre

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Hermite

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Ultraspherical

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Hermite

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13: 28.23 Expansions in Series of Bessel Functions

§28.23 Expansions in Series of Bessel Functions

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14: 9.19 Approximations

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§9.19(ii) Expansions in Chebyshev Series

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MacLeod (1994) supplies Chebyshev-series expansions to cover $\mathrm{Gi}\left(x\right)$ for and $\mathrm{Hi}\left(x\right)$ for . The Chebyshev coefficients are given to 20D.

15: 33.19 Power-Series Expansions in $r$

§33.19 Power-Series Expansions in $r$

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16: 33.6 Power-Series Expansions in $\rho$

§33.6 Power-Series Expansions in $\rho$

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17: 30.8 Expansions in Series of Ferrers Functions

§30.8 Expansions in Series of Ferrers Functions

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30.8.6 $$a_{n,k}^{-m}({\gamma }^2)=\frac{(n-m)!(n+m+2k)!}{(n+m)!(n-m+2k)!}{a}_{n,k}^m({\gamma }^2).$$

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Symbols:

$!$: factorial (as in $n!$), $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter and $a_{n,k}^m({\gamma }^2)$: coefficients

Referenced by:

§30.11(i)

Permalink:

http://dlmf.nist.gov/30.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §30.8(i), §30.8 and Ch.30

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18: 28.15 Expansions for Small $q$

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

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19: 30.17 Tables

§30.17 Tables

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20: 8.25 Methods of Computation

… ►Although the series expansions in §§8.7, 8.19(iv), and 8.21(vi) converge for all finite values of $z$, they are cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. …