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1: 4.6 Power Series
Binomial Expansion
Note that (4.6.7) is a generalization of the binomial expansion (1.2.2) with the binomial coefficients defined in (1.2.6).
2: 28.8 Asymptotic Expansions for Large q
28.8.4 U m ( ξ ) D m ( ξ ) - 1 2 6 h ( D m + 4 ( ξ ) - 4 ! ( m 4 ) D m - 4 ( ξ ) ) + 1 2 13 h 2 ( D m + 8 ( ξ ) - 2 5 ( m + 2 ) D m + 4 ( ξ ) + 4 !  2 5 ( m - 1 ) ( m 4 ) D m - 4 ( ξ ) + 8 ! ( m 8 ) D m - 8 ( ξ ) ) + ,
28.8.5 V m ( ξ ) 1 2 4 h ( - D m + 2 ( ξ ) - m ( m - 1 ) D m - 2 ( ξ ) ) + 1 2 10 h 2 ( D m + 6 ( ξ ) + ( m 2 - 25 m - 36 ) D m + 2 ( ξ ) - m ( m - 1 ) ( m 2 + 27 m - 10 ) D m - 2 ( ξ ) - 6 ! ( m 6 ) D m - 6 ( ξ ) ) + ,
3: 8.17 Incomplete Beta Functions
8.17.5 I x ( m , n - m + 1 ) = j = m n ( n j ) x j ( 1 - x ) n - j , m , n positive integers; 0 x < 1 .
4: 2.6 Distributional Methods
2.6.6 S ( x ) 2 π 3 s = 0 ( - 1 ) s ( - 1 3 s ) x - s - ( 1 / 3 ) , x .
2.6.7 S ( x ) 2 π 3 s = 0 ( - 1 ) s ( - 1 3 s ) x - s - ( 1 / 3 ) - s = 1 3 s ( s - 1 ) ! 2 5 ( 3 s - 1 ) x - s ;
5: 2.10 Sums and Sequences
2.10.7 j = 1 n - 1 j α ζ ( - α ) + n α + 1 α + 1 s = 0 ( α + 1 s ) B s n s , n ,
6: 1.10 Functions of a Complex Variable
Note that (1.10.4) is a generalization of the binomial expansion (1.2.2) with the binomial coefficients defined in (1.2.6). …
7: 24.4 Basic Properties
§24.4(iv) Finite Expansions
24.4.12 B n ( x + h ) = k = 0 n ( n k ) B k ( x ) h n - k ,
24.4.13 E n ( x + h ) = k = 0 n ( n k ) E k ( x ) h n - k ,
24.4.14 E n - 1 ( x ) = 2 n k = 0 n ( n k ) ( 1 - 2 k ) B k x n - k ,
24.4.15 B 2 n = 2 n 2 2 n ( 2 2 n - 1 ) k = 0 n - 1 ( 2 n - 1 2 k ) E 2 k ,
8: 18.18 Sums
§18.18(i) Series Expansions of Arbitrary Functions
Legendre
Laguerre
Hermite
Ultraspherical
9: 2.11 Remainder Terms; Stokes Phenomenon
§2.11(i) Numerical Use of Asymptotic Expansions
§2.11(iii) Exponentially-Improved Expansions
In this way we arrive at hyperasymptotic expansions. …
10: 13.8 Asymptotic Approximations for Large Parameters
For the parabolic cylinder function U see §12.2, and for an extension to an asymptotic expansion see Temme (1978). … For other asymptotic expansions for large b and z see López and Pagola (2010). For more asymptotic expansions for the cases b ± see Temme (2015, §§10.4 and 22.5)For an extension to an asymptotic expansion complete with error bounds see Temme (1990b), and for related results see §13.21(i). …
p k ( z ) = s = 0 k ( k s ) ( 1 - b + s ) k - s z s c k + s ( z ) ,