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1: 4.17 Special Values and Limits
Table 4.17.1: Trigonometric functions: values at multiples of 1 12 π .
θ sin θ cos θ tan θ csc θ sec θ cot θ
π / 6 1 2 1 2 3 1 3 3 2 2 3 3 3
π / 3 1 2 3 1 2 3 2 3 3 2 1 3 3
2 π / 3 1 2 3 1 2 3 2 3 3 2 1 3 3
4.17.3 lim z 0 1 cos z z 2 = 1 2 .
2: 18.13 Continued Fractions
18.13.1 1 x + 1 2 x + 1 2 x + ,
18.13.2 1 2 x + 1 2 x + 1 2 x + .
18.13.3 a 1 x + 1 2 3 2 x + 2 3 5 3 x + 3 4 7 4 x + ,
18.13.4 a 1 1 x + 1 2 1 2 ( 3 x ) + 2 3 1 3 ( 5 x ) + 3 4 1 4 ( 7 x ) + ,
18.13.5 1 2 x + 2 2 x + 4 2 x + 6 2 x + .
3: 24.2 Definitions and Generating Functions
Table 24.2.5: Coefficients b n , k of the Bernoulli polynomials B n ( x ) = k = 0 n b n , k x k .
k
5 0 1 6 0 5 3 5 2 1
6 1 42 0 1 2 0 5 2 3 1
14 7 6 0 691 30 0 455 6 0 1001 10 0 143 2 0 1001 30 0 91 6 7 1
Table 24.2.6: Coefficients e n , k of the Euler polynomials E n ( x ) = k = 0 n e n , k x k .
k
5 1 2 0 5 2 0 5 2 1
13 5461 2 0 26949 2 0 22165 2 0 7293 2 0 1287 2 0 143 2 0 13 2 1
4: 12.7 Relations to Other Functions
12.7.8 U ( 2 , z ) = z 5 / 2 4 2 π ( 2 K 1 4 ( 1 4 z 2 ) + 3 K 3 4 ( 1 4 z 2 ) K 5 4 ( 1 4 z 2 ) ) ,
For these, the corresponding results for U ( a , z ) with a = 2 , ± 3 , 1 2 , 3 2 , 5 2 , and the corresponding results for V ( a , z ) with a = 0 , ± 1 , ± 2 , ± 3 , 1 2 , 3 2 , 5 2 , see Miller (1955, pp. 42–43 and 77–79). …
12.7.12 u 1 ( a , z ) = e 1 4 z 2 M ( 1 2 a + 1 4 , 1 2 , 1 2 z 2 ) = e 1 4 z 2 M ( 1 2 a + 1 4 , 1 2 , 1 2 z 2 ) ,
12.7.13 u 2 ( a , z ) = z e 1 4 z 2 M ( 1 2 a + 3 4 , 3 2 , 1 2 z 2 ) = z e 1 4 z 2 M ( 1 2 a + 3 4 , 3 2 , 1 2 z 2 ) .
12.7.14 U ( a , z ) = 2 1 4 1 2 a e 1 4 z 2 U ( 1 2 a + 1 4 , 1 2 , 1 2 z 2 ) = 2 3 4 1 2 a z e 1 4 z 2 U ( 1 2 a + 3 4 , 3 2 , 1 2 z 2 ) = 2 1 2 a z 1 2 W 1 2 a , ± 1 4 ( 1 2 z 2 ) .
5: 26.3 Lattice Paths: Binomial Coefficients
( m n ) is the number of ways of choosing n objects from a collection of m distinct objects without regard to order. ( m + n n ) is the number of lattice paths from ( 0 , 0 ) to ( m , n ) . …The number of lattice paths from ( 0 , 0 ) to ( m , n ) , m n , that stay on or above the line y = x is ( m + n m ) ( m + n m 1 ) . For numerical values of ( m n ) and ( m + n n ) see Tables 26.3.1 and 26.3.2.
Table 26.3.1: Binomial coefficients ( m n ) .
m n
6: 13.15 Recurrence Relations and Derivatives
13.15.3 ( κ μ 1 2 ) M κ 1 2 , μ + 1 2 ( z ) + ( 1 + 2 μ ) z M κ , μ ( z ) ( κ + μ + 1 2 ) M κ + 1 2 , μ + 1 2 ( z ) = 0 ,
13.15.12 ( κ μ 1 2 ) z W κ 1 2 , μ + 1 2 ( z ) + 2 μ W κ , μ ( z ) ( κ + μ 1 2 ) z W κ 1 2 , μ 1 2 ( z ) = 0 ,
13.15.16 d n d z n ( e 1 2 z z μ 1 2 M κ , μ ( z ) ) = ( 1 2 + μ κ ) n ( 1 + 2 μ ) n e 1 2 z z μ 1 2 ( n + 1 ) M κ 1 2 n , μ + 1 2 n ( z ) ,
13.15.24 d n d z n ( e 1 2 z z μ 1 2 W κ , μ ( z ) ) = ( 1 ) n e 1 2 z z μ 1 2 ( n + 1 ) W κ + 1 2 n , μ + 1 2 n ( z ) ,
13.15.25 d n d z n ( e 1 2 z z μ 1 2 W κ , μ ( z ) ) = ( 1 ) n e 1 2 z z μ 1 2 ( n + 1 ) W κ + 1 2 n , μ 1 2 n ( z ) ,
7: 12.13 Sums
12.13.2 U ( a , x + y ) = e 1 2 x y 1 4 y 2 m = 0 ( a 1 2 m ) y m U ( a + m , x ) ,
12.13.3 V ( a , x + y ) = e 1 2 x y + 1 4 y 2 m = 0 ( a 1 2 m ) y m V ( a m , x ) ,
12.13.4 V ( a , x + y ) = e 1 2 x y 1 4 y 2 m = 0 y m m ! V ( a + m , x ) .
12.13.5 U ( a , x cos t + y sin t ) = e 1 4 ( x sin t y cos t ) 2 m = 0 ( a 1 2 m ) ( tan t ) m U ( m + a , x ) U ( m 1 2 , y ) , a 1 2 , 0 t 1 4 π .
12.13.6 n ! U ( n + 1 2 , z ) = i n e 1 2 z 2 erfc ( z / 2 ) U ( n 1 2 , i z ) + m = 1 1 2 n + 1 2 U ( 2 m n 1 2 , z ) , n = 0 , 1 , 2 , .
8: 12.4 Power-Series Expansions
12.4.3 u 1 ( a , z ) = e 1 4 z 2 ( 1 + ( a + 1 2 ) z 2 2 ! + ( a + 1 2 ) ( a + 5 2 ) z 4 4 ! + ) ,
12.4.4 u 2 ( a , z ) = e 1 4 z 2 ( z + ( a + 3 2 ) z 3 3 ! + ( a + 3 2 ) ( a + 7 2 ) z 5 5 ! + ) .
12.4.5 u 1 ( a , z ) = e 1 4 z 2 ( 1 + ( a 1 2 ) z 2 2 ! + ( a 1 2 ) ( a 5 2 ) z 4 4 ! + ) ,
12.4.6 u 2 ( a , z ) = e 1 4 z 2 ( z + ( a 3 2 ) z 3 3 ! + ( a 3 2 ) ( a 7 2 ) z 5 5 ! + ) .
9: 10.57 Uniform Asymptotic Expansions for Large Order
Asymptotic expansions for j n ( ( n + 1 2 ) z ) , y n ( ( n + 1 2 ) z ) , h n ( 1 ) ( ( n + 1 2 ) z ) , h n ( 2 ) ( ( n + 1 2 ) z ) , i n ( 1 ) ( ( n + 1 2 ) z ) , and k n ( ( n + 1 2 ) z ) as n that are uniform with respect to z can be obtained from the results given in §§10.20 and 10.41 by use of the definitions (10.47.3)–(10.47.7) and (10.47.9). Subsequently, for i n ( 2 ) ( ( n + 1 2 ) z ) the connection formula (10.47.11) is available. For the corresponding expansion for j n ( ( n + 1 2 ) z ) use
10.57.1 j n ( ( n + 1 2 ) z ) = π 1 2 ( ( 2 n + 1 ) z ) 1 2 J n + 1 2 ( ( n + 1 2 ) z ) π 1 2 ( ( 2 n + 1 ) z ) 3 2 J n + 1 2 ( ( n + 1 2 ) z ) .
10: 28.6 Expansions for Small q
28.6.1 a 0 ( q ) = 1 2 q 2 + 7 128 q 4 29 2304 q 6 + 68687 188 74368 q 8 + ,
28.6.2 a 1 ( q ) = 1 + q 1 8 q 2 1 64 q 3 1 1536 q 4 + 11 36864 q 5 + 49 5 89824 q 6 + 55 94 37184 q 7 83 353 89440 q 8 + ,
28.6.3 b 1 ( q ) = 1 q 1 8 q 2 + 1 64 q 3 1 1536 q 4 11 36864 q 5 + 49 5 89824 q 6 55 94 37184 q 7 83 353 89440 q 8 + ,
28.6.6 a 3 ( q ) = 9 + 1 16 q 2 + 1 64 q 3 + 13 20480 q 4 5 16384 q 5 1961 235 92960 q 6 609 1048 57600 q 7 + ,
28.6.7 b 3 ( q ) = 9 + 1 16 q 2 1 64 q 3 + 13 20480 q 4 + 5 16384 q 5 1961 235 92960 q 6 + 609 1048 57600 q 7 + ,