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1: 24.1 Special Notation
Euler Numbers and Polynomials
Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations E n , E n ( x ) , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
2: 24.21 Software
§24.21(ii) B n , B n ( x ) , E n , and E n ( x )
3: 24.6 Explicit Formulas
§24.6 Explicit Formulas
24.6.4 E 2 n = k = 1 n 1 2 k - 1 j = 1 k ( - 1 ) j ( 2 k k - j ) j 2 n ,
24.6.5 E 2 n = 1 2 n - 1 k = 0 n - 1 ( - 1 ) n - k ( n - k ) 2 n j = 0 k ( 2 n - 2 j k - j ) 2 j ,
24.6.10 E n = 1 2 n k = 1 n + 1 ( n + 1 k ) j = 0 k - 1 ( - 1 ) j ( 2 j + 1 ) n .
24.6.12 E 2 n = k = 0 2 n 1 2 k j = 0 k ( - 1 ) j ( k j ) ( 1 + 2 j ) 2 n .
4: 24.14 Sums
§24.14 Sums
24.14.4 k = 0 n ( n k ) E k E n - k = - 2 n + 1 E n + 1 ( 0 ) = - 2 n + 2 ( 1 - 2 n + 2 ) B n + 2 n + 2 .
24.14.6 k = 0 n ( n k ) 2 k B k E n - k = 2 ( 1 - 2 n - 1 ) B n - n E n - 1 .
24.14.9 ( 2 n ) ! ( 2 j ) ! ( 2 k ) ! ( 2 ) ! E 2 j E 2 k E 2 = 1 2 ( E 2 n - E 2 n + 2 ) .
24.14.12 det [ E r + s ] = ( - 1 ) n ( n + 1 ) / 2 ( k = 1 n k ! ) 2 .
5: 24.10 Arithmetic Properties
§24.10 Arithmetic Properties
§24.10(ii) Kummer Congruences
24.10.5 E n E n + p - 1 ( mod p ) ,
24.10.6 E 2 n E 2 n + w ( mod 2 ) ,
§24.10(iv) Factors
6: 24.5 Recurrence Relations
§24.5 Recurrence Relations
24.5.4 k = 0 n ( 2 n 2 k ) E 2 k = 0 , n = 1 , 2 , ,
24.5.5 k = 0 n ( n k ) 2 k E n - k + E n = 2 .
§24.5(iii) Inversion Formulas
b n = k = 0 n / 2 ( n 2 k ) E 2 k a n - 2 k .
7: 24.20 Tables
§24.20 Tables
Wagstaff (1978) gives complete prime factorizations of N n and E n for n = 20 ( 2 ) 60 and n = 8 ( 2 ) 42 , respectively. …
8: 24.2 Definitions and Generating Functions
§24.2(ii) Euler Numbers and Polynomials
E 2 n + 1 = 0 ,
Table 24.2.1: Bernoulli and Euler numbers.
n B n E n
Table 24.2.4: Euler numbers E n .
n E n
9: 24.9 Inequalities
§24.9 Inequalities
24.9.3 4 - n | E 2 n | > ( - 1 ) n E 2 n ( x ) > 0 ,
24.9.7 8 n π ( 4 n π e ) 2 n ( 1 + 1 12 n ) > ( - 1 ) n E 2 n > 8 n π ( 4 n π e ) 2 n .
24.9.10 4 n + 1 ( 2 n ) ! π 2 n + 1 > ( - 1 ) n E 2 n > 4 n + 1 ( 2 n ) ! π 2 n + 1 1 1 + 3 - 1 - 2 n .
10: 24.11 Asymptotic Approximations
§24.11 Asymptotic Approximations
24.11.3 ( - 1 ) n E 2 n 2 2 n + 2 ( 2 n ) ! π 2 n + 1 ,
24.11.4 ( - 1 ) n E 2 n 8 n π ( 4 n π e ) 2 n .