Bernoulli numbers ♦ 11—20 of 59 matching pages ♦ SearchAdvancedHelp (0.004 seconds) 11—20 of 59 matching pages 11: 24.4 Basic Properties … ► 24.4.7 ∑ k = 1 m k n = B n + 1 ( m + 1 ) − B n + 1 n + 1 , ⓘ Symbols: B n : Bernoulli numbers, B n ( x ) : Bernoulli polynomials, k : integer, m : integer and n : integer A&S Ref: 23.1.4 Referenced by: §24.17(iii) Permalink: http://dlmf.nist.gov/24.4.E7 Encodings: pMML, png, TeX See also: Annotations for §24.4(iii), §24.4 and Ch.24 … ► 24.4.25 B n ( 0 ) = ( − 1 ) n B n ( 1 ) = B n . ⓘ Symbols: B n : Bernoulli numbers, B n ( x ) : Bernoulli polynomials and n : integer A&S Ref: 23.1.20 Referenced by: §24.17(ii), §24.4(vi) Permalink: http://dlmf.nist.gov/24.4.E25 Encodings: pMML, png, TeX See also: Annotations for §24.4(vi), §24.4 and Ch.24 … ► 24.4.27 B n ( 1 2 ) = − ( 1 − 2 1 − n ) B n , ⓘ Symbols: B n : Bernoulli numbers, B n ( x ) : Bernoulli polynomials and n : integer A&S Ref: 23.1.21 Referenced by: §2.10(i), §24.17(ii) Permalink: http://dlmf.nist.gov/24.4.E27 Encodings: pMML, png, TeX See also: Annotations for §24.4(vi), §24.4 and Ch.24 … ► 24.4.31 B n ( 1 4 ) = ( − 1 ) n B n ( 3 4 ) = − 1 − 2 1 − n 2 n B n − n 4 n E n − 1 , n = 1 , 2 , … . ⓘ Symbols: B n : Bernoulli numbers, B n ( x ) : Bernoulli polynomials, E n : Euler numbers and n : integer A&S Ref: 23.1.22 Permalink: http://dlmf.nist.gov/24.4.E31 Encodings: pMML, png, TeX See also: Annotations for §24.4(vi), §24.4 and Ch.24 … ► §24.4(ix) Relations to Other Functions … 12: 24.9 Inequalities §24.9 Inequalities … ► 24.9.1 | B 2 n | > | B 2 n ( x ) | , 1 > x > 0 , ⓘ Symbols: B n : Bernoulli numbers, B n ( x ) : Bernoulli polynomials, n : integer and x : real or complex A&S Ref: 23.1.13 Referenced by: §24.9 Permalink: http://dlmf.nist.gov/24.9.E1 Encodings: pMML, png, TeX See also: Annotations for §24.9 and Ch.24 ► 24.9.2 ( 2 − 2 1 − 2 n ) | B 2 n | ≥ | B 2 n ( x ) − B 2 n | , 1 ≥ x ≥ 0 . ⓘ Symbols: B n : Bernoulli numbers, B n ( x ) : Bernoulli polynomials, n : integer and x : real or complex Referenced by: §24.9 Permalink: http://dlmf.nist.gov/24.9.E2 Encodings: pMML, png, TeX See also: Annotations for §24.9 and Ch.24 … ► 24.9.6 5 π n ( n π e ) 2 n > ( − 1 ) n + 1 B 2 n > 4 π n ( n π e ) 2 n , ⓘ Symbols: B n : Bernoulli numbers, π : the ratio of the circumference of a circle to its diameter, e : base of natural logarithm and n : integer Referenced by: §24.9, §24.9 Permalink: http://dlmf.nist.gov/24.9.E6 Encodings: pMML, png, TeX See also: Annotations for §24.9 and Ch.24 … ► 24.9.8 2 ( 2 n ) ! ( 2 π ) 2 n 1 1 − 2 β − 2 n ≥ ( − 1 ) n + 1 B 2 n ≥ 2 ( 2 n ) ! ( 2 π ) 2 n 1 1 − 2 − 2 n ⓘ Symbols: B n : Bernoulli numbers, π : the ratio of the circumference of a circle to its diameter, ! : factorial (as in n ! ), n : integer and β : quantity Referenced by: §24.9 Permalink: http://dlmf.nist.gov/24.9.E8 Encodings: pMML, png, TeX See also: Annotations for §24.9 and Ch.24 … 13: 24.7 Integral Representations … ► §24.7(i) Bernoulli and Euler Numbers … ► 24.7.1 B 2 n = ( − 1 ) n + 1 4 n 1 − 2 1 − 2 n ∫ 0 ∞ t 2 n − 1 e 2 π t + 1 d t = ( − 1 ) n + 1 2 n 1 − 2 1 − 2 n ∫ 0 ∞ t 2 n − 1 e − π t sech ( π t ) d t , ⓘ Symbols: B n : Bernoulli numbers, π : the ratio of the circumference of a circle to its diameter, d x : differential of x , e : base of natural logarithm, sech z : hyperbolic secant function, ∫ : integral, n : integer and t : real or complex Permalink: http://dlmf.nist.gov/24.7.E1 Encodings: pMML, png, TeX See also: Annotations for §24.7(i), §24.7 and Ch.24 … ► 24.7.3 B 2 n = ( − 1 ) n + 1 π 1 − 2 1 − 2 n ∫ 0 ∞ t 2 n sech 2 ( π t ) d t , ⓘ Symbols: B n : Bernoulli numbers, π : the ratio of the circumference of a circle to its diameter, d x : differential of x , sech z : hyperbolic secant function, ∫ : integral, n : integer and t : real or complex Permalink: http://dlmf.nist.gov/24.7.E3 Encodings: pMML, png, TeX See also: Annotations for §24.7(i), §24.7 and Ch.24 ► 24.7.4 B 2 n = ( − 1 ) n + 1 π ∫ 0 ∞ t 2 n csch 2 ( π t ) d t , ⓘ Symbols: B n : Bernoulli numbers, π : the ratio of the circumference of a circle to its diameter, d x : differential of x , csch z : hyperbolic cosecant function, ∫ : integral, n : integer and t : real or complex Permalink: http://dlmf.nist.gov/24.7.E4 Encodings: pMML, png, TeX See also: Annotations for §24.7(i), §24.7 and Ch.24 ► 24.7.5 B 2 n = ( − 1 ) n 2 n ( 2 n − 1 ) π ∫ 0 ∞ t 2 n − 2 ln ( 1 − e − 2 π t ) d t . ⓘ Symbols: B n : Bernoulli numbers, π : the ratio of the circumference of a circle to its diameter, d x : differential of x , e : base of natural logarithm, ∫ : integral, ln z : principal branch of logarithm function, n : integer and t : real or complex Referenced by: §24.7(i) Permalink: http://dlmf.nist.gov/24.7.E5 Encodings: pMML, png, TeX See also: Annotations for §24.7(i), §24.7 and Ch.24 … 14: 24.13 Integrals … ► 24.13.4 ∫ 0 1 / 2 B n ( t ) d t = 1 − 2 n + 1 2 n B n + 1 n + 1 , ⓘ Symbols: B n : Bernoulli numbers, B n ( x ) : Bernoulli polynomials, d x : differential of x , ∫ : integral, n : integer and t : real or complex Permalink: http://dlmf.nist.gov/24.13.E4 Encodings: pMML, png, TeX See also: Annotations for §24.13(i), §24.13 and Ch.24 … ► 24.13.6 ∫ 0 1 B n ( t ) B m ( t ) d t = ( − 1 ) n − 1 m ! n ! ( m + n ) ! B m + n . ⓘ Symbols: B n : Bernoulli numbers, B n ( x ) : Bernoulli polynomials, d x : differential of x , ! : factorial (as in n ! ), ∫ : integral, m : integer, n : integer and t : real or complex A&S Ref: 23.1.12 Referenced by: §24.13(i) Permalink: http://dlmf.nist.gov/24.13.E6 Encodings: pMML, png, TeX See also: Annotations for §24.13(i), §24.13 and Ch.24 … ► 24.13.8 ∫ 0 1 E n ( t ) d t = − 2 E n + 1 ( 0 ) n + 1 = 4 ( 2 n + 2 − 1 ) ( n + 1 ) ( n + 2 ) B n + 2 , ⓘ Symbols: B n : Bernoulli numbers, E n ( x ) : Euler polynomials, d x : differential of x , ∫ : integral, n : integer and t : real or complex Permalink: http://dlmf.nist.gov/24.13.E8 Encodings: pMML, png, TeX See also: Annotations for §24.13(ii), §24.13 and Ch.24 ► 24.13.9 ∫ 0 1 / 2 E 2 n ( t ) d t = − E 2 n + 1 ( 0 ) 2 n + 1 = 2 ( 2 2 n + 2 − 1 ) B 2 n + 2 ( 2 n + 1 ) ( 2 n + 2 ) , ⓘ Symbols: B n : Bernoulli numbers, E n ( x ) : Euler polynomials, d x : differential of x , ∫ : integral, n : integer and t : real or complex Permalink: http://dlmf.nist.gov/24.13.E9 Encodings: pMML, png, TeX See also: Annotations for §24.13(ii), §24.13 and Ch.24 … ► 24.13.11 ∫ 0 1 E n ( t ) E m ( t ) d t = ( − 1 ) n 4 ( 2 m + n + 2 − 1 ) m ! n ! ( m + n + 2 ) ! B m + n + 2 . ⓘ Symbols: B n : Bernoulli numbers, E n ( x ) : Euler polynomials, d x : differential of x , ! : factorial (as in n ! ), ∫ : integral, m : integer, n : integer and t : real or complex A&S Ref: 23.1.12 Referenced by: §24.13(ii) Permalink: http://dlmf.nist.gov/24.13.E11 Encodings: pMML, png, TeX See also: Annotations for §24.13(ii), §24.13 and Ch.24 … 15: 24.2 Definitions and Generating Functions §24.2 Definitions and Generating Functions … ► B 2 n + 1 = 0 , … ► Table 24.2.1: Bernoulli and Euler numbers. ► ► n B n E n … ► ► … ► Table 24.2.3: Bernoulli numbers B n = N / D . ► ► n N D B n … ► ► … 16: 24.11 Asymptotic Approximations §24.11 Asymptotic Approximations … ► 24.11.1 ( − 1 ) n + 1 B 2 n ∼ 2 ( 2 n ) ! ( 2 π ) 2 n , ⓘ Symbols: B n : Bernoulli numbers, ∼ : asymptotic equality, π : the ratio of the circumference of a circle to its diameter, ! : factorial (as in n ! ) and n : integer Referenced by: §24.11 Permalink: http://dlmf.nist.gov/24.11.E1 Encodings: pMML, png, TeX See also: Annotations for §24.11 and Ch.24 ► 24.11.2 ( − 1 ) n + 1 B 2 n ∼ 4 π n ( n π e ) 2 n , ⓘ Symbols: B n : Bernoulli numbers, ∼ : asymptotic equality, π : the ratio of the circumference of a circle to its diameter, e : base of natural logarithm and n : integer Permalink: http://dlmf.nist.gov/24.11.E2 Encodings: pMML, png, TeX See also: Annotations for §24.11 and Ch.24 … ► 24.11.4 ( − 1 ) n E 2 n ∼ 8 n π ( 4 n π e ) 2 n . ⓘ Symbols: E n : Euler numbers, ∼ : asymptotic equality, π : the ratio of the circumference of a circle to its diameter, e : base of natural logarithm and n : integer Referenced by: §24.11 Permalink: http://dlmf.nist.gov/24.11.E4 Encodings: pMML, png, TeX See also: Annotations for §24.11 and Ch.24 … 17: 5.15 Polygamma Functions … ► 5.15.8 ψ ′ ( z ) ∼ 1 z + 1 2 z 2 + ∑ k = 1 ∞ B 2 k z 2 k + 1 , ⓘ Symbols: B n : Bernoulli numbers, ∼ : Poincaré asymptotic expansion, ψ ( z ) : psi (or digamma) function, k : nonnegative integer and z : complex variable A&S Ref: 6.4.12 Referenced by: §5.15 Permalink: http://dlmf.nist.gov/5.15.E8 Encodings: pMML, png, TeX See also: Annotations for §5.15 and Ch.5 ► 5.15.9 ψ ( n ) ( z ) ∼ ( − 1 ) n − 1 ( ( n − 1 ) ! z n + n ! 2 z n + 1 + ∑ k = 1 ∞ ( 2 k + n − 1 ) ! ( 2 k ) ! B 2 k z 2 k + n ) . ⓘ Symbols: B n : Bernoulli numbers, ∼ : Poincaré asymptotic expansion, ψ ( z ) : psi (or digamma) function, ! : factorial (as in n ! ), n : nonnegative integer, k : nonnegative integer and z : complex variable A&S Ref: 6.4.11 Referenced by: §5.15, Erratum (V1.1.3) for Additions Permalink: http://dlmf.nist.gov/5.15.E9 Encodings: pMML, png, TeX Addition (effective with 1.1.3): This equation was added. Suggested 2021-09-09 by Calvin Khor See also: Annotations for §5.15 and Ch.5 ►For B 2 k see §24.2(i). … 18: 25.11 Hurwitz Zeta Function … ► 25.11.22 ζ ′ ( 1 − 2 n , 1 2 ) = − B 2 n ln 2 n ⋅ 4 n − ( 2 2 n − 1 − 1 ) ζ ′ ( 1 − 2 n ) 2 2 n − 1 , n = 1 , 2 , 3 , … . ⓘ Symbols: B n : Bernoulli numbers, ζ ( s , a ) : Hurwitz zeta function, ζ ( s ) : Riemann zeta function, ln z : principal branch of logarithm function and n : nonnegative integer Keywords: derivative Source: Miller and Adamchik (1998, (17), p. 205) Permalink: http://dlmf.nist.gov/25.11.E22 Encodings: pMML, png, TeX See also: Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25 ► 25.11.23 ζ ′ ( 1 − 2 n , 1 3 ) = − π ( 9 n − 1 ) B 2 n 8 n 3 ( 3 2 n − 1 − 1 ) − B 2 n ln 3 4 n ⋅ 3 2 n − 1 − ( − 1 ) n ψ ( 2 n − 1 ) ( 1 3 ) 2 3 ( 6 π ) 2 n − 1 − ( 3 2 n − 1 − 1 ) ζ ′ ( 1 − 2 n ) 2 ⋅ 3 2 n − 1 , n = 1 , 2 , 3 , … . ⓘ Symbols: B n : Bernoulli numbers, ζ ( s , a ) : Hurwitz zeta function, ζ ( s ) : Riemann zeta function, π : the ratio of the circumference of a circle to its diameter, ψ ( z ) : psi (or digamma) function, ln z : principal branch of logarithm function and n : nonnegative integer Keywords: derivative Source: Miller and Adamchik (1998, (18), p. 205) Permalink: http://dlmf.nist.gov/25.11.E23 Encodings: pMML, png, TeX See also: Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25 … ► 25.11.34 n ∫ 0 a ζ ′ ( 1 − n , x ) d x = ζ ′ ( − n , a ) − ζ ′ ( − n ) + B n + 1 − B n + 1 ( a ) n ( n + 1 ) , n = 1 , 2 , … , ℜ a > 0 . ⓘ Symbols: B n : Bernoulli numbers, B n ( x ) : Bernoulli polynomials, ζ ( s , a ) : Hurwitz zeta function, ζ ( s ) : Riemann zeta function, d x : differential of x , ∫ : integral, ℜ : real part, n : nonnegative integer, x : real variable and a : real or complex parameter Keywords: definite integral, integral representation Source: Adamchik (1998, (15), p. 196) Permalink: http://dlmf.nist.gov/25.11.E34 Encodings: pMML, png, TeX See also: Annotations for §25.11(viii), §25.11 and Ch.25 … ► 25.11.43 ζ ( s , a ) − a 1 − s s − 1 − 1 2 a − s ∼ ∑ k = 1 ∞ B 2 k ( 2 k ) ! ( s ) 2 k − 1 a 1 − s − 2 k . ⓘ Symbols: B n : Bernoulli numbers, Γ ( z ) : gamma function, ζ ( s , a ) : Hurwitz zeta function, ( a ) n : Pochhammer’s symbol (or shifted factorial), ∼ : Poincaré asymptotic expansion, ! : factorial (as in n ! ), k : nonnegative integer, a : real or complex parameter and s : complex variable Keywords: asymptotic approximation Source: Paris (2005b, (1.3), (1.4), p. 298) Referenced by: §25.11(xii), Erratum (V1.0.9) for Chapters 7, 25 Permalink: http://dlmf.nist.gov/25.11.E43 Encodings: pMML, png, TeX Notational Change (effective with 1.0.9): We have rewritten the original summation ∑ k = 1 ∞ B 2 k ( 2 k ) ! Γ ( s + 2 k − 1 ) Γ ( s ) a 1 − s − 2 k more concisely as ∑ k = 1 ∞ B 2 k ( 2 k ) ! ( s ) 2 k − 1 a 1 − s − 2 k using the Pochhammer symbol. See also: Annotations for §25.11(xii), §25.11 and Ch.25 … ► 25.11.44 ζ ′ ( − 1 , a ) − 1 12 + 1 4 a 2 − ( 1 12 − 1 2 a + 1 2 a 2 ) ln a ∼ − ∑ k = 1 ∞ B 2 k + 2 ( 2 k + 2 ) ( 2 k + 1 ) 2 k a − 2 k , ⓘ Symbols: B n : Bernoulli numbers, ζ ( s , a ) : Hurwitz zeta function, ∼ : Poincaré asymptotic expansion, ln z : principal branch of logarithm function, k : nonnegative integer and a : real or complex parameter Keywords: asymptotic approximation Source: Elizalde (1986, (18), p. 349) Permalink: http://dlmf.nist.gov/25.11.E44 Encodings: pMML, png, TeX See also: Annotations for §25.11(xii), §25.11 and Ch.25 … 19: 25.1 Special Notation … ► ► k , m , n nonnegative integers. … ► B n , B n ( x ) Bernoulli number and polynomial (§24.2(i)). … ► … 20: 24.8 Series Expansions … ► 24.8.6 B 4 n + 2 = ( 8 n + 4 ) ∑ k = 1 ∞ k 4 n + 1 e 2 π k − 1 , n = 1 , 2 , … , ⓘ Symbols: B n : Bernoulli numbers, π : the ratio of the circumference of a circle to its diameter, e : base of natural logarithm, k : integer and n : integer Referenced by: §24.8(ii) Permalink: http://dlmf.nist.gov/24.8.E6 Encodings: pMML, png, TeX See also: Annotations for §24.8(ii), §24.8 and Ch.24 ► 24.8.7 B 2 n = ( − 1 ) n + 1 4 n 2 2 n − 1 ∑ k = 1 ∞ k 2 n − 1 e π k + ( − 1 ) k + n , n = 2 , 3 , … . ⓘ Symbols: B n : Bernoulli numbers, π : the ratio of the circumference of a circle to its diameter, e : base of natural logarithm, k : integer and n : integer Permalink: http://dlmf.nist.gov/24.8.E7 Encodings: pMML, png, TeX See also: Annotations for §24.8(ii), §24.8 and Ch.24 … ► 24.8.8 B 2 n 4 n ( α n − ( − β ) n ) = α n ∑ k = 1 ∞ k 2 n − 1 e 2 α k − 1 − ( − β ) n ∑ k = 1 ∞ k 2 n − 1 e 2 β k − 1 , n = 2 , 3 , … . ⓘ Symbols: B n : Bernoulli numbers, e : base of natural logarithm, k : integer, n : integer, α : parameter and β : parameter Referenced by: §24.8(ii) Permalink: http://dlmf.nist.gov/24.8.E8 Encodings: pMML, png, TeX See also: Annotations for §24.8(ii), §24.8 and Ch.24 …