It's a part of mathematical functions.

1. 23. Bernoulli and Euler Polynomials-
Riemann Zeta Function
EMILIE HAYNGWORTH' KARLGOLDBERQ'
V. AND
Contents
Psge
Mathematical Properties . . . . . . . . . . . . . . . . . . . . 804
23.1. Bernoulli and Euler Polynomials and the Euler-Maclaurin
Formula . . . . . . . . . . . . . . . . . . . . . . 804
23.2. Riemann Zeta Function and Other Sums of Reciprocal
Powers.. . . . . . . . . . . . . . . . . . . . . . 807
References . . . . . . . . . . . . . . . . . . . . . . . . . . 808
Table 23.1. Coefficients of the Bernoulli and Euler Polynomials . . . 809
B,(z) and E&), n=0(1)15
Table 23.2. Bernoulli and Euler Numbers . . . . . . . . . . . . . 810
B, and E,, n=O, 1, 2(2)60, Exact and B, to 10s
Table 23.3. Sums of Reciprocal Powers . . . . . . . . . . . . . . 81 1
20D
A("=& (2k+l)"'
n=1(1)42
Table 23.4. Sums of Positive Powers . . . . . . . . . . . . . . . 813
m
Ck",n=1(1)10, m=1(1)100
k-1
Table 23.5. P/n!,z=2(1)9, n=1(1)50, 10s . . . . . . . . . . . . 818
The authors acknowledge the assistance of Ruth E. Capuano in the preparation and
checking of the tables.
1 National Bureau of Standards. (Presently, Auburn University.)
ZNational Bureau of Standards.

2. 23. Bernoulli and Euler Polynomials-Riemann Zeta
Function
Mathematical Properties
23.1. Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula
Generating Functions
23.1.1 -
lez'
-
e'-1
-CBn(z) a
OD
n-0
t"
Bernoulli and Euler Numbers
23.1.2 Bn=B,(0) n=O, 1, . . . En=2"En (;)=integer n=O, 1 , . . .
1 1 1
23.1.3 BO=l, Bl=-ij, B2=6, B4=-- Eo=l,E,=-l, E4=5
30
n=l,2,. . .
n=O, 1 , . . .
Expanaiona
n=O, 1 , . . .
n=O, 1 , . . .
n=O, 1 , . . . En(l-z)=(-l)"E.(~) n=O, 1 , . . .
n=O, 1 , . . . =E,(z)-22"
(- l)*+'En(-z) n=O, 1 , . . .
23.1.10
n=o, 1 , . . .
m=1,2,. . . m=1, 3 , . . .
En(mz)=-- n+l mn F-(-1)kBn+l(z+k)
m- 1
-0
n=O, 1 , . . .
m=2,4, . . .
804

3. BERNOULLI AND EULER POLYNOMIALS,RIEMANN ZETA FUNCTION 805
Integrals
23.1.12 1 Bn(t)Bm(t)dt=(-l)m-l -
m !n!
(m+n)! Bm+n 1 En(t)Em(t)dt
m!n!
= (-1)"4(2m+"+2-1)
m,n=l, 2, . . . (m+n+2)! Bm+n+z
I m,n=O, 1 , . . .
(The polynomials are orthogonal for m+n o d d . )
Inequalities
23.1.13 IB2nI>IB2s(z)1 n=l, 2,. . ., 4-"lE2,1>(-1)"E2,(~)>0 n=1,2, . . ., ~>z>O
23.1.14
4(2n- 1) !
(
2(2n+1)! L)>(-l)n+lBzn+l (z)>O
(27r)2"+' 1 -2-"
n2"
n=1,2, . . . , +>z>O
n=1,2,. . ., i>z>O
23.1.15
I
n=1,2,. . . I n=O,l,. . .
Fourier Expandona
I
n>l,l>z>O n>O, 1 2 x 2 0
n=l,l>z>O n=O, l>z>O
23.1.17
&n-1(z> =
(-1)"2(2n-I)!
(2?y)*n--1 z7 sin 2km
n>l, 122'>0
E2n-1(4=
(-1)*4(2n-l)!
?y2"
5cos (2k+l)7rs
t-o (2k+1)2"
n=1,2,. . ., 1 2 ~ 1 0
n=l,l>z>O
Special Values
23.1.19 Bzn+l
=O n=1,2,.. . EZn+,=0 n=O,l,. . .
23.1.20 B (0)= (- 1) "3 (1)
, 2, E" =- n (1 1
(0) E
=B, n=O,1, . . . = -2(n+I)-l(2"+l- 1) &+I n=1,2,. . .

4. 806 BERNOULLI AND EULER POLYNOMIALS,RIEMANN ZETA FUNCTION
E 2 n -1 = --E2n-1(3>
= -(2n)-l(I -31-h) (2h-1)B2,
n=1,2,. . .
Symbolic Operationn
23.1.25 p(B(z)+ 1) -p(B(z)) =p'(~)
23.1.26 B.(z+h)=(B(z)+h)" n=O, 1, . . .
Here p(z) denotes a polynomial in x and after expanding we set {B(z)}"=B.(z) {E(x)}"=E.(z).
and
Relatione Between the Polynomials Equivalent to this is
23.1.27
23.1.31
23.1.28
n=2,3,. . .
23.1.29
Let &(z) =B,(z-[z]). The Euler Summation
Euler-Maclaurin Formulas Formula is
Let F(z) have its 6rst 2n derivatives continuous
on an interval (a, a ) . Divide the interval into 23.1.32
m equal parts and let h=(b--a)/m. Then for
some e, I>e>o, depending on F(2")(z) (a, b),
on
we have
23.1.30

5. BERNOULLI AND EULER POLYNOMIALS, RIEMANN ZETA FUNCTION 807
23.2. Riemann Zeta Function and Other Sums 23.2.13 {'(0)=--3 I n 2r
of Reciprocal Powers
23.2.14 {(-2n) =O n=l,2,. . .
(D
23.2.1 {(s)=C
k= 1
k-* as>1
23.2.15 {(1--2n)=-- Bz. n=1,2,. . .
2n
23.2.2 =n (l-p-')--l as>1
(242"
P
23.2.16 { ( 2 n ) = 7 1Z l
B. n=1,2,. .
2(2n).
(product over all primes p).
23.2.11
23.2.3
n=1,2,. . .
Sums of Reciprocal Powerr
* 23.2.4 The sums referred to are
(D
23.2.5
23.2.18 {(n) =Ek-" n=2,3, . . .
k-1
where 23.2.19
v(n)=F (-1)"1k-m=(1-21-m){(n) 11=1,2,. . .
-1
as>o 23.2.20
a0
23.2.6 =2*n'-I sin (37rs)r(l--s){(l-s) x(n)=C (2k+l)-"=(1-2-"){(n) n=2,3,. . .
k-0
23.2.7 dx 23.2.21
23.2.8 -
B(n) =%
k-
(- 1 >k (2k+1) n=1,2,. . .
These sums can be calculated from the Bernoulli
0
and Euler polynomials by means of the last two
23.2.9 =2k-'+(s-l)-ln1-*-sL
k-1
x- [XI
x'+ 1
dX
formulas for special values of the zeta function
(note that r)(l)=ln 2), and
(?r/2)2"+1 n=O, 1, . . .
23.2.22 8(2n+l)=
23.2.10 2(2n)! I 21
E.
product over all zeros p of {(s) with gp>O. 23.2.23
The contour C in the fourth formula starts at
infinity on the positive real axis, circles the origin
(-1)"1F2n ' Ezn-l(x) sec(m)dx
B(2n)=4(2n-1)!
once in the positive direction excluding the points 71=1,2,. . .
f 2 n i r for n=1, 2, . . ., and returns to the
starting point. Therefore p(s) is regular for all p(2) is known as Catalan's constant. Some
values of s except for a simple pole at s = l with other special values are
residue 1.
1 1 ?rz
Special Values
23.2.24 {(2)=1+3+3+ . . . = -6
2 3
23.2.11 S(0) = -3 1 1 r4
23.2.25 {(4)=1+-+-+ . . . =-
23.2.12 r(l)=- 24 34 90
'See page If.

6. 808 BERNOULLI AND EULER POLYNOMIALS, RIEMANN ZETA FUNCTION
1 1 _9
7 1 1 a4
23.2.26 ~(2)=1--+-5-3 ...- 23.229 X(4)=1+-+-+ . . . =-
22 12 34 54 96
1 1 U
23.2.30 /3(1)=1--+-- ... =$
3 5
1 1 n2 1 1 -P
_
23.2.28 X(2)=1+3+3+ . . ’ - 8
-- 23.2.31 /3(3)=1--+--- 5 3
33 * “-32
3 5
References
Testa Tables
[23.1] G. Boole, The calculus of finite differences, 3d ed. (23.121 G. Blanch and R. Siegel, Table of modified
(Hafner Publishing Co., New York, N.Y., 1932). Bernoulli polynomials, J. Research NBS 44,
[23.2] W. E. Briggs and S. Chowla, The power series co- 103-107 (1950) RP2060.
efficients of ~ ( s ) ,Amer. Math. hlonthly 62, (23.131 H. T. Davis, Tables of the higher mathematical
323-325 (1955). functions, vol. I1 (Principia Press, Bloomington,
[23.3] T. Fort, Finite differences (Clarendon Press, Ind., 1935).
Oxford, England, 1948). [23.14] R. Hensman, Tables of the generalized Riemann
[23.4] C. Jordan, Calculus of finite differences, 2d ed. Zeta function, Report No. T2111, Telecommuni-
(Chelsea Publishing Co., New York, N.Y., cations Research Establishment, Ministry of
1960). Supply, Great Malvern, Worcestershire, Eng-
[23.5] K. Knopp, Theory and application of infinite land (1948). l(s, a), s=-10(.1)0, a=0(.1)2,
series (Blackie and Son, Ltd., London, England, 5D; ( ~ - l ) r ( ~a), ~=0(.1)1,~=0(.1)2, 5D.
,
1951). [23.15] D. H. Lehmer, On the maxima and minima of
[23.6] L. M. Milne-Thomson, Calculus of finite differences Bernoulli polynomials, Amer. Math. Monthly
(Macmillan and Co., Ltd., London, England, 47, 533-538 (1940).
1951). [23.16] E. 0. Powell, A table of the generalized Riemann
[23.7] N. E. Norlund, Vorlesungen uber Differenzen- Zeta function in a particular case, Quart. J.
rechnung (Edwards Bros., Ann Arbor, Mich., Mech. Appl. Math. 5, 116-123 (1952). {($,a),
1945). a=1(.01)2(.02)5(.05)10, 10D.
t23.81 C. H. Richardson, An introduction to the calculus
of finite differences (D. Van Nostrand Co., Inc.,
New York, N.Y., 1954).
[23.9] J. F. Steffensen, Interpolation (Chelsea Publishing
Co., New York, N.Y., 1950).
[23.10] E. C. Titchmarsh, The zeta-function of Riemann
(Cambridge Univ. Press, Cambridge, England,
1930).
[23.11] A. D. Wheelon, A short table of summable series,
Report No. SM-14642, Douglas Aircraft Co.,
Inc., Santa Monica, Calif. (1953).