伯努力數之計算與應用

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Abstract

Up to the present, it is an important study for calculating Bernoulli

number.

There are many different methods to claculate Bernoulli

number. But for these methods, we must take lots of steps to calaulate

Bernoulli number. Based on this, our research applies Riemann–zeta

function and the extended function of the sums of powers of

consecu-tive integers to get an easier method. Then, we will calculate Bernoulli

number by using Matlab 7.1, and investigate the relationship between

Bernoulli nmuber and Stirling number of second kind.

Our results are as follows.

1. The formula of Bernoulli number is

B

2k

=

1

2k + 1

(

C

_2k2k+1

S

₁0

(−1) +

k

X

i=1

C

_2i+12k+1

S

_2k−2i0

(−1)

)

, k ∈ N.

2. When k is bigger, Bernoulli number will become bigger and be

alternated between plus and minus.

3. The relationship between Bernoulli number and Stirling number

of second kind is

B

m+1

=

m+1

X

k=1

(−1)

k

k + 1

· k! · S

2

(m + 1, k).

Keywords: the extended function of the sums of powers of

consecutive integers, Riemann–zeta function,

Bernoulli number, Stirling number of second kind.

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∞ X k=0 ˆ Sk(−1) Tk k! = 1 − eT eT _{− 1} = −1,

FJ

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5„pà-

:

ç

_{k = 1}

v

_,

;W4”

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ªJ)ƒ

ˆ S3(x) − ˆS3(−1) = 3 Z x −1 ˆ S2(y)dy = 3 Z x −1 y(y + 1)(2y + 1) 6 dy = x2(x + 1)2 4 ,

ÄÑ

_Sˆ_{3(−1) = 0 (}

;Wt

(2.7)),

/

_{S3(x) =} x2(x+1)2 4 ,

FJ

Sˆ3(x) = S3(x), ∀x ∈ R.

cq

_{k = p > 1}

v

_{, ˆSp(x) = Sp(x), ∀x ∈ R}

A

ÛÊ5?

k = p + 1

v

_,

;Wt

(2.6)

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₍₁₉₈₈₎

-

_^*ƒbD+

›‰b

,

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_{(2.4) ,}

ªJ)ƒ

ˆ S_l0(−1) = S_l0(−1) = B1, ∀l ∈ N,

¢;W4”

_2.1.1,



ˆ S_p+10 (x) − ˆS_p+10 (−1) = p Z x −1 ˆ S_p0(y)dy = p{ ˆSp(x) − ˆSp(−1)} = p{Sp(x) − Sp(−1)} = S_p+10 (x) − S_p+10 (−1),

FJ

Sˆ_p+10 (x) = S_p+10 (x), ∀x ∈ R.

yŸ })

ˆ Sp+1(x) − ˆSp+1(−1) = Sp+1(x) − Sp+1(−1), ∀x ∈ R,

ÇÕ

,

;Wt

(2.7)

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₍₁₉₈₈₎

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,

J£

_(2.2)

ª)

ˆ Sp+1(−1) = Sp+1(x), ∀x ∈ R,

]

ˆ Sp+1(x) = Sp+1(x), ∀x ∈ R.

Ĥ

_,

;Wbç¦Ñ¶

,

)„

_Sˆ_{k(x) = Sk(x), ∀x ∈ R, k ∈ N.}

;W

eT (x+1)_−eT eT₋₁

Ê

T = 0

5œ

 Ç

,

.Øl|

ˆ S0(x) = x, ˆS1(x) = x(x + 1) 2 , ˆS2(x) = x(x + 1)(2x + 1) 6 .

Í7ú

k ≥ 3 , ˆSk(x)

5l

,

N

˛ÿ‰)'õ¦
ÖÍMøl

Sˆk(x) ,

yøl

!‹D

Sk(x)

5ì2úÎ

,

ªJÀUð„

Sˆk(x) = Sk(x)

59õ

,

Oà°‡H

,

ú

_k

'×ív`

_,

Møªú

_Sˆ_k(x)

D

_Sk(x)

uÎn‘ví T

F

_,

ªJ)ƒ

_Sˆ_k(x)

x4”

_2.1.1

D4”

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Í7;W

Sk(x)

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Sˆk(x) = Sk(x), ∀x ∈ R, k ∈ N

6ÿu

∞ X k=0 Sk(x) Tk k! = eT (x+1)_{− e}T eT _{− 1} . 

L ‚à

_Sk(x)

_¨Æ5U‚ƒb

_Ck(x)

úL< ì£cb

k ,

I

_+›‰b

_{Bk(x) =} d dxSk(x) = S 0 k(x − 1) ,

†ø}

-Þ4”

:

4_{” 2.1.3.}

J

_k

Ñ×kk

₂

5£cbv

_,

†

_{Bk(0) = Bk(1)}

„p

:

ÄÑ

∞ X k=0 Sk(x) Tk k! = eT (x+1)_{− e}T eT _{− 1} ,

FJ

∞ X k=0 S_k0(x)T k k! = T eT (x+1) eT _{− 1} ,

]

∞ X k=0 Bk(x) Tk k! = T eT x eT _{− 1},

Ĥ

∞ X k=0 Bk(1) Tk k! = T eT et_{− 1}, (2.8) ∞ X k=0 Bk(0) Tk k! = T eT _{− 1}, (2.9)

â

_(2.8)



_(2.9)

sª)

T = ∞ X k=0 {Bk(1) − Bk(0)} Tk k!,

ª7#|

B0(1) = B0(0), B1(1) − B1(0) = 1, Bk(1) = Bk(0), ∀K ≥ 2,

)„

_

;W4”

2.1.3,

ì

_{2ƶ,5U‚ƒb}

_{Ck(x) ,∀k ≥ 2}

à-

_: Ck(x) = Bk  x 2π  , ∀x ∈ [0, 2π]

/

_{Ck(x) = Ck(x + 2π).}

LÔ U‚

_ƒb

Ck(x)

5

Z s‰²

â†/ð

(1991)

5

_{Z s}&»ød}

_,

ªøU‚

_ƒb[ýÑ

Ck(x) = X m∈z an,keinx, n ∈ Z, k ≥ 2,

w2

einx = cos (nx) + i sin (nx),

an,k = 1 2π Z 2π 0 e−inxCk(x)dx,

;W

Ck(x)

5ì2D }d‰b‰²

,

ª)

an,k = Z 1 0 e−2πinxBk(x)dx, n ∈ Z, k ≥ 2,

Q

6;Wcb

n

5¦

_Mª}Ñ

_{n 6= 0}

D

n = 0

s8$Vl

an,k.

ç

_{n = 0}

v

_, Z 1 0 e−2nπxiBk(x)dx = Z 1 0 Bk(x)dx = Sk(0) − Sk(−1) = 0, k ≥ 2,

]

_{a0,k = 0, ∀k ≥ 2.}

ç

_{n 6= 0}

v

_,

ÄÑ

Z 1 0 e−2πinxB0(x)dx = Z 1 0 e−2πinxdx = 0, Z 1 0 e−2nπxiB1(x)dx = Z 1 0 e−2nπxi  x − 1 2  dx = Z 1 0 xe−2nπxidx − 1 2 Z 1 0 e−2nπxidx(= 0) = −xe −2nπxi 2nπi     1 0 + 1 2nπi Z 1 0 e−2nπxidx(= 0) = − 1 2nπi,

l

R1 0 e −2πinx_B k(x)dx

5NbÞA

ƒb

,

ª)

∞ X k=0 Z 1 0 e−2nπxiBk(x)dx  Tk k! = Z 1 0 e−2nπxi ( _∞ X k=0 Bk(x) Tk k! ) dx = Z 1 0 e−2nπxi T e T x eT _{− 1}  dx = T eT _{− 1} Z 1 0 e(T −2nπi)xdx = T eT _{− 1} · 1 T − 2nπi h e(T −2nπi)x 1 0 i = T T − 2nπi,

FJ

Z 1 0 e−2nπxi T e T x eT _{− 1}  dx = − T 2nπi + ∞ X k=2 an,k Tk k!,

]

an,k = − k! (2nπi)k, n ∈ Z, n 6= 0, k ≥ 2,

;W

_{Z s‰²}

Ck(x) = P_n∈Zan,ke−2nπxi ,

FJ

Bk  x 2π  = X n∈Z,n6=0 − k! (2nπi)ke −inx , k ≥ 2,

¹

,

úL<

_{k ≥ 2, x ∈ [0, 1],} Bk(x) = X n∈Z,n6=0 − k! (2nπi)ke 2nπxi = − k! (2πi)k X n∈Z,n6=0 1 nke 2nπxi = − k! (2πi)k X n∈Z,n6=0 1 nk[cos (2nπx) + i sin (2nπx)] = − k! 2k_(πi)k ( _∞ X n=1 1 nk [cos (2nπx) + i sin (2nπx)] + ∞ X n=1 1 (−n)k[cos (2nπx) − i sin (2nπx)] ) . (2.10)

â

_(2.10),

ªJ)ƒ

_k

Ñ£

_Xbv

_, S_k0(x − 1) = − k! 2k−1_(−πi)k ∞ X n=1 1 nk cos (2nπx), ∀x ∈ [0, 1],

7ç

k

Ñ×k

1

5Jbv

, S_k0(x − 1) = − k! 2k−1_(−πi)k ∞ X n=1 1 nk sin (2nπx), ∀x ∈ [0, 1].

LŽ

_S0 k(x)

D-^*ƒb5É:

ç

_k

Ñ£

_Xbv

_, S_k0(0) = − k! 2k−1_(πi)k ∞ X n=1 1 nk = − k! 2k−1_(πi)kζ(k), (2.11)

7ç

k

Ñ×kø5Jbv

, S_k0(0) = 0, (2.12)

;W

Sk(x) = 1 k + 1{(x + 1) k+1_{− C}k+1 2 Sk−1(x) − · · · − Ckk+1S1(x) − x − 1},

ªJ)ƒ

_:

ç

_k

Ñ£

_Xbv

_, ζ(k) = −2 k−1_(πi)k (k + 1)! k − Ck+1 2 S 0 k−1(0) − C k+1 3 S 0 k−2(0) − · · · − C k+1 k S 0 1(0) , (2.13)

‚à

_(2.4)

J£

_{(2.11) ,}

ª)

ζ(2) = ∞ X n=1 1 n2 = S 0 2(0)π2 = π2 6 ,

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(2.3)



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_(2.5)



_(2.12)



_(2.13)

ª)

ζ(4) = ∞ X n=1 1 n4 = − π4 154 − C 5 2S 0 3(0) − C 5 3S 0 2(0) − C 5 4S 0 4(0) = π4 90,

./‚à

_(2.11),

)ƒ

S₄0(0) = − 1 30,

°Ü

_,

;W

(2.13)

ª)

ζ(6) = ∞ X n=1 1 n6 = 2π6 3156 − C 7 3S 0 4(0) − C57S 0 2(0) − C67S 0 1(0) = π6 945,

Y¤éR

,

ç

_k

×k

₆

5

_Xbv

_,

ªJ

_"Τj¶

_,

lMøl|

S₁0(0), S₂0(0), · · · , S_k−20 (0),

Í(yªø¥‚à

S₁0(0), S₂0(0), · · · , S_k−20 (0),

¹ª

)ƒ

ζ(k)

5

_M

ù

_{+›‰b5óÉû˝}

Ô úk

_{n = 1, 2, 3, · · · ;}



Bn = 2(2n)! (2π)2nS2n,

w2

_Bn

Ñ

_+›‰b

_{, S2n =}P∞ m=1 1 m2n

„p

:

lVû˝ø_x½b@àíζ
âk

ex = 1 + x +x 2 2! + · · · + xn n! + · · · ,

FJ

x ex_{− 1} = x x + x2 2! + · · · + xn n! + · · · = 1 1 + _2!x + x_3!2 + · · · + xn−1_n! + · · ·,

cq¥_¼Býúk—Düí

_x

_Mª[Ab

x ex_{− 1} = 1 + ∞ X n=1 βn n!x n ,

í$

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w[b¦A

βn n!

í$

,

¥ccuÑ7üì[bvœÑjZ

;WÉ[

 1 + x 2!+ x2 3! + · · · + xn−1 n! + · · ·  ×  1 + β1 1! + β2 2!x 2 + · · · + βn n!x n + · · ·  = 1,

Ç˝V®_j4

xn(n = 1, 2, · · · )

í[bkÉ
ªJ)|j˙

1 n!βn+ 1 (n − 1)!2!βn−1+ · · · + 1 (n − k + 1)!k!βn−k+1+ · · · + β1 1!n!+ 1 (n + 1)! = 0,

si

_J

_{(n + 1)!}

)

C₁n+1βn+ C2n+1βn−1+ · · · + Ckn+1βn+1−k+ · · · + Cnn+1β1+ 1 = 0,

‚àwDâùáóNíÉ[

,

¥<j˙¯Uí$,ªJŸA

: (β + 1)n+1− βn+1 _{= 0 (n = 1, 2, · · · ),}

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,

¾ |òá

βn+1

(

_,

4j

_βk

_Tà

_β k

H

,

)ƒüì

βn(n = 1, 2, · · · )

í̤j˙

2β1+ 1 = 0, 3β2 + 3β1+ 1 = 0, 4β3+ 6β2+ 4β1+ 1 = 0, 5β4+ 10β3+ 10β2+ 5β1+ 1 = 0, (2.14)

â

_(2.14)

ª)

β1 = − 1 2, β2 = 1 6, β3 = 0, β4 = − 1 30, β5 = 0, β6 = 1 42, β7 = 0, β8 = − 1 30, β9 = 0, β10= 5 66, β11 = 0, β12= − 691 2730, β13 = 0, β14= 7 6, (2.15)

;W

úk

_{|x| < 1,} x coth x = 1 + ∞ X n=1 (−1)n−12 2n_B n (2n)!x 2n_,

¥ê

Bn

Ñ

+›‰b

,B1 = 1 6, B2 = 1 30, · · · , (2.16)

ªø

x ex_{− 1}+ x 2 = x 2 ex_{+ 1} ex_{− 1} = x 2 ex2 + e− x 2 ex2 − e− x 2 = x 2coth x 2 = 1 + ∞ X m=1 (−1)m−1· Bm (2m)!x 2m_,

âk

_(2.15)

2

βn(n > 1)

íJbáÌÑÉ

,

â

_exx₋₁ + x 2

íÇø

x ex_{− 1} + x 2 = 1 + ∞ X n=2 βm m!x m ,

úk

_Xb—™í

_{β ,}



β2n= (−1)n−1Bn,

ku

B1 = 1 6, B2 = 1 30, B3 = 1 42, B4 = 1 30, B5 = 5 66, B6 = 691 2730, B7 = 7 6, · · · .

â

úk

_{|x| < 1,} πx coth πx = 1 + 2 ∞ X m=1 (−1)m−1S2mx2m,

¥ê

S2m= ∞ X n=1 1 n2m,

£

_{(2.16) ,}



πx coth πx = 1 + 2 ∞ X n=1 (−1)n−1S2nx2n, πx coth πx = 1 + ∞ X n=1 (−1)n−1(2π) 2n_B n (2n)! x 2n_,

*,Þsªø

S2n = (2π)2n 2(2n)!Bn. 

¡

¸Ä

(2006)

ê[5dı 

A Note on the Sums of Powers of Consecutive

Integers



d2Tƒ

: Sn(k) = 1n+ 2n+ · · · + (k − 1)n, k ∈ N, k > 1 = Z k 0 Bn(x)dx,

¢

Z k+1 k Bn(x)dx = Z k+1 0 Bn(x)dx − Z k 0 Bn(x)dx = Sn(k + 1) − Sn(k) = kn,

ø

_k

à

_x

¦H

Z x+1 x Bn(t)dt = xn,

øsi}

Bn(x + 1) − Bn(x) = nxn−1,

é

x = 0

ƒ

_{x = k + 1}

Ú‹

Bn(k) − Bn(0) = nSn−1(k) = n Z k 0 Bn−1(x)dx, Sn(k) = 1 n + 1[Bn+1(k) − Bn+1(0)], Bn(k) = n Z k 0 Bn−1(x)dx + Bn(0),

/

Bn = Bn(0) = Sn0(0),

é

Bx(0) = 1, Bn(x) = n Z x 0 Bn−1(t)dt + Bn, n = 1, 2, 3, · · · ,

I

_{n = 1, 2, 3, · · · ,} B1(x) = x + B1, B2(x) = x2+ 2B1x + B2, B3(x) = x3+ 3B1x2 + 3B2x + B3, .. .

ª)

Bn(x) = n X i=0 C_inBixn−i, Sn(k) = 1 n + 1 n X i=0 C_in+1Bikn+1−i,

ÇøjÞ

,

é

x = 0, Bn(1) − Bn(0) = 0,

FJ

Bn = Bn(0) = Bn(1) = n X i=0 C_inBi,

]

Bn+1 = n+1 X i=0 C_in+1Bi = n X i=0 C_in+1Bi+ Bn+1,

Ĥ

n X i=0 C_in+1Bi = 0,

FJ

Bn= − 1 n + 1 n−1 X i=0 C_in+1Bi.

 Ù;Æ

(2000)



_{4¸D+›‰bí¡j¶}

4¸

M − N

_{[ýí¡j¶+›‰b}

Bk

¸

+›‰Öá

Bk(y)

Êb

û˝2”ѽb

,

…bâà-

_ƒbì2

x ex_{− 1} = ∞ X k=0 Bk k!x k , (|x| < 2π), xexy ex_{− 1} = ∞ X k=0 Bk(y) k! x k_{, y ∈ R,} B0 = 1, B1 = − 1 2, B2k+1 = 0 (k ≥ 1). ùÜ 2.2.1. k ≥ 2 ,

†

Sk(n) = k Z n 0 Sk−1(x)dx + Bk· n. (2.17) ùÜ 2.2.2. (M − N )

[ýp

M = 2n + 1, N = n(n + 1) ,

†

S2k(n) = M N 2(2k + 1) k−1 X r=0 (−1)rA(k, r)Nk−r−1, (2.18) S2k+1(n) = N2 2 k−1 X r=0 (−1)r A(k, r) k + 1 − rN k−r−1_{, (2.19)}

w2

_{A(k, 0) = 1, A(k, 1) =} 1 3C 2 k, A(k, 2) = 7k−1 60 C 3 k, A(k, r) = r X j=1 (−1)j−1 1 2j + 1C 2j k+j−rA(k, r − j), (2.20) A(k, r) = 1 k + 1 r X j=1 (−1)j−1C_k+12j+1A(k − j, r − j), (2.21) A(k, r) = (k − r + 1)! (k + 1)!                    C_k+2−r3 C_k+2−r1 C_k+3−r5 C_k+3−r3 C_k+3−r1 · · · · C_k2r−1 · · · C1 k C_k+12r+1 · · · C3 k+1                    , (2.22) ùÜ 2.2.3. k ≥ 0 ,

†

Bk( 1 2) = 1 2k(2 − 2 k_)B k. (2.23) ùÜ 2.2.4.

qÅ—‘K

p − 1|2k

íF[bÑ

p1, p2, · · · , ps ,

†

1. B2k

í}‚u

p1p2· · · ps(s ≥ 2),

1/

,B2k = _p1pa₂2k···ps,a2k

ÑJb

; 2.

J/ÑJ

p ∈ {p1, p2, · · · , ps}

v

, p|(pB2k+ 1)

í}ä

, p||B2k

í}‚

ìÜ 2.2.1. k ≥ 2, 1 ≤ r ≤ k − 2

v

     A(k, k − 2) = 3A(k, k − 1) = (−1)k−1_{6(2k + 1)B} 2k, C2

2k−2r+1A(k, r) = Ck−r+12 A(k, r − 1) + C2k+12 A(k − 1, r).

(2.24)

„p

:

âùÜ

2.2.1

)

2kS2k−1(n) + B2k = d dnS2k(n), (2.25)

âùÜ

2.2.2,

1·<

_M2 _{= 4N + 1, dM = 2dn, dN = M dn}

₎

2k(2k + 1) k−2 X r=0 (−1)rA(k − 1, r) k − r N k−r + 2(2k + 1)B2k = k−1 X r=0 (−1)rA(k, r) d dn(N M k−r ) = k−1 X r=0 (−1)rA(k, r)[2Nk−r+ (k − r)M2Nk−r−1] = k−1 X r=0 (−1)rA(k, r)[(4k − 4r + 2)Nk−r+ (k − r)Nk−r−1] = k−1 X r=1 (−1)r[(4k − 4r + 2)A(k, r) − (k + 1 − r)A(k, r − 1)]Nk−r +(4k + 2)A(k, 0)Nk+ (−1)k−1A(k, k − 1).

ªœsi

_N

í°Ÿ

_á[b¹)„

_ ìÜ 2.2.2. k ≥ 1, 0 ≤ r ≤ k − 1 ,

†

A(k, r) = (−1 4) r r X j=0 (2 − 4j)B2jC2k+12j C k−r k−j. (2.26)

„p

:

âùÜ

2.2.3

£

_{M = 2n + 1}

)

M 2 + m X k=1 S2k(n) x2k (2k)! = 1 2 + ∞ X k=0 S2k(n) x2k (2k)! = 1 2 + n X r=1 ( ∞ X k=0 (rx)2k (2k)! ) = 1 2 + n X r=1 chrx = sh M 2 x 2shx₂ = shM x 2 ex2 ex_{− 1} = 1 xsh M x 2 · xex2 ex_{− 1} = 1 x " _∞ X k=0 1 (2k + 1)!  M x 2 2k+1# · " _∞ X k=0 Bk(1₂) k! x k # = M 2 " _∞ X k=0 M2k 4k_{(2k + 1)!}x 2k # " _∞ X k=0 2 − 2k 2k_{· k!}x k # ,

ªœJ,si

_x2k

_í[b)

S2k(n) = M 2 · 4k_{(2k + 1)} k X r=0 (2 − 4r)C_2k+12r B2rM2k−2r, (2.27)

â

_M2 _{= 4N + 1}

₎

S2k(n) = M (4k + 2)4k k X r=0 (2 − 4r)C_2k+12r B2r(4N + 1)k−r, S2k(n) = M 4k + 2 k−1 X r=0 " 1 4 r X j=0 (2 − 4j)B2jC_2k+12j C_k−jk−r # Nk−r, (2.28)

ªœ

(2.18)

D

(2.28) N

í[b†)ƒ

(2.26)

_

Q-Vu

_{+›‰bí0§¶}

,

íl

: ìÜ 2.2.3. k ≥ 2 ,

†

k X r=0 (2 − 4r)B2rC2k+12r = 0, (2.29) B2k = −8 3(2k + 1)4k k−2 X r=0 (2 − 4r)B2rC2k+12r Ck−r2 , (2.30) B2k = 2 (2k + 1)4k k−1 X r=0 (2 − 4r)B2rC2k+12r Ck−r1 , (2.31) B2k = (−1)k−1 (2k + 1)(k + 1)!                    C3 3 C31 C5 4 C43 C41 · · · · C_k2k−3 · · · C1 k C_k+12k−1 · · · C_k+13                    . (2.32)

„p

:

I

_{n = 0 ,}

†

_{M = 1, N = 0, S2k(n) = 0 ,}

â

_(2.27)

_¹)ƒ

_{(2.29) ,}

C6â

(2.26)

£

_{A(k, k) = 0}

6)ƒ

_(2.29)

â

_(2.22)

¸

_(2.26)

£

_{A(k, k − 2) = 3A(k, k − 1) = (−1)}k−1_{6(2k + 1)B} 2k

†

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_{(2.30) (2.31) (2.32)}

_

Ù;Æ

₍₁₉₉₇₎



_{4¸íˆD+›‰bílídı2J°)7+›‰b}

B0 ∼ B106 ,

Oç

k

œ×vÿÌ?щ7
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2.2.1

ªJõ|

,

úœ×í

k

M

6ªJ0§Ë|

A(k, r)

í

_M

_,

1/

_Éb°)ø

_{A(k, r)}

í

_M

_,

ÿøÔ°)7

s_

_{4¸t¸+›‰b}

B2k

í

M
7/

,

|(|í

B2k ,

ªàùÜ

2.2.4

'

ñqð„w£ü4

,

*76ÿð„7

A(k, r)

í£ü4
;W¥øj¶

,

T6¢½

h°)7‡

₁₀₆

_

_{+›‰b¸‡}

₁₀₇

_4¸t
¥

_³É|

_{B102 ∼ B106}

¸

S101(N )

í

M
7;W

S2k(N ) = _2k+1M S2k+10 (N )

¹ª|

S100(N )

í

M

' –Ó

(1998)

Ék

Pn i=1i k

_í°¸£w

_+›‰bíl

Ék

Pn i=1ik

íl

,

ÅqÕ˛'Öj¶

,

øOËz

,

ìÜí„p·'µ

Æ

_,

°¸tíl¾6'×
…d*ÌÏÜ|ê

,

#|7jZí]Rt

,

ìÜí„p6'¡

,

1„p7

Pn i=1i k

_°¸Ö

_{á[bDO±í+›‰5Èí}

É[
*7ßZû|7+›‰bí]Rt

ìÜ 2.2.4.

úL<£cb

_{k, n}

-É[A

_: n X i=1 ik = ak0nk+1+ ak1nk+ · · · + akkn,

w2

ak0 = 1 k + 1, akj = 1 k − j + 1[C j k− ak0Ck+1j+1− ak1Ckj− · · · − akjCk−j+22 ], (2.33) j = 1, 2, 3, · · · , k.

„p

:

I

_{fk(Z) − fk(Z − 1) = Z}k _,

_üìw[b

_,

_¹

ak0(x − 1 + 1)k+1+ ak1(x − 1 + 1)k+ · · · + akk(x − 1 + 1) −[ak0(x − 1)k+1+ ak1(x − 1)k+ · · · + akk(x − 1)] = (x − 1 + 1)k,

FJ

ak0[Ck+11 (x − 1)k+ Ck+12 (x − 1)k−1+ · · · + Ck+1k+1] +ak1[Ck1(x − 1) k−1_{+ C}2 k(x − 1) k−1_{+ · · · + C}k k] +ak2[Ck−11 (x − 1) k−2_{+ C}2 k−1(x − 1) k−3_{+ · · · + C}k−1 k−1] + · · · + akk = (x − 1)k+ Ck1(x − 1)k−1+ C_k2(x − 1)k−2+ · · · + 1,

ªœsi[b)

: ak0 = 1 k + 1, akj = 1 k − j + 1 C j k− ak0Ck+1j+1− ak1Ckj − ak2Ck−1j−1− · · · − akj+1Ck−j+22  , j = 1, 2, 3, · · · , k.

üì[b(í

_fk(x)

Å—

_{fk(x) − fk(x − 1) ≡ x}k _{, −∞ < x < +∞ ,}

_éÍ

fk(0) = 0, fk(1) = fk(1) − fk(0) = 1 ,

FJ

fk(n) = [fk(n) − fk(n − 1)] + [fk(n − 1) − fk(n − 2)] + · · · + [fk(2 − fk(1)] = n X i=1 ik,

FJ

n X i=1 ik = m X i=0 akjnk+1−j.  ìÜ 2.2.5. a_k1 = 1₂

„p

:

â]Rt

(2.33)

)

ak1 = 1 k − 1 + 1 C 1 k− ak0C_k+11+1
= 1 2.  ìÜ 2.2.6. ak0+ ak+ · · · + a_k(1 k)×2 = 1 2

„p

:

ÄÑ

f (0) − f (−1) = 0,

FJ

f (−1) = 0,

â

fk(1) + fk(−1) 2 = fk(1) − fk(−1) 2 = 1 2,

¹)…A

 ìÜ 2.2.7. ak 2i+1 = 0, i = 1, 2, · · · ,k−1₂

„p

:

ç

_k

Ñ

_Xbv

_, fk(−n) = [−fk(1 − n) − fk(−n)] − [fk(2 − n) − fk(1 − n)] − · · · − [fk(−1) − fk(−2)] + fk(−1) = − n X i=1 ik,

FJ

fk(n) + fk(−n) = nk, ⇒ 2ak1nk+ 2ak3nk−2+ · · · + 2ak k−1n2 = nk, ⇒ 2ak3nk−2+ 2ak5nk−4+ · · · + 2ak k−1n2 = 0, ⇒ ak3 = ak5 = · · · = ak k−1 = 0,

ç

_k

ÑJbv

_,

ÄÑ

f (−n) = n−1 X i=1 ik,

FJ

fk(n) − fk(−n) = nk = 2ak1nk+ 2ak3nk−2+ · · · + 2akkn.

°šË

_{ak3 = ak5 = · · · = akk = 0 ,}

‚àìÜ

_2.2.5

 ìÜ

_2.2.7

ø

_(2.33)

“Ñ

akj = 1 k − j + 1  j − 1 2(j + 1)C j k− ak2Ck−1j−1− ak4Ck−3j−3− · · · − ak j−2Ck−j+33  , (2.34)

˚

_(2.34)

Ñd²]Rt

,

-Þy„ø_Z]Rt

_ ìÜ 2.2.8. fk(x)

D

fk−1(x)

Öá[bÅ-É[

: akj = kak−j k − j + 1, j = 0, 1, · · · , k − 1.

„p

: [fk(x) − fk(x − 1)]0 = kxk−1 = k[fk−1(x) − fk−1(x − 1)],

¹

[(k + 1)ak0xk+ kak1(x − 1)k−1+ · · · + akk] −[(k + 1)ak0(x − 1)k+ kak1(x − 1)k−1+ · · · + akk] = k[ak−1,0(x − 1)k−1+ · · · + ak−1,k−1(x)] −k[ak−1,0(x − 1)k+ ak−1,1(x − 1)k−1+ · · · + ak−1,k−1(x − 1)],

ªœsi[b)

akj = kak−1 j k + 1 − j, j = 0, 1, 2, · · · , k − 1. (2.35)

˚

(2.35)

Ñó²]Rt

_

½µ‚à

(2.35)

ª)

ùÜ 2.2.5. akj = 1 k + 1C j k+1ajj, (k > j). (2.36)

yø

_(2.36)

Hp

_{(2.33) ,}

1‚àìÜ

_2.2.7

)

ùÜ 2.2.6. aii =              0,

ç

_i

Ñ×k

1

íJbv

1 2,

ç

i = 1

v

1 i+1 _i−1 2 C 4

i+1a22− Ci+14 aii− Ci+1i+2ai−2,i−2 ,

ç

i

Ñ×k

1

íJbv

p

f0(n) = 10+ 20+ · · · + n0 = n ,

‚à

(2.35)

)

fk(x)

í[búi[

:

1 1 1 2 1 2 1 3 1 2 1 6 1 4 1 2 1 4 0 1 5 1 2 1 3 0 −1 30 1 6 1 2 5 12 0 −1 12 0 1 7 1 2 1 2 0 −1 6 0 1 42 1 8 1 2 7 12 0 −7 24 0 1 2 0 1 9 1 2 2 3 0 −7 15 0 2 9 0 −1 30 ... ... ... ... ... ... ... ... ... zpW 2.2.1.

;Wúi$[ªø

n X i=1 i8 = 1 9n 8₊1 2n 8₊ 2 3n 7₋ 7 15n 5₊ 2 9n 3₋ 1 30n,

9õ,

a00 = 1, a11= 1 2, a22 = 1 6, a44 = −1 30, · · · ,

Zu

_{O±í+›‰b}

ìÜ 2.2.9.

I

Bi =      aii, i 6= 1. −aii, i = 1.

†-É[A

z ez_{− 1} = ∞ X n=0 BnZn n! , |z| < 2π. (2.37)

„p

_:

5?

∞ X n=0 BnZn n !  ez _{− 1} z  = 1,

¹

∞ X n=0 BnZn n ! _∞ X n=0 1 (n + 1)!z n_{= 1} ! ,

ø,˝iO

_z

í4bÇ

,

)

_{B0 = 1, zn}

í[bÑ

4n= 1 (n + 1)!B0+ 1 n! B1 1! + 1 (n − 1)! B2 2! + · · · + 1 1! Bn n! ,

ø

(2.36) (

yâìÜ

2.2.5,

ìÜ

2.2.6,

ìÜ

2.2.7

)

_{4n = 0)}

Hp

4n

í[®

,

)

4n = 1 n!(an0− an1+ an2+ ann),

ÄÑ

z ez _{− 1}

Ê

|z| < 2π

j&

,

FJìÜ

2.2.9

A
Ĥ

,

˚Å—É[

(2.37)

í[b

Bn

Ñ

+›‰b

Q6‚àùÜ

2.2.6

)

_{+›‰bílt}

_,

à-B2i = 1 2i + 1  2i − 1 2 − C 2

2i+1B2− C2i+14 B4− · · · − C2i+12i−2B2i−2

 , i = 1, 2, · · · . zpW 2.2.2. B2 = 1 2 + 1 × 2 − 1 2 = 1 6, B4 = 1 2 × 2 + 1  2 × 2 − 1 2 − C 2 5B2  = − 1 30, B6 = 1 7  5 2 − C 2 7B4  = 1 42, · · · .

ú

_{ùéÍ,Šb5óÉû˝}

Ô

_ÛÅ

₍₂₀₀₅₎

_{Õ¯í•}DùéÍ,Šb}

íl

,

lV

_{ÜÍ,Šbí ¯<2£4”
ùéÍ,ŠbªJ*s.}

°íiVj„
øÑ}º½æ

,

ø

_n

_–í7[p

m

_ó°í]ä

,

b°

®].˛

,

†.°í[7j¶bÑ

S2(n, m)

Çø†ÑÕ¯í•}

,

ø

Ö

n

_

jÖíÕ¯/ß}A

m

_Ìå

_{ݲäÕíF.°•}íbñ}

,

_¹

S2(n, m)

ùéÍ,Šb

S2(n, m)

-Þí4”

: 4_{” 2.3.1. S2(n, 0) = 0, S2(n, 1) = 1, S2(n, 2) = 2}n−1 _{− 1, S} 2(n, n − 1) = Cn 2, S2(n, n) = 1

4_{” 2.3.2.}

Å—à-í]Rt

: S2(n, m) = mS2(n − 1, m) + S2(n − 1, m − 1).

ÛÊVõÕ¯

A = {1, 2, 3, 4, 5}

,Öý_.°í

_{gÉ[
.°í•}}

bÑ

S2(5, 1) + S2(5, 2) + S2(5, 3) + S2(5, 4) + S2(5, 5) = 1 + 15 + 25 + 10 + 1 = 52,

Ä7

_A

,

₅₂

_.°ígÉ[

Q-V„pJ-ís_ä

: S2(n, 2) = 2n−1− 1, S2(n, n − 1) = C2n.

„p

_: S2(n, 2) = S2(n − 1, 1) + 2S2(n − 1, 2) = 1 + [S2(n − 2, 1) + 2S2(n − 3, 2)] = 1 + 2 + 4S2(n − 3, 2) = · · · = 1 + 2 + 22+ · · · + 2n−2S2(2, 2) = 2n−1− 1. S2(n, n − 1) = S2(n − 1, n − 2) + (n − 1)S2(n − 1, n − 1) = S2(n − 2, n − 3) + (n − 2)S2(n − 2, n − 2) + (n − 1) = C₂n. 

7jêw ¯<2£4”5(

_,

yV

_{w…àùéÍ,ŠbíNbÞAƒb}

Vl

_{S2(n, m).}

âk

(ex− 1)m ₌  x +x 2 2! + · · · m = ∞ X n=0 an xn n!, (2.38)

w2

an= X n! n1!n2! · · · nm! ,

uú

n1+ n2+ · · · + nm = n

íø~£cbjV°

â

_n

_.°í7/ß[ƒ

m

_.°í]ä³

,

[7j¶u

m!S2(n, m),



an=      0, n < m, P n! n1!n2!···nm! =P C n n1n2···nm = m!S2(n, m), n ≥ m.

ø¥!‹Hp

_(2.38)

)

(ex− 1)m ₌ X_m!S 2(n, m) xn n!, (ex− 1)m _{= C}m me mx_{− C}m m−1e (m−1)x_{+ C}m m−2e (m−2)x_{− · · · + (−1)}m_Cm 0 · 1 = C_nm  1 + m 1!x + m2 2! x 2_{+ · · ·}  −Cm m−1  1 + m − 1 1! x + (m − 1)2 2! x 2_{+ · · ·}  + (−1)mC₀m· 1,

ªœ,si

xn n!

í[b)

m!S2(n, m) = Cmmm n_{− C}m m−1(m − 1) n + C_m−2m (m − 2)n− · · · + (−1)m−1C₁m· 1,

*7)ƒ

S2(n, m) = 1 m! m−1 X k=0 (−1)kC_km(m − k)n.

ø,5!‹TÑ@à

_,

1/V„p-Þs

_: S2(n, n − 1) = C2n, S2(n, n − 2) = C3n+ 3C n 4.

„p

:

ÄÑ

S2(n, n) = 1, X (−1)kC_kn−1 = (n − 1)!C₂n,

FJ

S2(n, n − 1) = C2n ⇒ X(−1)kC₂n−2(n − 2 − k)n ⇒ (n − 2)! {Cn 3 + 3C n 4} ⇒ S2(n, n − 2) = C3n+ 3C4n.



Ž Ü

(2005)

_{ùéÍ,ŠbíÀ4”}

ìÜ 2.3.1. S(n, n − 2) = Cn 3 + 3C4n

„p

_:

â

_{S(n, k) = S(n − 1, k − 1) + kS(n − 1, k),}

w2

_{1 ≤ k ≤ 1}

í]RÉ[

_,

)

S(n, n − 2) = S(n − 1, n − 3) + (n − 2)S(n − 1, n − 2) = S(n − 1, n − 3) + (n − 2)C₂n = S(n − 2, n − 4) + (n − 3)S(n − 2, n − 3) + (n − 2)C₂n−1 = · · · = S(3, 1) + 2C₂3+ 3C₂4+ · · · + (n − 3)C₂n−2+ (n − 2)C₂n−1 = 1 2[1 2_{+ 2}2_{+ · · · + (n − 2)}2_{+ 1}3 _{+ 2}3_{+ · · · + (n − 2)}3_] = 1 2  (n − 2)(n − 1)(2n − 3) 6 + 3(n − 2)2_{(n − 1)}2 12  = n(n − 1)(n − 2) 6 + 3(n − 1)(n − 2)(n − 3) 24 = C₃n+ 3C₄n.  ìÜ 2.3.2. S(n, n − 3) = Cn 4 + 10 + C5n+ 15C6n

„p

:

àbç¦Ñ¶

:

ç

_{n = 4}

v

_,

éÍA

cqç

n = k − 1

vA

,

_¹

S(k − 1, k − 4) = C₄k−1+ 10C₅k−1+ 15C₆k−1,

†

S(k, k − 3) = S(k − 1, k − 4) + (k − 3)S(k − 1, k − 3) = C₄k−1+ C₃k−1+ 10C₅k−1+ 10C₄k−1+ 15C₅k−1+ 15C₅k−1 +(k − 4)C₃k−1+ (3k − 19)C₄k−1− 15Ck−1 6 = C₄k+ 10C₅k+ 15C₆k.  ìÜ 2.3.3. mn_{= C}m k · S(n, k) · k!,

w2

m, n

u£cb

„p

:

íl

,

¯,ø_}º½æq)ƒìÜí„p

,

_¹z

n

_.°í7[p

m

_

.°í]ä³

,

or˛]

,

†[7íj¶buÖý

j¶ø

_:

ø

_n

_.°í7}pÑ

_{x1, x2, · · · xn, m}

_.°í]ä}pd

b1, b2, · · · , bm,

©_7

xi(1 ≤ i ≤ n)

·ªJ[p

m

_.°í]ä

bj(1 ≤ j ≤ m)

2íLSø_

,

_{¹©_7·}

m

.°í[¶

,

Ä7

n

_.°í7u

mn

_

[¶

j¶ù

:

lVõ

n

_.°í7[p

m

_.°í]ä³

,

.

_or˛]íj

bÑ

_{S(n, m)m!}

9õ,

_,

lwÑ]äuêró°í

_,

z

n

_7ZAí ¯

A{x1, x2, · · · , xn}

•}A

m

_ݲäÕ

A1, A2, · · · , Am ,

Í(ø©_

Ai(1 ≤ i ≤ m)

[p©ø_]ä³

,

ZAø_³˛]í}ºj
¥³

A1, A2, · · · , Am

u

_A

íø_

m

}

_’

_,

wjbÑ

_{S(n, m)}

7¥³í

m

_]äuêr.°í

FJ

, A

í

_m

_ä

Õí.°§ZAí}ºju.°í

,

_¹Ñ

S(n, m)m!

*

_m

_.°í]äL²ø_íj¶bÑ

Cm 1 ,

z7r¶[p

,

”ìÑ˛]

,

†j¶bÑ

_C1 mS(n, 1)1!

â¤ø

m

2²

k

_.°í]äz

n

_7r¶[p

,

b

°²|í

k

_.°í

_]ä.or˛í

_,

„

_{²í·Ñ˛]
;W ¶ŸÜø}

_,

j

¶bÑ

_Cm k S(n, k)k!

;W‹¶ŸÜø

,

z

n

_.°í7[p

m

_.°í]ä

³

,

_or˛]

,

†[7íj¶bu

Pm k=1Ckm· S(n, k) · k!

âj¶ø£j¶ùªøìÜA

_

¡ †±3

(2001)

ùéÍ,Šbí4”£w@à

cq

A

uø_Ý˛Õ¯

, A1, A2, · · · , Ak

uÕ¯

A

íݲäÕ¯

,

à‹

Ai∩ Aj = φ(i 6= j), A1 ∪ A2 ∪ · · · ∪ Ak = A,

†˚

A1, A2, · · · , Ak

ÑÕ¯

A

íø

_

_k

_éí}•
ø

_n

_jÖíÕ¯}Ñ

k

_{éí.°}•í_bpÑ}

S(n, k),

†

˚

S(n, k)

Ñ

_{ùéÍ,Šb
.°}•í,b}

₍

}ÑL<b

₎

pÑ

Tn ,

ku

Tn =Pn_k=1S(n, k)

cq

_{A1, A2, · · · , Ak}

uÕ¯

_A

íø_

_k

_é}•

_,

à‹

_{|Ai| ≥ r(i = 1, 2, · · · , k),}

w2

_{1 ≤ r ≥ n,}

_{¥ší}•_bpÑ}

_{Sr(n, k),}

†˚

Sr(n, k)

Ñ

2ùéÍ,

Šb
w,bpÑ

: Tr(n) = n X k=1 Sr(n, k).

Ê7

_{jêùéÍ,ŠbJ£2ùéÍ,Šbí–1¸pU(}

_,

Q-V

²w…w4”
à-

: 4_{” 2.3.3. S(n, 1) = S(n, n) = 1, S(n + 1, k) = S(n, k − 1) + kS(n, k)}

„p

:

5?

n + 1

_jÖíÕ¯

A ,

Ê}¥_Õ¯Ñ

k

_äÕí}é2

,

/_j

Ö

a ,

ªJÀÖ Aø_äÕ

{a} ,

ku¥}é

S(n, k − 1)

_

_;

C6jÖ

a

‹pƒwF˛% A

k

_äÕ2íø_

,

¥}é

kS(n, k)

_

_,

⋶ŸÜ

S(n + 1, k) = S(n, k − 1) + kS(n, k).

4_{” 2.3.4.} P∞ n=k S(n,k) zn+1 = 1 z(z−1)(z−2)···(z−k)

„p

_:

p

Fk(z) = ∞ X n=k S(n, k) zn+1 ,

ku

(z − k)Fk(z) = (z − k) ∞ X n=k−1 S(n + 1, k) zn+2 = ∞ X n=k−1 S(n + 1, k) − kS(n, k) zn+1 = ∞ X n=k−1 S(n, k − 1) zn+1 = Fk−1(z),

*7

Fk(z) Fk−1(z) = 1 z − k,

FJ

Fk(z) = F0(z) F1(z)F2(z) · · · Fk(z) F0(z)F1(z) · · · Fk−1(z) = 1 z(z − 1) · · · (z − k). (2.39)  4_{” 2.3.5.}

ç

_{n ≥ 1}

v

_{, S(n, k) =} 1 k Pk j=0(−1)jCjk(k − j)n

„p

:

ø

_(2.39)

¬i}jA¶M}(k

1 k!· 1 z − k − 1 1!(k − 1)!· 1 z − k + 1 + · · · + (−1)k−1 (k − 1)!1!· 1 z − 1+ (−1)k k!z ,

O

_z−1

_í4Ç

_,

_15?

_z−n−1

_í[b

_¹ª



4_{” 2.3.6. S(n, k)}

í

_ƒb

P∞ n=0S(n, k)x n₌ xk (1−x)(1−2x)···(1−kx), k ≥ 1

„p

:

q

Mk(x) = ∞ X n=0 S(n, k)xn,

‚à]RÉ[

S(n, 1) = S(n, n) = 1, S(n + 1, k) = D(n, k − 1) + kS(n, k),

·<ƒ

S(0, k) = 0,



Mk(x) = (kx)Mk(x) + xMk−1(x) ⇒ Mk(x) = x 1 − kxMk−1(x),

·<ƒ

_{M0(x) = 1,}

)

∞ X n=0 S(n, k)xn= x k (1 − x)(1 − 2x) · · · (1 − kx), k ≥ 1.  4_{” 2.3.7. S(n, k)}

íNb

_ƒb

: P∞ n=0 S(n,k) n! x n₌ 1 k!(e x_{− 1)}k

„p

:

õÒ,

,

‚à4”

2.3.5

ªJ)

∞ X n=0 S(n, k) n! x n ₌ 1 k! ∞ X n=0 " _k X j=0 (−1)jC_jk(k − j)n # xn n! = 1 k! k X i=0 (−1)k−iC_ik ∞ X n=0 in_xn n! ! = 1 k! k X i=0 (−1)k−iC_ikeix = 1 k!(e x_{− 1)}k_{. (2.40)}

 4_{” 2.3.8. S(n, k) =} 4k0n k!

4_{” 2.3.9.}

úkL<õb

_x



_: xn= S(n, 1)x + S(n, 2)x(x − 1) + · · · + S(n, n)x(x − 1) · · · (x − n + 1).

„p

:

úk

_ƒb

_{F (x) = x}n_,

_‚àÏ}t

F (x) = n X k=0 C_kx4kF (0),

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_ 4_{” 2.3.10.}

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T0 = 1,

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P∞ n=0 Tnxn n! = e ex−1

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: ∞ X n=0 Tnxn n! = 1 + ∞ X n=1 n X k=1 S(n, k) n! x n = 1 + ∞ X k=1 ∞ X n=k S(n, k) n! x n = ∞ X k=0 (ex_{− 1)}k k! = eex−1.  4_{” 2.3.11. Tn+1} =Pn k=0C n kTn−k

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: ex· eex₋₁ = ∞ X n=0 xn n! ∞ X n=0 Tn n!x n = ∞ X n=0 " _n X k=0 C_knTn−k # xn n!

= eex−1
t = ∞ X n=0 Tn n!x n !t = ∞ X n=0 Tn+1 n! x n_,

ªœsi

xn

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Tn+1 = n X k=0 C_knTn−k.  4” 2.3.12.

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_{n ≥ 1}

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_{, Tn=} 1 e P∞ k=1 kn k!

J-†Ñ

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à-}

_: 4_{” 2.3.13. Sr}(n, k) = Cn r−1Sr(n, k − 1) + kS(n, k), 1 ≤ k ≤ n, 0 ≤ r ≤ n,

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Sr(n, k) = Cr−1n Sr(n, k − 1) + kS(n, k), 1 ≤ k ≤ n, 0 ≤ r ≤ n. (2.41)  4_{” 2.3.14.} ∞ X n=0 Sr(n, k) n! x n₌ 1 k!  ex− 1 − x −x 2 2! − · · · − xr−1 (r − 1)! 

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j&[ý

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: S2(m + 1, n) = S2(m, n − 1) + nS2(m, n), (2.42)

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_{S2(m, 1) = S2(m, m) = 1, m, n ∈ N.}

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m − n

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S2(m + 1, n) = m−n+1 X i=0 ni· S2(m − i, n − 1). (2.43)

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S2(m + 1, n)

bí

j&[ýt

: ìÜ 2.3.4. S2(m + 1, n) = (−1) n n! Pn j=1(−1) j _{· C}n j · jm+1, m, n ∈ N.

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:

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:

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S2(m + 1, k + 1) = m−k X i=0 (k + 1)i(−1) k k! k X j=1 (−1)j · Ck j · jm−i = (−1) k k! k X j=1 (−1)j · Ck j m−k X i=0 jm−i· (k + 1)i = (−1) k k! k X j=1 (−1)j · C_jk· jk m−k X i=0 jm−k−i· (k + 1)i = (−1) k (k + 1)! k X j=1 (−1)j· Ck+1 j · j k_{·(k + 1)}m−k+1_{− j}m−k+1

= (−1) k (k + 1)! " (k + 1)m−k+1 k X j=1 (−1)j· Ck+1 j · j k₋ k X j=1 (−1)j· Ck+1 j · j m+1 # = (−1) k (k + 1)! ( (k + 1)m−k+1 "_k+1 X j=0 (−1)j · Ck+1 j · j k_{− (−1)}k+1_{(k + 1)}k # − k X j=1 (−1)j· C_jk+1· jm+1 ) .

@àøíì

k+1 X j=0 (−1)j· Ck+1 j · j k _{= 0,}

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S2(m + 1, k + 1) = (−1)k+1 (k + 1)! " _k X j=1 (−1)j· Ck+1 j · j m+1_{+ (−1)}k+1_{· (k + 1)}m+1 # = (−1) k+1 (k + 1)! k+1 X j=1 (−1)j · Ck+1 j · j m+1_.

Ä7ç

n = k + 1

v

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2.3.4

6uA
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2.3.4

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n, m ∈ N

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ùÜ 3.1.1.

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_{T , |T | < 2π,} ∞ X k=0 Sk(x) Tk k! = eT (x+1)− eT eT _{− 1} .

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Sk(x) = 1 k + 1 ( (x + 1)k+1− k X i=1 C_i+1k+1Sk−i(x) − 1, ) k ≥ 1,

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(x + 1)k+1 = 1 + k X i=0 C_i+1k+1Sk−i(x) = 1 + (k + 1)! k X i=0 1 (i + 1)! Sk−i(x) (k − i)!,

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(x + 1)k+1 (k + 1)! = 1 (k + 1)! + k X i=0 1 (i + 1)! Sk−i(x) (k − i)!,

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k X i=0 Ti+1 (i + 1)! Sk−i(x)Tk−i (k − i)! ,

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∞ X i=0 Si(x)Ti i! ,

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(Cauchy product),

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∞ X k=0 (x + 1)kT k k! = e T + (eT − 1) ∞ X k=0 Sk(x) Tk k!.  ùÜ 3.1.2.

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S_k0(−1) = S_k0(0).

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3.1.1

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_{fn}∞ n=0,

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∞ X k=0 S_k0(x)T k k! = T eT (x+1) eT _{− 1} ,

ªø¥¦

x = 0

ª)

T eT eT _{− 1} = ∞ X k=0 S_k0(0)T k k!, T eT _{− 1} = ∞ X k=0 S_k0(−1)T k k!, (3.1)

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(3.1)

ª)

T = ∞ X k=0 {S_k0(0) − S_k0(−1)} T k k!, S₀0(0) = S₀0(−1), S₁0(0) − S₁0(−1) = 1, S_k0(0) = S_k0(−1), ∀k ≥ 2.  ùÜ 3.1.3.

úLS£cb

_{k ,}

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S_2k0 (x − 1) = −(2k)! 22k−1_(πi)2k ∞ X n=1 1 n2kcos(2nπx), ∀x ∈ [0, 1], (3.2) S_2k+10 (x − 1) = −(2k + 1)! 22k_(πi)2k+1 ∞ X n=1 i n2k+1sin(2nπx), ∀x ∈ [0, 1]. (3.3)

„p

_:

âùÜ

_3.1.3

ª

_{“¨|U‚ƒb}

_{Ck(x), k ≥ 2.}

ì2à-

:Ck(x) = Sk0( x 2π− 1)

ú

x ∈ [0, 2π]

J£

Ck(x) = Ck(x + 2π)

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x ∈ [0, 2π] c

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Ck(x)

í

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Ck(x) =P_n∈Zan,keinx ,

/

an,k = 1 2π Z 2π 0 e−inxCk(x)dx.

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an,k = Z 1 0 e−2nπxiS_k0(x − 1)dx, n ∈ Z, k ≥ 2.

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Z 1 0 e−2nπxiS₀0 (x − 1) dx = Z 1 0 e−2nπxidx = 0, Z 1 0 e−2nπxiS₁0 (x − 1) dx = − 1 2nπi, |T | ≤ 2π,

/

T T − 2nπi = Z 1 0 e−2nπxi T e T x eT _{− 1}dx = Z 1 0 e−2nπxi ( _∞ X k=0 S_k0(x − 1)T k k! ) dx = ∞ X k=0 Tk k! Z 1 0 e−2nπxiS_k0(x − 1)dx,

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Z 1 0 e−2nπxi T e T x eT _{− 1}dx = − T 2nπi+ ∞ X k=2 an,k Tk k!.

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,

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_{|T | < 2π ,}

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T T − 2nπi = − ∞ X k=1  T 2nπi k ,

<N

− T 2nπi+ ∞ X k=2 an,k Tk k! = − ∞ X k=1  T 2nπi k ,

à¤

an,k = − k! (2nπi)k, n ∈ Z, n 6= 0, k ≥ 2.

ç

_{n = 0}

v

_,

ÄÑ

Z 1 0 e−2nπxiS_k0(x − 1)dx = Sk(0) − Sk(1) = 0, k ≥ 2,

úLS£cb

_{k ≥ 2,}

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_{Ck(x) =}P n∈Zan,ke inx_,

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S_k0  x 2π − 1  = X n∈Z,n6=0 −k! (2nπi)ke inx_,

Ĥ

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úLS£cb

_{k ≥ 2, x ∈ [0, 1],} S_k0 (x − 1) = X n∈Z,n6=0 −k! (2nπi)ke 2nπxi ₌ −k! (2πi)k X n∈Z,n6=0 1 nke 2nπxi_. 

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(1988



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_{^*ƒbD+›‰b}

,



ζ(2k) = ∞ X n=1 1 n2k = (−1) k−1₂2k−1B2kπ2k (2k)! , k ∈ N, (3.4)

¢‚àùÜ

3.1.3,

)ƒ

S_2k0 (−1) = (−1) k−1_(2k)! 22k−1_(π)2k ζ(2k), S 0 2k+1(−1) = 0, k ∈ N,

Ĥû|

_S0 2k(−1) = B2k, k ∈ N

W¤y!¯

(1.1)

û|

B0 = S00(−1), B1 = S10(−1), B2k = 1 2k + 1 ( C_2k2k+1S₁0(−1) + k X i=1 C_2i+12k+1S_2k−2i0 (−1) ) , k ∈ N. (3.5)

ù

‚à

_Matlab

_l+›‰b

;W

(3.5),

û˝6‚à

Matlab 7.1

_l|+›‰b

B2

B

B50

í

M

,

w

Mà-

: 1. B2 k = 1, S₀0 = 1.

FJ

B2 = − 1 3 ( C₂3· −1 2+ 1 X i=1 C_2i+13 S_2−2i0 (−1) ) = 1.666666666666667e − 001. 2. B4 k = 2, ζ(2) = 1 6π 2 , S₂0 = 1.666666666666667e − 001.

FJ

_{B4 = −}1 5 ( C₄5· −1 2+ 2 X i=1 C_2i+15 S_4−2i0 (−1) ) = −3.333333333333340e − 002. 3. B6 k = 3, ζ(4) = 1 90π 4 , S₄0 = −3.333333333333340e − 002.

FJ

_{B6 = −}1 7 ( C₆7· −1 2+ 3 X i=1 C_2i+17 S_6−2i0 (−1) ) = 2.380952380952404e − 002. 4. B8 k = 4, ζ(6) = 1 945π 6_{, S}0 6 = 2.380952380952404e − 002.

FJ

B8 = − 1 9 ( C₈9· −1 2+ 4 X i=1 C_2i+19 S_8−2i0 (−1) ) = −3.333333333333469e − 002.

5. B10 k = 5, ζ(8) = 1 9450π 8_{, S}0 8 = −3.333333333333469e − 002.

FJ

B10 = − 1 11 ( C₁₀11· −1 2+ 5 X i=1 C_2i+111 S_10−2i0 (−1) ) = 7.575757575758833e − 002. 6. B12 k = 6, ζ(10) = 1 93555π 10_{, S}0 10= 7.575757575758833e − 002.

FJ

B12 = − 1 13 ( C₁₂13· −1 2+ 6 X i=1 C_2i+113 S_12−2i0 (−1) ) = −2.531135531137224e − 001. 7. B14 k = 7, ζ(12) = 691 638512875π 12_{, S}0 12 = −2.531135531137224e − 001.

FJ

B14 = − 1 15 ( C₁₄15· −1 2+ 7 X i=1 C_2i+115 S_14−2i0 (−1) ) = 1.166666666669796e + 000. 8. B16 k = 8, ζ(14) = 2 18243225π 14_{, S}0 14= 1.166666666669796e + 000.

FJ

B16 = − 1 17 ( C₁₆17· −1 2+ 8 X i=1 C_2i+117 S_16−2i0 (−1) ) = −7.092156862821243e + 000. 9. B18 k = 9, ζ(16) = 3617 325641566250π 16_{, S}0 16 = −7.092156862821243e + 000.

FJ

B18 = − 1 19 ( C₁₈19· −1 2 + 9 X i=1 C_2i+119 S_18−2i0 (−1) ) = 5.497117794722328e + 001. 10. B20 k = 10, ζ(18) = 43867 38979295480125π 18 , S₁₈0 = 5.497117794722328e + 001.

FJ

B20 = − 1 21 ( C₂₀21· −1 2 + 10 X i=1 C_2i+121 S_20−2i0 (−1) ) = −5.291242425151527e + 002. 11. B22 k = 11, ζ(20) = 174611 1531329465290625π 20_{, S}0 20 = −5.291242425151527e + 002.

FJ

B22 = − 1 23 ( C₂₂23· −1 2 + 11 X i=1 C_2i+123 S_22−2i0 (−1) ) = 6.192123192661362e + 003. 12. B24 k = 12, ζ(22) = 155366 13447856940643125π 22_{, S}0 22 = 6.192123192661362e + 003.

FJ

B24 = − 1 25 ( C₂₄25· −1 2 + 12 X i=1 C_2i+125 S_24−2i0 (−1) ) = −8.658025335156410e + 004. 13. B26 k = 13, ζ(24) = 236364091 201919571963756521875π 24_, S₂₄0 = −8.658025335156410e + 004.

FJ

B26 = − 1 27 ( C₂₆27· −1 2 + 13 X i=1 C_2i+127 S_26−2i0 (−1) ) = 1.425517182341780e + 006.

14. B28 k = 14, ζ(26) = 1315862 11094481976030578125π 26 , S₂₆0 = 1.425517182341780e + 006.

FJ

B28 = − 1 29 ( C₂₈29· −1 2+ 14 X i=1 C_2i+129 S_28−2i0 (−1) ) = −2.729823226851125e + 007. 15. B30 k = 15, ζ(28) = 6785560294 564653660170076273671875π 28_, S₂₈0 = −2.729823226851125e + 007.

FJ

B30 = − 1 31 ( C₃₀31· −1 2+ 15 X i=1 C_2i+131 S_30−2i0 (−1) ) = 6.015809797412372e + 008. 16. B32 k = 16, ζ(30) = 6892673020804 5660878804669082674070015625π 30_, S₃₀0 = 6.015809797412372e + 008.

FJ

_{B32 = −}1 33 ( C₃₂33· −1 2+ 16 X i=1 C_2i+133 S_32−2i0 (−1) ) = −1.511632640519539e + 010. 17. B34 k = 17, ζ(32) = 7709321041217 62490220571022341207266406250π 32_, S₃₂0 = −1.511632640519539e + 010.

FJ

B34 = − 1 35 ( C₃₄35· −1 2+ 17 X i=1 C_2i+135 S_34−2i0 (−1) ) = 4.296158524259357e + 011.

18. B36 k = 18, ζ(34) = 151628697551 12130454581433748587292890625π 34 , S₃₄0 = 4.296158524259357e + 011.

FJ

B36 = − 1 37 ( C₃₆37· −1 2 + 18 X i=1 C_2i+137 S_36−2i0 (−1) ) = −1.371180959826842e + 013. 19. B38 k = 19, ζ(36) = 26315271553053477373 20777977561866588586487628662044921875π 36_, S₃₆0 = −1.371180959826842e + 013.

FJ

B38 = − 1 39 ( C₃₈39· −1 2+ 19 X i=1 C_2i+139 S_38−2i0 (−1) ) = 4.883543134530475e + 014. 20. B40 k = 20, ζ(38) = 308420411983322 2403467618492375776343276883984375π 38_, S₃₈0 = 4.883543134530475e + 014.

FJ

_{B40 = −} 1 41 ( C₄₀41· −1 2 + 20 X i=1 C_2i+141 S_40−2i0 (−1) ) = −1.930005581237949e + 016. 21. B42 k = 21, ζ(40) = 261082718496449122051 20080431172289638826798401128390556640625π 40_, S₄₀0 = −1.930005581237949e + 016.

FJ

B42 = − 1 43 ( C₄₂43· −1 2+ 21 X i=1 C_2i+143 S_42−2i0 (−1) ) = 8.422996050255715e + 017.

22. B44 k = 22, ζ(42) = 3040195287836141605382 2307789189818960127712594427864667427734375π 42_, S₄₂0 = 8.422996050255715e + 017.

FJ

B44 = − 1 45 ( C₄₄45· −1 2 + 22 X i=1 C_2i+145 S_44−2i0 (−1) ) = −4.045434872315526e + 019. 23. B46 k = 23, ζ(44) = 5060594468963822588186 37913679547025773526706908457776679169921875π 44 , S₄₄0 = −4.045434872315526e + 019.

FJ

B46 = − 1 47 ( C₄₆47· −1 2 + 23 X i=1 C_2i+147 S_46−2i0 (−1) ) = 2.139462175522717e + 021. 24. B48 k = 24, ζ(46) = 103730628103289071874428 7670102214448301053033358480610212529462890625π 46_, S₄₆0 = 2.139462175522717e + 021.

FJ

B48 = − 1 49 ( C₄₈49· −1 2 + 24 X i=1 C_2i+149 S_48−2i0 (−1) ) = −1.264407313304302e + 023. 25. B50 k = 25, ζ(48) = 5609403368997817686249127547 4093648603384274996519698921478879580162286669921875π 48_, S₄₈0 = −1.264407313304302e + 023.

FJ

B50 = − 1 51 ( C₅₀51· −1 2 + 25 X i=1 C_2i+151 S_50−2i0 (−1) ) = 8.884654931324014e + 024.

ú

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Bn= n X k=1 1 k + 1 k X j=1 (−1)j· Ck j · j n_{, (3.6)}

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_(3.6)

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Bm+1 = m+1 X k=1 1 k + 1 k X j=1 (−1)j · Ck j · j m+1_{. (3.7)}

¢ÄÑìÜ

_2.3.4 S2(m + 1, n) = (−1)n n! n X j=1 (−1)j· C_jn· jm+1, ⇒ S2(m + 1, k) = (−1)k k! k X j=1 (−1)j · Ck j · j m+1_,

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k X j=1 (−1)j· Ck j · jm+1 = k! · (−1)k· S2(m + 1, k), (3.8)

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Bm+1 = m+1 X k=1 (−1)k k + 1 · k! · S2(m + 1, k). (3.9)

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(3.9),

û˝6ªJ)ƒ

:

ç

_{m = 1}

v

_, B2 = 2 X k=1 (−1)k k + 1 · k! · S2(2, k) = −1 2· 1! · S2(2, 1) + 1 3 · 2! · S2(2, 2) = 1 6.

ç

_{m = 3}

v

_, B4 = 4 X k=1 (−1)k k + 1 · k! · S2(4, k) = −1 2· 1! · S2(4, 1) + 1 3 · 2! · S2(4, 2) + − 1 4· 3! · S2(4, 3) + 1 5 · 4! · S2(4, 4) = − 1 30.

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