N
(ai)"
i=0
n " N ai = 0 i > n
a = (ai)" b = (bi)"
i=0 i=0
a b c = a " b
i
ci = ajbi-j = a0bi + a1bi-1 + · · · + aib0
j=0
i " N
x y x+y
ai = bi = ci =
i i i
c = a " b
(ai)n
i=0
n
a A(t) = aiti
i=0
B
B
 B
i
A B C D
a b c d A
b c B a
C a b d D
c d
a b c d
A
B
C
D
[5] i =
1, 2, . . . , 5 i i (i + 1)
(k1, k2, . . . , k5)
,
i ki
m × n
m n [n]
îÅ‚ Å‚Å‚
4 1 5 3 2
ðÅ‚ ûÅ‚
1 2 4 5 3
2 5 3 4 1
.
B
B ri
i B
B B
r0 = 1 rB(t) (ri )i=0,1,...
B
"
B
rB(t) = ri ti.
i=0
m × n
min(m, n) rB(t)
" B
rB(t)
" rB(t)
" B
B1 B2
B = ,
B3 B4
B2 B3
rB(t) = rB (t)rB (t)
1 4
B s
B B1 B
s B2
B s
rB(t) = rB (t) + t · rB (t).
1 2
i
s
B1 B2
s ri ri-1
Å»
B B B
B n × n
n
Å»
B
B B B
n! - (n - 1)!r1 + (n - 2)!r2 - · · · + (-1)n0!rn .
A
n
B i = 1, 2, . . . , n
Ai A i
B n
Å»
B A
B
n
|A| - | Ai| = |A| - |Ai| + |Ai )" Aj| - · · · + (-1)N |A1 )" A2 )" · · · )" An|.
i=1 i i |Ai )" Ai )" · · · )" Ai |
1 2 k
n i1 i2, . . . , ik
B k
i1 i2, . . . , ik
B (n - k)! (n - k)
k i1 < i2 < · · · < ik
B
(n - k)!rk
[n]
n B
n × n
rB(t)
Å»
n
n
rB(t) = (1 + t)n = ti.
Å»
i
i=0
1 + t
n n
n n!
(-1)i(n - i)! = (-1)i .
i i!
i=0 i=0
 a = (ai)"
i=0
"
A(t) = aiti = a0 + a1t + a2t2 + . . . .
i=0
(ai)
A(t)
A(t) B(t)
a b a " b
a0b0 + (a0b1 + a1b0)t + (a0b2 + a1b1 + a2b0)t2 + . . . ,
A(t) B(t)
t t
t
"
A(t) = aiti [ti]A(t)
i=0
ti
[ti]A(t) = ai.
A(t) = 0 i

ai = 0 A(t) n A(t) = n

A(t) · B(t) = A(t) + B(t)
"
A(t) = aiti A(t) =
i=0
0 a0 = 0

"
A(t) = aiti
i=0
"
a0 = 0 B(t) = biti A(t)B(t) = 1

i=0
a0b0 = 1
a0bi + a1bi-1 + · · · + aib0 = 0 i > 0,
(bi)
1
b0 =
a0
1
bi = - (a1bi-1 + · · · + aib0) i > 0.
a0
"
1
= iti.
1 - t
i=0
 x Px(t)
"
x
i
i=0
"
x
Px(t) = ti
i
i=0
x(x - 1) x(x - 1) . . . (x - i + 1)
= 1 + xt + t2 + · · · + ti + . . . .
2 i!
x = n
Pn(t) = (1 + t)n
Px(t)Py(t) = Px+y(t)
x, y " R Px(t) x
1
(1 + t) x = -n x = n
n
"
1 i + n - 1
(1 + t)-n = = (-1)i ti
(1 + t)n n - 1
i=0
"
1 i + n - 1
(1 - t)-n = = ti
(1 - t)n n - 1
i=0
"
"
(n - 1)(2n - 1) · · · ((i - 1)n - 1)
n
(1 + t)1/n = 1 + t = 1 - (-1)i ti
nii!
i=1
"
"
(n - 1)(2n - 1) · · · ((i - 1)n - 1)
n
(1 - t)1/n = 1 - t = 1 - ti
nii!
i=1
t -t
n
tn
(1 + t + t2 + · · · + ti + . . . )(1 + t2 + t4 + · · · + t2i + . . . )(1 + t5 + t10 + · · · + t5i + . . . ) =
1
= .
(1 - t)(1 - t2)(1 - t5)
Å„Å‚
0 n = 0;
òÅ‚
Fn = 1 n = 1;
ół
Fn-1 + Fn-2 n e" 2.
 F(t) = F0 + F1t + F2t2 + . . .
t2F(t) = F0t2 + f1t3 + · · · + Fn-2tn+ . . .
tF(t) = F0t+F1t2 + F2t3 + · · · +Fn-1tn+ . . .
(t + t2)F(t) = F2t2 + F3t3 + · · · + Fntn + . . .
= F(t) - t,
t
F(t) = .
1 - t(t + 1)
"
F(t) = t ti(1 + t)i =
i=0
" i
i
= t ti tj =
j
i=0 j=0
ëÅ‚ öÅ‚
"
n - j
íÅ‚ Å‚Å‚
= t tn =
j
n=0 j
ëÅ‚ öÅ‚
"
n - 1 - j
íÅ‚ Å‚Å‚
= t tn
j
n=0 j
n - 1 n - 2 n - 3
Fn = + + + . . . .
0 1 2
"
1 Ä… 5
1 - t - t2 = (1 - 1t)(1 - 2t), 1,2 = ,
2
1 1
F(t)
1-1t 1-2t
Ä…1 Ä…2
F(t) = + =
1 - 1t 1 - 2
"
= (Ä…1n + Ä…2n)tn.
1 2
n=0
 (un)"
n=0
un+r + ar-1un+r-1 + · · · + a1un+1 + a0un = 0,
a0 = 0

1 s
r + ar-1r-1 + · · · + a1 + a0 = ( - 1)k . . . ( - s)k ,
i = j i = j (un)"

n=0
(njn)" 1 d" i d" s 0 d" j d" ki - 1
i n=0
tn+r
n
" " " "
un+rtn+r+ar-1t un+r-1tn+r-1+· · ·+a1tr-1 un+1tn+1+a0tr untn = 0.
n=0 n=0 n=0 n=0
"
U(t) = untn (un)"
n=0 n=0
r-1 r-2
U(t) - aiti +ar-1 U(t) - aiti +· · ·+a1tr-1 (U(t) - u0)+a0trU(t) = 0,
i=0 i=0
W(t)
U(t) = ,
1 + ar-1t + · · · + a1tr-1 + a0tr
W(t) t r - 1
Ä…1 Ä…2 Ä…k
1
U(t) = + + · · · + +
1
1 - 1t (1 - 1t)2 (1 - 1t)k
Ä…k +1 Ä…k +2 Ä…k +k2
1 1 1
+ + + · · · + + · · · +
2
1 - 2t (1 - 2t)2 (1 - 2t)k
Ä…k +···+ks-1+1 Ä…k +···+ks-1+2 Ä…k +···+ks-1+ks
1 1 1
+ + + · · · + ,
s
1 - st (1 - st)2 (1 - 2t)k
Ä…i
"
1 n + j - 1
= ntn,
i
(1 - it)j j - 1
n=0
"
n+j-1
(un)" n
n=0 i
j-1
n=0
n+j-1
1 d" i d" s 0 d" j d" ki
j-1
j - 1 n (un)"
n=0
 n
" n = 0 n > 0 (L, R) L R
n - 1
cn n
Å„Å‚
c0 = 1,
ôÅ‚
òÅ‚
n-1
cn = cjcn-1-j, n > 0.
ôÅ‚
ół
j=0
cn
n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 . . .
cn 1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 . . .
C(t)
tn n e" 1
ëÅ‚ öÅ‚
" " n-1
íÅ‚
cntn = t cjcn-1-jłł tn-1,
n=1 n=1 j=0
C(t) - c0 = t · C2(t),
t · C2(t) - C(t) + 1 = 0.
C(t)
C(t)
"
1 Ä… 1 - 4t
t.
"
"
1 · 3 · · · · · (2i - 3)
1 - 4t = 1 - (4t)i
2ii!
i=1
"
2 2i - 2
= 1 - ti.
i i - 1
i=1
C(t) +
n " N
1 2n
cn = .
n + 1 n
 n × n
(a1, a2, . . . , a2n) Ä…1
a1 + · · · + ar
sn n
sn
"
sntn
n=0
n
n - 1
n > 1
(pn)"
n=0
n
sn = sn-1 + pn-2,
pn
pn = pn-1 + sn-2,
n e" 2
s0 = s1 = 1 p0 = 0 p1 = 1
sn
pn
" "
S(t) = sntn P (t) = pntn
n=0 n=0
S(t) - s0 - s1t = t(S(t) - s0) + t2 · P (t)
P (t) - p0 - p1t = t(P (t) - p0) + t2 · S(t),
(t - 1)S(t) + t2 · P (t) = -1
t2 · S(t) + (t - 1)P (t) = -t.
S(t)
1 - t + t3
S(t) = . (")
1 - 2t + t2 - t4
(1 - 2t + t2 - t4)S(t)
n e" 4
[tn](1 - 2t + t2 - t4)S(t) = 0,
sn = 2sn-1 - sn-2 + sn-4 n e" 4. ("")
sn
s0 = s1 = 1 s2 = 1
s3 = 2
sn
n-4
sn = sn-1 + sk + 1, n e" 4,
k=0
 010 . . . 01
10 . . . 01
n-2 n
sn = 1 + (1 + sl-k-3)
k=1 l=k+3
k l
sn
1
60 - · 12
3
s10 ("")
s10 = 72
sn
S(t) (")
("")
sn k
r
k k-1
r + k - 1
k
"
n - 2k + 1
sn = .
2k
k=0
" "
n - 2k + 1
tn
2k
n=0 k=0
(")
" " " "
n - 2k + 1 n - 2k + 1
tn = tn
2k 2k
n=0 k=0 k=0 n=0
" " "
n - 2k + 1
= tn + tn
2k
n=0 k=1 n=4k-1
" "
1 r + 2k
= + tr+4k-1
1 - t 2k
k=1 r=0
"
1 t4k-1
= +
1 - t (1 - t)2k+1
k=1
1 - t + t3
=
(1 - t)2 - t4
"
f(t) = aiti
i=0
N
fN (t) = aiti N = 0, 1, . . .
i=0
(fn(t))" f(t) =
n=0
"
aiti i " N Ni n > Ni
i=0
[ti]fn(t) = ai f(t) = limn" fn(t)
"
aiti =
i=0
limN" N aiti
i=0
(fn(t))"
n=0
lim fn(t) = ".
n"
"
N
(1 + fn(t))
n=0
N=0
"
(1 + fn(t)).
n=0
d
dt
" "
d
aiti = iaiti-1
dt
i=0 i=1
"
= (i + 1)ai+1ti.
i=0
d d d
(f(t) + g(t)) = (f(t)) + (g(t))
dt dt dt
d d d
(f(t) · g(t)) = (f(t)) · g(t) + f(t) · (g(t)).
dt dt dt
f(t)
g(t) h(t)
d
f(t)
dt
d d
f(g(t)) = h(g(t)) · g(t).
dt dt
exp ln
et exp(t)
"
1
et = exp(t) = tn.
n!
n=0
t f(t)
"
1
ef(t) = exp(f(t)) = f(t)n.
n!
n=0
exp(f(t) + g(t)) = exp(f(t)) · exp(g(t)).
d
exp(t) = exp(t).
dt
"
(-1)n+1
ln(1 + t) = tn.
n
n=1
"
f(t) = aiti
i=1
"
(-1)n+1
ln(1 + f(t)) = (f(t))n.
n
n=1
f(t) g(t)
ln(f(t)g(t)) = ln(f(t)) + ln(g(t)).
"
d 1
ln(1 + t) = (-1)n+1tn-1 = .
dt 1 + t
n=1
exp ln
ln(exp(f(t)) = f(t), exp(ln(1 + f(t)) = 1 + f(t)
f(t)