Séries génératrices

n(an)n
n an
a0+a1x+a2x2+. . . =X
k≥0
akxk.
n
n
kk∈{0,...,n}
n
X
k=0 n
kxk= (1 + x)n.
x
x
x
n
X
k=0
anxn
∞
X
k=n
anxn

x= 0
A(x) = X
n≥0
xn
A(x)−1 = xA(x) (1 −x)A(x)=1
A(x) 1 −x
X
n≥0
xn=1
1−x.
1
1−x
x1
N≥0
1 + x+x2+. . . +xN=1−xN+1
1−x
x > 1x≤ −1N
x∈]−1,1[ xN+1
1
1−x
Pn≥0xn]−1,1[ x7→ Pn≥0xn
x7→ 1
1−x
P
QQ(0) = 0
F0= 0 F1= 1 Fn+2 =Fn+1 +Fn
F(x)
xn+2 n≥0
F(x)−x= (x+x2)F(x),

F(x) = x
1−x−x2.
X
m≥0
amxm! X
n≥0
bnxn!=X
m,n≥0
ambnxm+n=X
n≥0 n
X
k=0
akbn−k!xn
(Pn
k=0 akbn−k)n
xkxn−k=xn
(an)n
(Sn)Sn=Pn
k=0 akSn
Pk≥0xk=1
1−x(an)n
(Sn)n
1−x
Pi≥0aixia06= 0
Pj≥0bjxj
X
i≥0
aixi! X
j≥0
bjxj!=X
k≥0 k
X
i=0
aibk−1!xk= 1.
ai
bia0b0= 1 a06= 0
b0=1
a0.
a0b1+a1b0= 0.
a06= 0 b1=−a1b0
a0
b0, . . . , bn−1n
a0bn+. . . +anbn= 0 a06= 0 bn

P
QP Q Q(0) 6= 0
X
n≥0
anxn!0 =X
n≥1
nanxn−1.
A(x)
Pn≥1nxn−11
1−xPn≥1nxn−1=1 (1−x)2.
(1 −x)−k
k
r k
r
kr
k=r(r−1) . . . (r−k+ 1) k!.
n
(1 + x)n=X
k≥0n
kxk.
k
−1
k=(−1)(−2) . . . (−k) k!= (−1)k−2 k=(−2)(−3) . . . (−2−k+ 1) k!= (−1)k(k+1).
(1 −x)−1(1 −x)−2
−n
kn
n k
−n
k= (−1)kn+k−1 k.
f g
f(g(x))
f(g(x)) f(g(0))
x= 0

f(g(x)) f g(0)
f(g(x)) g(0) 6= 0
g g(0) = 0 f(g(x))
n n
n
f=Pk≥0akxkg(x) = xh(x)h(x)
f(g(x)) = a0+a1xh(x) + a2x2h(x)2+. . . +anxnh(x)n+an+1xn+1h(x)n+1 +. . .
n n+1
f(x) = P(x)
Q(x)
(an)ann
b(1 −ax)−k
P Q P =EQ +R E R
deg R < deg Q f =E+R
Q
Q
QQi(1 −aix)mi
(1 −aix)−j1≤j≤mi.
bi,j
f(x) = E(x) + X
iX
1≤j≤mi
bi,j
(1 −aix)j.
x
bi,mi(1 −aixi)mix=1
ai.
x x bi,1
(Fn)
x
1−x−x2.