Mathématiques/Compléments de cours/Récurrences linéaires à

EC
Pq
q∈NPqP(n)n∈N
P q P0
k∈Nek(nk)e0
P7−→ P(n)(P(n))
P= 0
Cq[X]Pq(e0, e1, . . . , eq)Cq[X]
Pq
∆
u= (un) ∆(u)=(u′
n)u′ n=un+1 −un
n∈N
∆E
∆P0
q∈N∆(Pq+1) = Pq
∆Pq+1 Ker ∆ = P0⊂ Pq+1
P0
∆(Pq+1) dim Pq+1 −1 = q+ 1
∆(Pq+1)
(e0, . . . , eq+1) ∆(e0) = 0 k∈[[1, q + 1]] ∆(ek) = (n+ 1)k−nk
k−1nPq∆(Pq+1)⊂ Pq
u E q ∈N
∆(u)∈ Pq⇐⇒ u∈ Pq+1
u∈ Pq+1 =⇒∆(u)∈ Pq
u∈E∆(u)∈ Pq
v∈ Pq+1 ∆(v) = ∆(u)w∈Ker ∆ = P0
u=v+w w ∈ P0⊂ Pq+1 u=v+w∈ Pq+1
q∈N∗Ker(∆q) = Pq−1
q q = 1
q>1u∈E
u∈Ker ∆q+1 ⇐⇒ ∆q∆(u)= 0 ⇐⇒ ∆(u)∈Ker ∆q=Pq−1⇐⇒ u∈ Pq
▹ ◃

F
(R)∀n∈Nun+q=aq−1un+q−1+aq−2un+q−2+· · · +a1un+1 +a0un=
q−1
k=0
akun+k
a0aq−1a0̸= 0
T u =
(un)∈E T (u) (vn)vn=un+1 n
T E k ∈N
u= (un)Tk(u) (un+k)
Tq(u) =
q−1
k=0
akTk(u) [P(T)](u) = 0
P=Xq−q−1
k=0 akXk(R)
F P (T)
PC[X]P=
r
i=1
(X−bi)mibi
P mi
F= Ker P(T) =
r
i=1
Ker[(X−bi)mi](T)=
r
i=1
Ker(T−biIdE)mi
Ker(T−bIdE)m
b= 1
T−IdE= ∆
Ker(T−IdE)m=Pm−1
a0̸= 0 P
b̸= 0
ΦE u v
∀n∈NΦ(u)n=vn=bnun
Φ (un)7−→ (un/bn)
u= (un)∈E v = Φ(u)w= [T−bIdE](v) = [(T−bIdE)◦Φ](u)n
wn=vn+1 −bvn=bn+1un+1 −bn+1un=b.bn(un+1 −un)
(T−bIdE)◦Φ = bΦ◦(T−IdE)T−bIdE=bΦ◦(T−IdE)◦Φ−1
▹ ◃

(T−bIdE)m=bmΦ◦(T−IdE)m◦Φ−1b̸= 0
Φu
u∈Ker(T−bIdE)m⇐⇒ bm[Φ ◦(T−IdE)m◦Φ−1](u) = 0
⇐⇒ [(T−IdE)m◦Φ−1](u) = 0
⇐⇒ Φ−1(u)∈Ker(T−IdE)m=Pm−1
uKer(T−bIdE)m
P m −1un/bn=P(n)un=P(n)bnn
Cm−1[X]m
FKer(T−biIdE)mimi=
deg P=q q (R)
∀n∈Nun=
r
i=1
Pi(n)bn
i
biPi
mibi
(bnnk)b
k∈Nb
(R)
(bn
i)
a0= 0
m
a0=a1=· · · =am−1= 0 am̸= 0
FKer Tm
m
m u0um−1
q−m
m
un+mn+m>m
▹ ◃