Caractérisation des suites linéaires récurrentes

E k S(E)
(un)n∈NE
σS(E)u= (un)v
∀n∈N, vn=un+1.
P∈k[X]P(σ)
u= (un)∈ S(E)r>0
q0, q1, . . . , qr−1∈k
∀n∈N, un+r+qr−1un+r−1+· · · +q1un+1 +q0un= 0.
u∈ S(E)
Iu:= {P∈k[X], P (σ)(u) = 0}Iuk[X]
u Iu6={0}
πu∈k[X]
Q∈k[X]
Q(σ)(u) = 0 ⇐⇒ πu|Q.
πuu
u∈ S(km)n un= (u1,n, . . . , um,n)
u ui∈ S(R),16i6m
πuπui
E=k=CP=Qi(X−λi)αi∈C[X] ker P(σ)
ker P(σ) = M
i
ker(σ−λiid)αi.
iker(σ−λiid)αi
k=R
(2n+ 3n)
(n2·2n)
h2n+ 3n
n2·2ni
(n!)
R(E)
E=k u = (un)∈ S(k)m>0
Hm(u)(1) Mm+1(k)
Hm(u) := [ui+j−2]16i, j 6m+1 .
u∈ S(R)
Dm(u) := d´et (Hm(u))
H1839 1873

ϕ∈ S(R)
ϕ0= 0, ϕ1= 1 ∀n∈N, ϕn+2 =ϕn+1 +ϕn.
Dm(ϕ)m>0πϕϕ
u πud>1
m>d Dm(u)=0
u∈ S(k)d>1
?? Dd−1(u)6= 0 ∀m>d, Dm(u) = 0.
u
u ??
Hd(u)
q1, q2,...qs∈k
Hd(u)·
qd
qd−1
q1
1
=
0
0
0
0
m>s
λm:= um+q1um−1+· · · +qd−1um−d+1 +qdum−d.
d6m62d λm= 0
Dd+1(u) =
u0u1. . . ud−10 0
u1u2. . . ud0 0
ud−1ud. . . u2d−20 0
udud+1 . . . u2d−10λ2d+1
ud+1 ud+2 . . . u2dλ2d+1 λ2d+2
.
λ2d+1 = 0
m>d+ 1
λd=λd+1 =· · · =λ2d=· · · =λm+d−1= 0.
Dm(u) =
u0u1. . . ud−10 0 . . . 0
ud−1. . . . . . u2d−20 0 . . . 0
ud. . . . . . u2d−10. . . 0λm+d
0∗
0λm+d
um. . . . . . um+d−1λm+d∗. . . ∗
.
λm+d
u
πu=Xd+q1Xd−1+· · · +qd−1X+qd.