Phép biến đổi laplace rời rạc với phương trình sai phân

f
∗
(n)
h = 0 y = f(x)
x
0
, x
1
= x
0
+ h, x
2
= x
0
+ 2h, . . . , x
n
= x
0
+ nh, . . . (n ∈ N)
y
0
, y
1
, . . . ., y
n
, . . .
y

0
=f(x
0
) = f
0
;
y
1
=f(x
1
) = f (x
0
+ h) = f
1
;
y
2
=f(x
2
) = f (x
1
+ h) = f
2
;
. . .
=f(x
n
) = f (x
n−1
+ h) = f

n
;
. . .
∆y = f(x + h) − f(x) = ∆f(x)
f(x) x.
∆y
0
= ∆f(x
0
) = y
1
− y
0
; ∆y
1
= ∆f(x
1
) = f (x
1
+ h) − f(x
1
) = y
2
− y
1
; . . .
∆y
n−1
= ∆f(x
n−1

) = y
n
− y
n−1
∆y
n
= y
n+1
− y
n
.
y = f(x) x
k
=
x
0
+ k, k ∈ N
∆y
k
= ∆f
k
= f(x
k+1
) − f(x
k
)
f(x) x
k
f(x) ∈ R[x]
f(x + 1) − f(x) = 0 ∀x.

(x + 1) − f(x) = x ∀x.
x = 0, 1, 2 . . . n . . . f(x) = f(0)
f(x) f(x) = f(0) ∀x.
∆f(x) = 0 ∀x
f(x)
g(x) g(x + 1) − g(x) = x ∀x.
g(x) = ax
2
+ bx
a(x + 1)
2
+ b(x + 1) − ax
2
− bx = x ∀x ⇔ 2ax + a + b = x.
a =
1
2
, b =
−1
2
g(x) =
1
2
x
2
−
1
2
x.
(f(x + 1) − g(x + 1)) − (f(x) − g(x)) = 0 ∀x

∆(f(x) − g(x)) = 0.
f(x)−g(x) = C f(x) =
1
2
x
2
−
1
2
x+C.
f(x)
f(x)
f(x)
∆
n
f(x
k
) f(x) x
k
∆
n
y
k
= ∆
n
f(x
k
) = ∆(∆
n−1
f(x

k
)) = ∆
n−1
f(x
k+1
) − ∆
n−1
f(x
k
).
∆
2
y
k
= ∆
2
f
k
= ∆f(x
k+1
) − ∆f(x
k
) = f (x
k+2
) − 2f(x
k+1
) + f(x
k
)
y = f(x) x

k
x = x
k
x
0
x
1
x
2
x
3
x
4
. . .
y
k
= f(x
k
) y
0
y
1
y
2
y
3
y
4
∆y
k

∆y
0
∆y
1
∆y
2
∆y
3
. . .
∆
2
y
k
∆
2
y
0
∆
2
y
1
∆
2
y
3
. . .
. . . . . .
f(n)
f(n)
n 0 1 2 3 4 5 . . .

f(n) −1 2 13 44 107 214
∆f(n) 3 11 31 63 107 . . .
∆
2
f(n) 8 20 32 44 . . .
∆
3
f(n) 12 12 12 . . .
∆
4
f(n) 0 0 . . .
f(n)
f(n) = an
3
+ bn
2
+ cn + d (a = 0).
f(0) = −1, f(1) = 2, f (2) = 13, f (3) = 44





d = −1
a + b + c + d = 2
8a + 4b + 2c + d = 13
27a + 9b + 3c + d = 44
a = 2, b = −2, c = 3, d = −1
f(n) = 2n
3

− 2n
2
+ 3n − 1.
∆C = 0,
∆
k
y
n
=
k

i=0
(−1)
i
C
i
k
y
n+k−i
∆
k
(αf + βg) = α∆
k
f + β∆
k
g.
α, β f, g x
∀k, m ∈ N, m < k
y
k

= f(x
k
) =
k−m

i=0
C
i
k−m
∆
i
f(x
m
).
n − k
n = k
k > n
∆(f
k
.g
k
) = f
k
.∆g
k
+ g
k+1
.∆f
k
.

n

k=1
∆y
k
= y
n+1
− y
1
.
F (n, y
n
, ∆y
n
, . . . , ∆
k
y
n
) = 0
F (n, y
n
, y
n+1
, . . . , y
n+k
)
y(n)
∆
k
y

n
=
k

i=0
(−1)
i
C
i
k
y
n+k−i
∆y
n
= y
n+1
− y
n
∆
2
y
n
= y
n+2
− C
1
2
y
n+1
+ y

n
∆
3
y
n
= y
n+3
− C
1
3
y
n+2
+ C
2
3
y
n+1
− y
n
. . .
y
k
= f(x
k
) =
k−m

i=0
C
i

k−m
∆
i
f(x
m
).
m = n k = n + 1, n + 2, . . .
y
n+1
= f(x
n+1
) = f (x
n
) + ∆f(x
n
) = y
n
+ ∆y
n
y
n+2
= f(x
n+2
) = f (x
n
) + C
1
2
∆f(x
n

) + ∆
2
f(x
n
) = y
n
+ 2∆y
n
+ ∆
2
y
n
y
n+3
= y
n
+ C
1
3
∆y
n
+ C
2
3
∆
2
y
n
+ ∆
3

y
n
. . .
y
n+k
=
k

i=0
C
i
k
∆
i
y
n
.
y
n
∆
3
f(n) + 4∆
2
f(n) + 5∆f(n) + 2f(n) = 0.
∆f(n) = f(n + 1) − f(n)
∆
2
f(n) = f(n + 2) − 2f(n + 1) + f(n)
∆
3

f(n) = f(n + 3) − 3f(n + 2) + 3f(n + 1) −f(n)
f(n + 3) + f (n + 2) = 0
b
0
y
n+k
+ b
1
y
n+k−1
+ ··· + b
n
y
n
= F (n)
b
0
, b
1
, . . . , b
n
b
0
, b
k
= 0 y(n)
F (n)
b
0
f(n + k) + b

1
f(n + k −1) + ···+ b
n
f(n) = F (n).
• F (n) = 0
• F (n) = 0
• y(n) (n ∈ N)
• ˜y(n)
b
0
y
n+k
+ b
1
y
n+k−1
+ b
2
y
n+k−2
+ ··· + b
k
y
n
= 0
y
0
, y
1
, . . . , y

k−1
C
1
, C
2
, C
3
, . . . ., C
k
˜y(n)
˜y(i) = y
i
, i =
0, k −1
• y
∗
(n)
(a
n
)
F (z)
F (z) = a
1
+
a
1
z
+ . . . +
a
n

z
n
+ . . . =

n≥0
a
n
z
n
(1).
|a
n
| ≤ M e
αn
M > 0, α
|z| > R = e
α
F (z) ∞ F (z)
F (∞) = 0.
(a
n
)
∞
a
n
=
1
2πi

L

F (z)z
n−1
dz (2)
L
F (z)
(a
n
) f(n) a
n
= f(n)
f(n) |f(n)| ≤ Me
αn
f(n) = 0 n < 0.
z = e
p
, p = s + iθ
F (z) = F (e
p
) = F
∗
(p)
F
∗
(p) = f
0
+ f
1
.e
−p
+ f

2
.e
−2p
+ . . . . + f
n
.e
−np
+ . . . =

≥
f e
−np
(3)
f(n) L
D
F
∗
(p) = L
D
[f(n)]
F
∗
(p)  f(n)
F
∗
(p) f(n) L
D
F
∗
(p) 2πi

−π < Imp ≤ π Rep > α.
L
D
f(n) =
1
2πi
γ+πi

γ−πi
F
∗
(p)e
np
dp.
f(n)
F (z) =
1
(z −a)(z −b)
.
a = b F (z) =
1
(z −a)(z −b)
z
1
= a, z
2
= b.
f(n) =
1
2πi


L
F (z)z
n−1
dz
=Res[F (z)z
n−1
; a] + Res[F (z)z
n−1
; b] =
a
n−1
a − b
+
b
n−1
b − a
.
a = b F (z) z = a
f(n) =
1
2πi

L
F (z)z
n−1
dz = Res[F (z)z
n−1
; a]
=

1
(2 − 1)!
lim
z→a
d
dz
[(z −a)
2
z
n−1
(z −a)
2
)]
= (n − 1)a
n−2
.
f(n) = e
−n
.
F
∗
(p) =

n≥0
f(n)e
−np
=

n≥0
e

−n
e
−np
= lim
n→ ∞
n−1

k=0
e
−k(p+1)
= lim
n→ ∞
1 − e
−n(p+1)
1 − e
−(p+1)
=
1
1 − e
−(p+1)
=
e
p+1
e
p+1
− 1
=
e
p
e

p
− e
−1
.
f(n), g(n) f
j
(n), g
j
(n)
F
∗
(p), G
∗
(p), F
j
∗
(p), G
j
∗
(p)
n

j=1
α
j
f
j
(n) 
n


j=1
α
j
F
∗
(p)
f(n − k)  e
−kp
F
∗
(p), f(n − k) = 0 (n < k)
f(n + k)  e
kp
[F
∗
(p) −
k−1

r=0
f(r)e
−r.p
]
e
p
0
.n
f(n)  F
∗
(p − p
0

)
•
d[F
∗
(p)]
dp
 −n.f(n).
d
k
[F
∗
(p)]
(dp)
k
 (−1)
k
.n
k
.f(n)
f(0) = 0,
f(t)
t
|
t=0
= lim
t→0
+
f(t)
t
= 0

f(n)
n

+∞

p
F
∗
(p)dp.
f(t)
t
|
t=0
= lim
t→0
+
f(t)
t
= a a +
+∞

p
F
∗
(p)dp 
f(n)
n
f(n)  F
∗
(p)

n−1

k=0
f(k) 
F
∗
(p)
e
p
− 1
f(n)
F
∗
(p)
A
e
p
− e
p
0
 Ae
p
0
(n−1)
,
A
(e
p
− e
p

0
)
m
 A.e
p
0
(n−m)
.
(n − 1)
[m−1]
(m − 1)!
.
f(n)  F
∗
(p) ϕ(n)  Φ
∗
(p) f(n)∗ϕ(n) =
n

k=0
f(n − k)ϕ(k)
f(n) ∗ ϕ(n)  F
∗
(p)Φ
∗
(p).
f(n) =

m
[F

∗
(p).e
(n−1)p
; p
m
].
p
m
[F
∗
(p).e
(n−1)p
; p
m
] = lim
p→p
m
[F
∗
(p)(e
p
− e
p
m
).e
(n−1)p
].
p
m
[F

∗
(p).e
(n−1)p
; p
m
] =
1
(r −1)!
lim
p→p
m
d
r−1
de
(r−1)p
[F
∗
(p)(e
p
− e
p
m
)
r
.e
(n−1)p
].
F
∗
(p)

F
∗
(p) =
G(p)
H(p)
=
c
0
e
mp
+ c
1
e
(m−1)p
+ ··· + c
m
b
0
e
rp
+ b
1
e
(r−1)p
+ . . . ··· + b
r
.
z = e
p
F (z) =

G(z)
H(z)
=
c
0
z
m
+ c
1
z
m−1
+ ··· + c
m
b
0
z
r
+ b
1
z
r−1
+ . . . ··· + b
r
m ≤ r .
F (z)
G(z)
H(z)
F (z) z
k
f(n) =


k
G(z
k
)
H

(z
k
)
z
n−1
k
F (z) z
k
m
k
f(n) =

k
1
(m
k
− 1)!
lim
z→z
k
d
m
k

−1
dz
m
k
−1
[(z −z
k
)
m
k
F (z)z
n−1
]
F
∗
(p) =
e
p
e
2p
− 3e
p
+ 2
.
e
2p
− 3e
p
+ 2 = 0 ⇔ [
e

p
=2
e
p
=1
⇔ [
p
2
= ln 2
p
1
= 0
p
1
= 0 p
2
= ln 2
F
∗
(p)
[F
∗
(p).e
(n−1)p
; 0] = lim
p→0
[F
∗
(p)(e
p

− 1).e
(n−1)p
] = lim
p→0
e
pn
e
p
− 2
= −1
[F
∗
(p).e
(n−1)p
; ln 2] = lim
p→ln 2
[F
∗
(p)(e
p
− 2).e
(n−1)p
] = lim
p→ln 2
e
pn
e
p
+ 1
= 2

n
f(n) = [F
∗
(p).e
(n−1)p
; 0] + [F
∗
(p).e
(n−1)p
; ln 2] = −1 + 2
n
.
F
∗
(p) =
e
p
e
2p
− 1
.
p
1
= 0 p
2
= πi
−π ≤ imp ≤ π
[F
∗
(p).e

(n−1)p
; 0] = lim
p→0
[F
∗
(p)(e
p
− 1).e
(n−1)p
] = lim
p→0
e
pn
e
p
+ 1
=
1
2
[F
∗
(p).e
(n−1)p
; πi] = lim
p→πi
[F
∗
(p)(e
p
+ 1).e

(n−1)p
] = lim
p→πi
e
pn
e
p
− 1
e
πin
e
πi
− 1
=
(−1)
n−1
2
f(n) =
1
2
+
(−1)
n−1
2
.
F
∗
(p) =
e
p

(e
p
− 1)(e
p
− 2)(e
p
− 3)
.
z = e
p
F (z) =
z
(z −1)(z −2)(z − 3)
=
z
z
3
− 6z
2
+ 11z −6
z
1
= 1, z
2
= 2, z
3
= 3
f(n) =
3


k=1
z
n
k
3z
2
k
− 12z
k
+ 11
=
1
2
− 2
n
+
1
2
.3
n
.
F
∗
(p) =
e
p
(e
p
− 1)
2

(e
p
− 2)
.
z = e
p
F (z) =
z
(z −1)
2
(z −2)
F (z) z
1
= 2
z
2
= 1
[(z −2)(z −1)
2
]

(2)
= (2 − 1)
3
= 1
1
(2 − 1)!
lim
z→1
d

2−1
dz
2−1
[(z −1)
2
z
(z −2)(z −1)
2
z
n−1
] = lim
z→1
d
dz
[
z
n
z −2
]
= lim
z→1
nz
n−1
(z −2) −z
n
(z −2)
2
= −n − 1.
f(n) = 2
n

− n − 1.
F
∗
(p) =
e
3p
e
4p
+ 4
.
F
∗
(p) = e
p
e
2p
e
4p
+ 4
=
e
p
4
(
e
p
e
2p
− 2e
p

+ 2
−
e
p
e
2p
+ 2e
p
+ 2
)
e
an
sin αn 
e
p
.e
a
. sin α
e
2p
− 2e
p
.e
a
. cos α + e
2a
.
a α e
a
. cos α = 1 e

a
. cos α = −1 a = ln
√
2 α =
π
4
α =
3π
4

√
2

n
sin
π
4
n 
e
p
.
√
2. sin
π
4
e
2p
− 2e
p
.

√
2. cos
π
4
+ 2
e
p
e
2p
− 2e
p
+ 2

√
2

n
sin
3π
4
n 
e
p
.
√
2. sin
3π
4
e
2p

− 2e
p
.
√
2. cos
3π
4
+ 2
e
p
e
2p
+ 2e
p
+ 2
.
1
4
(
e
p
e
2p
− 2e
p
+ 2
−
e
p
e

2p
+ 2e
p
+ 2
) 
1
4

√
2

n

sin
π
4
n − sin
3π
4
n

F
∗
(p) = e
p
e
2p
e
4p
+ 4


1
4

√
2

n+1

sin
π
4
(n + 1) − sin
3π
4
(n + 1)

F
∗
(p) =
2e
2p
e
4p
+ 4
.
F
∗
(p) =
2e

2p
e
4p
+ 4
=
e
p
2
(
1
e
2p
− 2e
p
+ 2
−
1
e
2p
+ 2e
p
+ 2
).
1
e
2p
− 2e
p
+ 2
= e

−p
e
p
e
2p
− 2e
p
+ 2


√
2

n−1
sin
π
4
(n − 1)
1
e
2p
+ 2e
p
+ 2
= e
−p
e
p
e
2p

+ 2e
p
+ 2


√
2

n−1
sin
3π
4
(n − 1).
F
∗
(p) 

√
2

n−1
2
[sin
π
4
(n − 1) − sin
3π
4
(n − 1)].
F

∗
(p) =
e
p
(e
p
− e)
3
.
n
2
e
a.n

e
p
.e(e
p
+ e)
(e
p
− e)
3
n
2
e
n−1

e
p

(e
p
+ e)
(e
p
− e)
3
n
[2]
e
n

2.e
p
.e
2
(e
p
− e)
3
1
2
n
[2]
e
n−1

e
p
.e

(e
p
− e)
3
e
2p
(e
p
− e)
3
=
e
p
(e
p
+ e)
(e
p
− e)
3
−
e
p
e
(e
p
− e)
3
 n
2

e
n−1
−
1
2
n
[2]
e
n−1
= e
n−1

n
2
−
n(n − 1)
2

F
∗
(p) =
e
p
(e
p
− e)
3
= e
−p
e

2p
(e
p
− e)
3
 e
n−2

(n − 1)
2
−
(n − 1)(n − 2)
2

.
F
∗
(p) 
n(n − 1)
2
e
n−2
F
∗
(p) =
e
p
(e
p
− e)

3
F (z) =
z
(z −e)
3
z = e
f(n) =
1
2!
lim
z→e
d
2
dz
2
[(z −e)
3
.F (z).z
n−1
]
=
1
2!
lim
z→e
d
2
dz
2
(z

n
) =
n.(n − 1)
2
e
n−2
f(n) F
∗
(p)
E(n) =

0 , n < 0
1 , n ≥ 0
e
p
e
p
− 1
n
e
p
(e
p
− 1)
2
n
2
e
p
(e

p
+ 1)
(e
p
− 1)
3
n
3
e
p
(e
2
p + 4e
p
+ 1)
(e
p
− 1)
4
n
4
e
p
(e
3
p + 11e
2
p + 11e
p
+ 1)

(e
p
− 1)
5
n
[
2] = n(n − 1)
2e
p
(e
p
− 1)
3
n
[
k] = n(n − 1) . . . (n − k+!) (k!)
e
p
(e
p
− 1)
k+1
e
an
e
p
e
p
− e
a

ne
an
e
p
.e
a
(e
p
− e
a
)
2
n
2
e
an
e
p
.e
a
(e
p
+ e
a
)
(e
p
− e
a
)

3
n
[2]
e
an
2e
p
.e
a
(e
p
− e
a
)
3
n
[k]
e
an
k!e
p
.e
ka
(e
p
− e
a
)
k+1
sin αn

e
p
. sin α
e
2
p − 2e
p
cos α + 1
cos αn
(e
p
− cos α)e
p
e
2
p − 2e
p
cos α + 1
sinh(αn)
e
p
. sinh α
e
2
p − 2e
p
chα + 1
cosh(αn)
(e
p

− cosh α)e
p
e
2
p − 2e
p
chα + 1
e
an
sinαn
e
a
.e
p
sin α
e
2
p − 2e
p
.e
a
cos α + e
2
a
e
an
cosαn
(e
p
− e

a
cosα)e
p
e
2
p − 2e
p
cosα + e
2
a
n
[
k]
k!
e
an
e
p
.e
ka
(e
p
− e
α
)
k+1
n
[
k]
k!

a
an
e
p
.a
k
(e
p
− a)
k+1
f(n) =

C, n 0
0, n = 0
C
f(n) = E(n) =

0, n < 0
1, n ≥ 0
F
∗
(p)

≥
f e
−np
=

≥
e

−np
=
e
p
e
p
− 1
, p
f(n) = n
F
∗
(p) =

≥
f e
−np
=

≥
e
−np
n = −
d
dp
[

≥
e
−np
]

= −
d
dp
[
e
p
e
p
− 1
] =
e
p
(e
p
− 1)
2
, (Re p > 0)
f(n) = n
2
= n.g(n) g(n) = n
n.g(n) 
d
dp
(G
∗
(p)) = −
d
dp
(
e

p
(e
p
− 1)
2
)
=
e
p
(e
p
+ 1)
(e
p
− 1)
3
n
2

e
p
(e
p
+ 1)
(e
p
− 1)
3
.
f(n) = n

3
= n.n
2
n
2

e
p
(e
p
+ 1)
(e
p
− 1)
3
n.n
2
 −
d
dp
(
e
p
(e
p
+ 1)
(e
p
− 1)
3

)
e
p
(e
2p
+ 4e
p
+ 1)
(e
p
− 1)
4
n
3

e
p
(e
2p
+ 4e
p
+ 1)
(e
p
− 1)
4
.
n
4
= n.n

3
 −
d
dp
(
e
p
(e
2p
+ 4e
p
+ 1)
(e
p
− 1)
4
)
e
p
(e
3p
+ 11e
2p
+ 11e
p
+ 1)
(e
p
− 1)
5

.
n
4

e
p
(e
3p
+ 11e
2p
+ 11e
p
+ 1)
(e
p
− 1)
5
f(n) = n
[2]
= n(n−1) = n
2
−n
n
2
−n 
e
p
(e
p
+ 1)

(e
p
− 1)
3
−
e
p
(e
p
− 1)
2
=
2.e
p
(e
p
− 1)
3
n
[2]

2.e
p
(e
p
− 1)
3
.
f(n) = n
[k]

= n(n − 1) . . . .(n − k + 1)
k
f(n) 
k!.e
p
(e
p
− 1)
k+1
k
k > 2, k ∈ N k
n
[k+1]

(k + 1)!.e
p
(e
p
− 1)
k+2
n
[k+1]
= n(n − 1) . . . .(n − k + 1).(n − k) = n
[ ]
.n − n
[k]
.k
 −
d
dp

[
k!.e
p
(e
p
− 1)
k+1
] − k.
k!.e
p
(e
p
− 1)
k+1
= −k!
−e
p
− k.e
p
(e
p
− 1)
k+2
= (k + 1)!
e
p
(e
p
− 1)
k+2

n
[k]

k!.e
p
(e
p
− 1)
k+1
.
f(n) = e
a.n
f(n)  F
∗
(p) =

≥
e
−np
e
a.n
=

≥
e
−n(p−a)
=
1
1 − e
−(p−a)

=
e
p
e
p
− e
a
p − a) > 0
ne
a.n

e
p
.e
a
(e
p
− e
a
)
2
e
−iαn
= cos αn − i sin αn i = −1)
e
−iαn

e
p
e

p
− e
−iα
=
e
p
e
p
− (cos α − i sin α)
e
p
[(e
p
− cos α) − i. sin α]
(e
p
− cos α)
2
+ sin
2
α
e
p
(e
p
− cos α)
e
2p
− 2e
p

. cos α + 1
−
e
p
. sin α
e
2p
− 2e
p
. cos α + 1
i
sin αn 
e
p
. sin α
e
2p
− 2e
p
. cos α + 1
sinh(αn) =
1
2
(e
αn
− e
−αn
) 
1
2

(
e
p
e
p
− e
α
−
e
p
e
p
− e
−α
)
=
1
2
(e
α
− e
−α
)e
p
e
2p
− e
p
(e
α

+ e
−α
) + 1
e
α
− e
−α
2
e
p
e
2p
− 2e
p
.
e
α
+ e
−α
2
+ 1
e
p
. sinh α
e
2p
− 2e
p
. cosh α + 1
sinh(αn) 

e
p
. sinh α
e
2p
− 2e
p
. cosh α + 1
.
cos α =
1
2
(e
iα
+e
−iα
) cos(iαn) =
1
2
(e
αn
+e
−αn
) = cosh(αn)
cos iα = cosh α
cosh αn = cos iαn 
e
p
(e
p

− cos iα)
e
2p
− 2e
p
. cos iα + 1
e
p
(e
p
− cosh α)
e
2p
− 2e
p
. cosh α + 1
cosh αn 
e
p
(e
p
− cosh α)
e
2p
− 2e
p
. cosh α + 1
.
e
an

f(n)  F
∗
(p −a) F
∗
(p)
f(n)
e
an
sin αn 
e
p−a
. sin α
e
2(p−a)
− 2
p−a
. cos α + 1
e
p
.e
a
. sin α
e
2p
− 2e
p
.e
a
. cos α + e
2a

e
an
sin αn 
e
p
.e
a
. sin α
e
2p
− 2e
p
.e
a
. cos α + e
2a
.
e
an
cos αn 
(e
p−a
− cos α)e
p−a
e
2(p−a)
− 2e
p−a
. cos α + 1
=

(e
p
− e
a
cosα)e
p
e
2p
− 2e
p
.e
a
. cos α + e
2a
n
[k]
k!
e
α.n
= C
k
n
e
αn
n
[k]
k!
e
α.n


e
p
.e
ka
(e
p
− e
a
)
k+1
.
n
[k]
k!
a
n
= C
k
n
a
n
n
[k]
.a
n
= n
[k]
.e
ln a
n

= n
[k]
.e
(ln a).n
 k!
e
p
.e
k lna
(e
p
− e
ln a
)
k+1
= k
e
p
.a
k
(e
p
− a)
k+1
n
[k]
k!
a
n
= C

k
n
a
n

e
p
.a
k
(e
p
− a)
k+1
.
f(n) =

C, n 0
0, n = 0
f(n)  F
∗
(p) =

n≥0
e
−pn
f(n) = e
0
.f(0) = C.
F
∗

(p) =
e
p
e
2p
− 7e
p
+ 10
5
n
− 2
n
3
F
∗
(p) =
e
p
e
2p
+ 1
sin
nπ
2
F
∗
(p) =
e
p
e

4p
− 1
1 + (−1)
n
4
−
1
2
sin(n + 1)
π
2
F
∗
(p) =
e
p
e
2p
+ 2ae
2p
+ 2a
2
a
n−1
2
n
2
sin
3nπ
4

F
∗
(p) =
e
p
(e
p
− e)
3
n(n − 1)
2
e
n−2
F
∗
(p) =
1
(e
p
− 3)(e
p
− 4)
4
n−1
− 3
n−1