The document discusses generating functions. It defines a generating function G(z) as a power series representation of a sequence <an> = a0, a1, a2, ... . Properties of generating functions include that differentiating or multiplying generating functions results in new generating functions, and that generating functions can reveal relationships between sequences.

2. Generating Function
< an >= a0 , a1 , a2 , a3 , ...
2 n
G(z) = a0 + a1 z + a2 z + ... = an z .
n≥0

3. Generating Function
2 n
G(z) = a0 + a1 z + a2 z + ... = an z .
n≥0
an = a0 , a1 , a2 , ...
an
G(z)

4. Generating Function
2 n
G(z) = a0 + a1 z + a2 z + ... = an z .
n≥0
an = a0 , a1 , a2 , ...
an
G(z)

5. Generating Function

7. G(z) =
H(z) =
n≥0
n
an z = a0 + a1 z + a2 z + ...
n
bn z = b0 + b1 z + b2 z + ...
2
2
A
n≥0
n n
αG(z) + βH(z) = α an z + β bn z
n≥0 n≥0
= (αan + βbn )z n .
n≥0
= + + + ... + + + + ...
=( + )+( + ) +( + ) + ...

8. m
z G(z) = a0 z + a1 z
=
m
n
an−m z .
m+1
2
G(z) = a0 + a1 z + a2 z + ...
+ a2 z m+2
+ ...
B
n≥m
2
= a0−m + a1−m z + a2−m z + ...
= 0·z 0 + 0·z 1 + 0·z 2 + ... + a0 z m + a1 z m+1 + ...
n≺0 an=0
n m n
= an−m z = z an z .
n≥0 n≥0
an−m = 0, ..., 0, a0 , a1 , a2 , ...
m

9. z −m
(G(z) −
nm
n
an z ) = z −m
n≥m
an z n
B
−m m m+1
=z (am z + am+1 z + ...)
2
= a0+m + a1+m z + a2+m z + ...
n
= an+m z .
n≥0
an+m = am , am+1 , am+2 , ...

10. G(z) = 1, 1, 1, ...
B
zG(z) = 0, 1, 1, ...
(1 − z)G(z) = 1
1 2
G(z) = = 1 + z + z + ...
1−z

11. G(z)
‣ zG(z)
Fn
A+B
Fn−1
‣ 2
z G(z) Fn−2
‣ 2
(1 − z − z )G(z) Fn − Fn−1 − Fn−2
n2 Fn − Fn−1 − Fn−2 = 0
(1 − z − z 2 )G(z) = F0 + (F1 − F0 )z + (F2 − F1 − F0 )z 2
=1
+ (F3 − F2 − F1 )z 3 + ...
=z
z
G(z) = 2
.
1−z−z

12. A+B
an = c1 an − 1 + ... + cm an−m G(z)
X
G(z) = m
.
1 − c1 z − ... − cm z
z
G(z) =
1 − z − z2

13. 2
G(z) = a0 + a1 z + a2 z + ...
H(z) = b0 + b1 z + b2 z 2 + ...
G(z)H(z) = (a0 + a1 z + a2 z 2 + ...)(b0 + b1 z + b2 z 2 + ...)
C
= (a0 b0 ) + (a0 b1 + a1 b0 )z + (a0 b2 + a1 b1 + a2 b0 )z 2 + ...
2
= c0 + c1 z + c2 z + ...
n
cn = ak bn−k
k=0
z m G(z)

14. G(z) = a0 + a1 z + a2 z 2 + ...
C
H(z) = b0 + b1 z + b2 z 2 + ... bn = 1
2
= 1 + z + z + ...
1
=
1−z
1
G(z)H(z) = G(z)
1−z
= a0 + (a0 + a1 )z + (a0 + a1 + a2 )z 2 + ...

15. F (z)G(z)H(z) = d0 , d1 , d2 , ...
dn =
a i bj c k .
C
i,j,k≥0
i+j+k=n
n
from F (z)G(z), cn = ak bn−k
k=0
ajk z k = zn a0k0 a1k1 . . .
j≥0 k≥0 n≥0 k0 ,k1 ,...≥0
k0 +k1 +...=n

16. z
D
G(z) = a0 + a1 z + a2 z 2 + a3 z 3 + ...
G(−z) = a0 − a1 z + a2 z 2 − a3 z 3 + ...
G(z) + G(−z) = 2a0 + 2a2 z 2 + 2a4 z 4 + ...
1 2 4
(G(z) + G(−z)) = a0 + a2 z + a4 z + . . .
2
G(z) = a0 + a1 z + a2 z 2 + a3 z 3 + ...
G(−z) = a0 − a1 z + a2 z 2 − a3 z 3 + ...
G(z) − G(−z) = 2a1 z + 2a3 z 3 + 2a5 z 5 + ...
1
(G(z) − G(−z)) = a1 z + a3 z 3 + a5 z 5 + . . .
2

17. Im
m
z
D
ri eiθ = cosθ + i sinθ
sinθ
θ
-r 0 r Re
cosθ
-ri

18. Im
m
z
D
ri e2π/im = cos(2π/m) + i sin(2π/m) =ω
sin(2π/m)
2π/m 1
n −kr k
an z = ω G(ω z)
-r 0 r Re n m
cos(2π/m) mod m=r 0≤km
0 ≤ r m.
-ri

19. Im
i
z
e2π/im = cos(2π/m) + i sin(2π/m) =ω
D
sin(2π/3)
√ √
= 3/2 120° 1 3
m=3 ω=− + i
-1 0 1 Re 2 2
cos(2π/3)
= 1/2
r=1
1 −k
n k
an z = ω G(ω z)
3
-i n mod 3=1 0≤k3
4 7
a1 z + a4 z + a7 z + . . .
1
= (G(z) + ω −1 G(ωz) + ω −2 G(ω 2 z)).
3

20. G(z) = a0 + a1 z + a2 z 2 + ... =
an z n
E
n≥0
G (z) = a1 + 2a2 z + 3a3 z 2 + ... = (k + 1)ak+1 z k .
k≥0
n
zG (z) = nan z . nan
n≥0

21. z
1 2 1 3
2
G(z) = a0 + a1 z + a2 z + ... =
n≥0
an z
1
n
k
E
G(t)dt = a0 z + a1 z + a2 z + . . . = ak−1 z .
0 2 3 k
k≥1
1 2
G(z) = = 1 + z + z + ...
1−z
1
2 k
G (z) = = 1 + 2z + 3z + ... = (k + 1)z .
(1 − z)2
k≥0
z 1
1 1 2 1 3 k
G(t)dt = ln = z + z + z ... = z .
0 1−z 2 3 k
k≥1

22. 0
z
G(t)dt = ln
1
1−z
1 2 1 3
= z + z + z ... =
2 3
E
1
k
k≥1
k
z .
1
G(z)H(z) = G(z) = a0 + (a0 + a1 )z + (a0 + a1 + a2 )z 2 + ...
1−z
1 1 1 2 1 1 3
ln = 0 + (0 + 1)z + (0 + 1 + )z + (0 + 1 + + )z + ...
1−z 1−z 2 2 3
3 2 11 3
= z + z + z + ... = Hk z k .
2 6
k≥0

23. r
(1 + z) = 1 + rz +
r(r − 1) 2
2
z + ... =
r
k
zk .
k≥0
F
r
1 −n − 1 k n+k
= (−z) = zk .
(1 − z)n+1 k n
k≥0 k≥0
r r(r − 2t − 1) 2 r − kt r k
x = 1 + rz + z + ··· = z .
2 k r − kt
k≥0

24. 1 2
exp z = ez = 1 + z + z + · · · =
2!
1
k!
zk .
k≥0
F
z n n 1 n+1 k zk
(e − 1) = z + z n+1 + · · · = n! .
n+1 n n k!
k

25. 1 2 1 3
ln(1 + z) = z − z + z − · · · =
2 3
(−1)k+1
k≥1
k
zk ,
F
1 1 m+k
ln( )= (Hm+k − Hm ) zk .
(1 − z)m+1 1−z k
k≥1
1 n n 1 n+1 k zk
(ln ) =z + z n+1 + · · · = n! .
1−z n+1 n n k!
k

26. z(z + 1) . . . (z + n − 1) =
k
k
n
k
z ,
F
z n k
= zk ,
(1 − z)(1 − 2z) . . . (1 − nz) n
k
z 1 1 2 Bk z k
z −1
= 1 − z + z + ··· = .
e 2 12 k!
k≥0
r(r + 2t) 2 r(r + kt)k−1
xr = 1 + rz + z + ··· = zk .
2 k!
k≥0

27. G(z) z n [z n ]G(z)
G
2 n
G(z) = a0 + a1 z + a2 z + · · · [z ]G(z) = an
n 1 G(z)dz
[z ]G(z) =
2πi |z|=r z n+1