1. Generating Functions II
Great Theoretical Ideas In Computer Science
S. Rudich
V. Adamchik
CS Spring 2006
Lecture 14
Mar 02, 2006
Carnegie Mellon University
Generating Functions II

2. What is a closed form the Fibonacci power series?
F0=0, F1=1,
Fn=Fn-1+Fn-2 for n2
What is a closed form the Fibonacci power series?

3. T0=1, T1=1,
Tn = T0 Tn-1 + T1 Tn-2+…+Tn-1 T0
Counting binary trees

4. Solving with Generating Functions
ak = ak-1 + ak-2 , kr2
a0=0, a1=1
We assume that f(x) is a generating function

5. ak = ak-1 + ak-2 , kr2
What are the generating functions for sequences ak, ak-1, ak-2 when k = 2, 3, 4,… ?

6. What is the generating function for ak , k = 2,3, … ?

7. What is the generating function for ak-1, k = 2,3, … ?

8. What is the generating function for ak-2, k = 0, 1, 2, … ?

9. Solving with Generating Functions
ak = ak-1 + ak-2 , k>1
a0=0, a1=1
In terms of generating functions

10. ak = ak-1 + ak-2

12. Simple Transformations
If f(x) is a generating function for ak, then
ak-1 -> x f(x)
ak-2 -> x2 f(x)

13. Find a GF for
a0=2, a1=4,a2=31
an=4an-1+3an-2-18an-3

14. T0=1, T1=1,
Tn = T0 Tn-1 + T1 Tn-2+…+Tn-1 T0
Counting binary trees

15. T0=1, T1=1, Tn = T0 Tn-1 + T1 Tn-2+…+Tn-1 T0
Let f(x) be a generating function Then summing-up the above recurrence, yields

16. Power series multiplication
Let us multiply formal power series p(x) = a0 + a1 x + a2 x2 + … q(x) = b0 + b1x + b2 x2 + … p(x)q(x) = a0b0 + (a0b1+a1b0) x + (a0b2+a1b1+a2b0) x2 +…

17. Power series multiplicationIf
then

18. T0=1, T1=1, Tn = T0 Tn-1 + T1 Tn-2+…+Tn-1 T0

19. T0=1, T1=1, Tn = T0 Tn-1 + T1 Tn-2+…+Tn-1 T0

20. Counting Binary Trees

21. Applying some calculus…
Catalan Numbers

22. Newton’s Binomial Theorem

24. Newton’s Binomial Theorem

25. Use generating functions to show that
Proving Identities
Use generating functions to show that

26. Consider the right hand side

27. Use the binomial theorem to obtain
What is the coefficient by xn?

28. Study Bee Solving recurrences via GFs Power series manipulations
Proving identities via GFs
Study Bee