How to solve a recursive equation?

Question

Mathematics

How to solve a recurrent equation of the form:
T( n) =4T(2n/3) + 1, T(1) = 3 ?
Considered one example:
T(n) = 2T(n/2) + 5n2
T(1) = 7
Solution:
For simplicity, we will assume that n is a power of 2, i.e. n = 2k
T(n) = 2T(n/2) + 5n2 =
= 2(2T (n/4) + 5 (n/2)2 ) + 5n2 =
= 2(2(2T (n/8) + 5 (n/4)2 ) + 5 (n/2)2 ) + 5n2 =…=
=2k T(1) + 2k-1 5 (n/2k-1 ) 2 + … + 2 5 (n /2)2 + 5n2
but I still can't enter what and where it comes from.
What sources of information can be used?
I was looking for something in Vilinkin's book (Combinatorics), but success passed me by.

Answer 1

Generating function method

Answer 2

What sources of information can be used?

googled: Fundamental theorem on recurrence relations . English Master theorem (analysis of algorithms) .