All Questions
5
questions
5
votes
3
answers
1k
views
Proving that odd partitions and distinct partitions are equal
I am working through The Theory of Partitions by George Andrews (I have the first paperback edition, published in 1998).
Corollary 1.2 is a standard result that shows that the number of partitions of $...
3
votes
1
answer
85
views
Showing $\prod_{n\geq 1} (1+q^{2n}) = 1 + \sum_{n\geq 1} \frac{q^{n(n+1)}}{\prod_{i=1}^n (1-q^{2i})}$
I want to show
\begin{align}
\prod_{n\geq 1} (1+q^{2n}) = 1 + \sum_{n\geq 1} \frac{q^{n(n+1)}}{\prod_{i=1}^n (1-q^{2i})}
\end{align}
I know one proof via self-conjugation of partition functions with ...
- 6,490
0
votes
1
answer
47
views
Infinite product formula for $\sum_{n \geq 0} p_e(n)\cdot x^n$
If $n$ is an integer and $p_e(n)$ is the number of partitions for $n$ such that all parts are even, what would be an infinite product formula for $\sum_{n \geq 0} p_e(n)\cdot x^n$
- 101
2
votes
1
answer
943
views
Proof that the series for the generating function of the partition function converges?
For $|q| < 1$, the generating function of the partition function $p(n)$ is given by
$$
\sum_{n=0}^\infty p(n) q^n
= \prod_{k=1}^\infty {1 \over 1-q^k}. \tag{1}
$$
I have an intuitive ...
- 826
7
votes
1
answer
171
views
Generating function for $r^\binom{n}{2}$
I'm trying to find a closed form of the generating function
$$
G(x) = \sum_{n \ge 0} r^\binom{n}{2} x^n
$$
for a real number $0 < r < 1$. I found that $G(x) = 1 + xG(rx)$. Any hints where to ...
- 3,078