Mathematics Asked by phy_math on December 29, 2020

I want to show

begin{align}

prod_{ngeq 1} (1+q^{2n}) = 1 + sum_{ngeq 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 Young Tableaux. But it seems not natural for me. [In the process of the proof it appears Durfee square, etc]

Is there any other (simple?) proof for this equality?

Substituting $sqrt{q}$ for $q$, we get $$ prod_{n=1}^{infty}(1+q^n)=sum_{k=0}^{infty}{frac{q^{binom{k+1}{2}}}{prod_{i=1}^{k}{(1-q^i)}}}. $$

The left-hand side is the generating function for partitions with distinct parts (a part of each size $n$ occurs $0$ or $1$ times).

On the other hand, if a partition with distinct parts has $k$ parts, then the $i$th smallest part is of size at least $i$. Given a partition $0<lambda_1<lambda_2<dots<lambda_k$, subtract $i$ from the size of the $i$th smallest part to get a partition with parts $0lelambda_1-1lelambda_2-2ledotslelambda_k-k$ with $le k$ parts (after exluding the $0$s), whose conjugate is a partition with the largest part $le k$. The "staircase" we subtracted has $1+2+dots+k=binom{k+1}{2}$ cells. That yields the summand on the right-hand side: $$ q^{binom{k+1}{2}}prod_{i=1}^{k}{frac{1}{1-q^i}}. $$

I think this is about as simple as it gets. The Durfee square need not be involved, as you can see.

Correct answer by Alexander Burstein on December 29, 2020

Get help from others!

Recent Questions

- How can I transform graph image into a tikzpicture LaTeX code?
- How Do I Get The Ifruit App Off Of Gta 5 / Grand Theft Auto 5
- Iv’e designed a space elevator using a series of lasers. do you know anybody i could submit the designs too that could manufacture the concept and put it to use
- Need help finding a book. Female OP protagonist, magic
- Why is the WWF pending games (“Your turn”) area replaced w/ a column of “Bonus & Reward”gift boxes?

Recent Answers

- Joshua Engel on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?
- Peter Machado on Why fry rice before boiling?
- Jon Church on Why fry rice before boiling?

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP