Monday, August 17, 2020

Andrews' approach to conjecture the Rogers-Ramanujan identities

 If something is worth doing, then I suppose it is worth doing again. 

I had previously written an article on how to discover the Rogers-Ramanujan identities. That was based on ideas of Dick Askey. In this talk I presented an introduction to partitions  presenting many results of Euler and ending with George Andrews' approach to discover the Rogers-Ramanujan identities. This approach was given in his Number Theory book, and it seems that it is not as well known as it should be. A notation that my collaborator Hartosh Singh Bal and I use to gain intuition is also explained here. 

 



Abstract


The two Rogers-Ramanujan identities were sent by Ramanujan to Hardy in a letter in 1913. As an example, here is the first Rogers-Ramanujan identity:

$$1+\frac{q}{(1-q)}+\frac{q^4}{(1-q)(1-q^2)}+\frac{q^9}{(1-q)(1-q^2)(1-q^3)}+\cdots $$

$$=\frac{1}{(1-q)(1-q^6)(1-q^{11})\dots}\times \frac{1}{(1-q^4)(1-q^{9})(1-q^{14})\dots}$$


They look less forbidding when interpreted in terms of partitions, which is how MacMahon considered them. A partition of a number  is a way of writing it as an unordered sum of other numbers. Unordered means that $2+3$ and $3+2$ are considered the same. For example, 

$$5 = 4+1 + 3+2 = 3+1+1 =  2+1+1+1$$ 

are partitions  of $5$. (Two partitions of $5$ are missing in this list; can you find them?) The theory of partitions is an attractive area of mathematics, where many complicated formulas are rendered completely obvious by making the `right picture'. However, while each side of the Rogers--Ramanujan identities are represented naturally in terms of partitions, they are still far from obvious.


In this talk, we will introduce partitions, explain how to enumerate them systematically, represent them graphically, and write their generating functions. We present an experimental approach  to discover the Rogers-Ramanujan identities. This approach is due to Professor George Andrews of Penn State University.


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