Let G(q) be the generating function for partitions such that if k is a part, then it occurs once and k+1 is not a part.

Let H(q) be the generating function for partitions with the same condition plus that 1 is not a part.

These are the left-hand sides of the Rogers-Ramanujan Identities.

$G(q)=\displaystyle\sum_{n=0}^\infty\frac{q^{n^2}}{(q;q)_n}=\frac{1}{(q;q^5)_\infty(q^4;q^5)_\infty}$

 

$H(q)=\displaystyle\sum_{n=0}^\infty\frac{q^{n^2+n}}{(q;q)_n}=\frac{1}{(q^2;q^5)_\infty(q^3;q^5)_\infty}$

I am intrigued by the following unreferenced statement in the wikipedia [page][1]: [1]: http://en.wikipedia.org/wiki/Rogers%E2%80%93Ramanujan_identities#Modular_functions%20%22page%22

If q = e^2πiτ, then q^−1/60G(q) and q^11/60H(q) are modular functions of τ.

1. Do modular forms shed any light on the Rogers-Ramanujan Identities, or is the connection (as far as we know) a curious coincidence?

2. Is there some class of modular forms whose Fourier series count natural collections of partitions such as those counted by the left-hand sides of the Rogers-Ramanujan Identities? In particular I have in mind the seemingly "non-local" condition that if k is part, then it is distinct and also k+1 is not parts.

3. In general, how does one tell if a certain generating function (that counts partitions of a certain type, say) is related (by a multiplicative factor like above) to a modular form of some weight for some group (maybe even with some character)?

Let G(q) be the generating function for partitions such that if k is a part, then it occurs once and k+1 is not a part.

Let H(q) be the generating function for partitions with the same condition plus that 1 is not a part.

These are the left-hand sides of the Rogers-Ramanujan Identities.

$G(q)=\displaystyle\sum_{n=0}^\infty\frac{q^{n^2}}{(q;q)_n}=\frac{1}{(q;q^5)_\infty(q^4;q^5)_\infty}$

$H(q)=\displaystyle\sum_{n=0}^\infty\frac{q^{n^2+n}}{(q;q)_n}=\frac{1}{(q^2;q^5)_\infty(q^3;q^5)_\infty}$

I am intrigued by the following unreferenced statement in the wikipedia page:

If q = e^2πiτ, then q^−1/60G(q) and q^11/60H(q) are modular functions of τ.

1. Do modular forms shed any light on the Rogers-Ramanujan Identities, or is the connection (as far as we know) a curious coincidence?

2. Is there some class of modular forms whose Fourier series count natural collections of partitions such as those counted by the left-hand sides of the Rogers-Ramanujan Identities? In particular I have in mind the seemingly "non-local" condition that if k is part, then it is distinct and also k+1 is not parts.

3. In general, how does one tell if a certain generating function (that counts partitions of a certain type, say) is related (by a multiplicative factor like above) to a modular form of some weight for some group (maybe even with some character)?

Let G(q) be the generating function for partitions such that if k is a part, then it occurs once and k+1 is not a part.

Let H(q) be the generating function for partitions with the same condition plus that 1 is not a part.

These are the left-hand sides of the Rogers-Ramanujan Identities.

$G(q)=\displaystyle\sum_{n=0}^\infty\frac{q^{n^2}}{(q;q)_n}=\frac{1}{(q;q^5)_\infty(q^4;q^5)_\infty}$

$H(q)=\displaystyle\sum_{n=0}^\infty\frac{q^{n^2+n}}{(q;q)_n}=\frac{1}{(q;q^5)_\infty(q^4;q^5)_\infty}$

I am intrigued by the following unreferenced statement in the wikipedia [page][1]: [1]: http://en.wikipedia.org/wiki/Rogers%E2%80%93Ramanujan_identities#Modular_functions%20%22page%22

If q = e^2πiτ, then q^−1/60G(q) and q^11/60H(q) are modular functions of τ.

1. Do modular forms shed any light on the Rogers-Ramanujan Identities, or is the connection (as far as we know) a curious coincidence?

2. Is there some class of modular forms whose Fourier series count natural collections of partitions such as those counted by the left-hand sides of the Rogers-Ramanujan Identities? In particular I have in mind the seemingly "non-local" condition that if k is part, then it is distinct and also k+1 is not parts.

3. In general, how does one tell if a certain generating function (that counts partitions of a certain type, say) is related (by a multiplicative factor like above) to a modular form of some weight for some group (maybe even with some character)?

Let G(q) be the generating function for partitions such that if k is a part, then it occurs once and k+1 is not a part.

Let H(q) be the generating function for partitions with the same condition plus that 1 is not a part.

These are the left-hand sides of the Rogers-Ramanujan Identities.

$G(q)=\displaystyle\sum_{n=0}^\infty\frac{q^{n^2}}{(q;q)_n}=\frac{1}{(q;q^5)_\infty(q^4;q^5)_\infty}$

$H(q)=\displaystyle\sum_{n=0}^\infty\frac{q^{n^2+n}}{(q;q)_n}=\frac{1}{(q^2;q^5)_\infty(q^3;q^5)_\infty}$

I am intrigued by the following unreferenced statement in the wikipedia [page][1]: [1]: http://en.wikipedia.org/wiki/Rogers%E2%80%93Ramanujan_identities#Modular_functions%20%22page%22

If q = e^2πiτ, then q^−1/60G(q) and q^11/60H(q) are modular functions of τ.

1. Do modular forms shed any light on the Rogers-Ramanujan Identities, or is the connection (as far as we know) a curious coincidence?

2. Is there some class of modular forms whose Fourier series count natural collections of partitions such as those counted by the left-hand sides of the Rogers-Ramanujan Identities? In particular I have in mind the seemingly "non-local" condition that if k is part, then it is distinct and also k+1 is not parts.

3. In general, how does one tell if a certain generating function (that counts partitions of a certain type, say) is related (by a multiplicative factor like above) to a modular form of some weight for some group (maybe even with some character)?