An analogue of Rogers-Ramanujan continued fraction

• A=2 (2,5) minimal model
  \(\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} \sim \sqrt\frac{2}{5+\sqrt{5}}\exp(\frac{\pi^2}{15t}+\frac{11t}{60})+o(1)\)
  \(\sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} \sim \sqrt\frac{2}{5-\sqrt{5}}\exp(\frac{\pi^2}{15t}-\frac{t}{60})+o(1)\)
  
• A=1/2 (3,5) minimal model
  \(\sum_{n=0}^\infty \frac {q^{\frac{n^2}{4}}} {(q;q)_n}\sim \frac{2}{\sqrt{5-\sqrt{5}}}\exp(\frac{\pi^2}{10t}-\frac{t}{40})+o(t^5)\)
  \(\sum_{n=0}^\infty \frac {q^{\frac{n^2}{4}+\frac{n}{2}}} {(q;q)_n} \sim \frac{2}{\sqrt{5+\sqrt{5}}}\exp(\frac{\pi^2}{10t}+\frac{t}{40})+o(t^5)\)
  

 

 

recurrence relation

• http://pythagoras0.springnote.com/pages/3004578
  

\(R(z)=\sum_{n\geq 0}\frac{z^nq^{n^2/4}}{(1-q)\cdots(1-q^n)}\)

\(H(q)=R(q)\)

 

 

(정리)

\(a=1,b=1/4\)

\(R(z)=R(zq^{1/a})+z^{a}q^{b}R(zq^{2b/a})\) i.e.

\(R(z)=R(zq)+zq^{1/4}R(zq^{1/2})\)