The Rogers–Ramanujan continued fraction is a continued fraction discovered by Rogers (1894) and independently by Srinivasa Ramanujan, and closely related to the Rogers–Ramanujan identities. It can be evaluated explicitly for a broad class of values of its argument.
Definition
Given the functions G ( q ) {\displaystyle G(q)} and H ( q ) {\displaystyle H(q)} appearing in the Rogers–Ramanujan identities, and assume q = e 2 π i τ {\displaystyle q=e^{2\pi i\tau }} ,
G ( q ) = ∑ n = 0 ∞ q n 2 ( 1 − q ) ( 1 − q 2 ) ⋯ ( 1 − q n ) = ∑ n = 0 ∞ q n 2 ( q ; q ) n = 1 ( q ; q 5 ) ∞ ( q 4 ; q 5 ) ∞ = ∏ n = 1 ∞ 1 ( 1 − q 5 n − 1 ) ( 1 − q 5 n − 4 ) = q j 60 2 F 1 ( − 1 60 , 19 60 ; 4 5 ; 1728 j ) = q ( j − 1728 ) 60 2 F 1 ( − 1 60 , 29 60 ; 4 5 ; − 1728 j − 1728 ) = 1 + q + q 2 + q 3 + 2 q 4 + 2 q 5 + 3 q 6 + ⋯ {\displaystyle {\begin{aligned}G(q)&=\sum _{n=0}^{\infty }{\frac {q^{n^{2}}}{(1-q)(1-q^{2})\cdots (1-q^{n})}}=\sum _{n=0}^{\infty }{\frac {q^{n^{2}}}{(q;q)_{n}}}={\frac {1}{(q;q^{5})_{\infty }(q^{4};q^{5})_{\infty }}}\\[6pt]&=\prod _{n=1}^{\infty }{\frac {1}{(1-q^{5n-1})(1-q^{5n-4})}}\\[6pt]&={\sqrt[{60}]{q\,j}}\,\,_{2}F_{1}\left(-{\tfrac {1}{60}},{\tfrac {19}{60}};{\tfrac {4}{5}};{\tfrac {1728}{j}}\right)\\[6pt]&={\sqrt[{60}]{q\left(j-1728\right)}}\,_{2}F_{1}\left(-{\tfrac {1}{60}},{\tfrac {29}{60}};{\tfrac {4}{5}};-{\tfrac {1728}{j-1728}}\right)\\[6pt]&=1+q+q^{2}+q^{3}+2q^{4}+2q^{5}+3q^{6}+\cdots \end{aligned}}}
and,
H ( q ) = ∑ n = 0 ∞ q n 2 + n ( 1 − q ) ( 1 − q 2 ) ⋯ ( 1 − q n ) = ∑ n = 0 ∞ q n 2 + n ( q ; q ) n = 1 ( q 2 ; q 5 ) ∞ ( q 3 ; q 5 ) ∞ = ∏ n = 1 ∞ 1 ( 1 − q 5 n − 2 ) ( 1 − q 5 n − 3 ) = 1 q 11 j 11 60 2 F 1 ( 11 60 , 31 60 ; 6 5 ; 1728 j ) = 1 q 11 ( j − 1728 ) 11 60 2 F 1 ( 11 60 , 41 60 ; 6 5 ; − 1728 j − 1728 ) = 1 + q 2 + q 3 + q 4 + q 5 + 2 q 6 + 2 q 7 + ⋯ {\displaystyle {\begin{aligned}H(q)&=\sum _{n=0}^{\infty }{\frac {q^{n^{2}+n}}{(1-q)(1-q^{2})\cdots (1-q^{n})}}=\sum _{n=0}^{\infty }{\frac {q^{n^{2}+n}}{(q;q)_{n}}}={\frac {1}{(q^{2};q^{5})_{\infty }(q^{3};q^{5})_{\infty }}}\\[6pt]&=\prod _{n=1}^{\infty }{\frac {1}{(1-q^{5n-2})(1-q^{5n-3})}}\\[6pt]&={\frac {1}{\sqrt[{60}]{q^{11}j^{11}}}}\,_{2}F_{1}\left({\tfrac {11}{60}},{\tfrac {31}{60}};{\tfrac {6}{5}};{\tfrac {1728}{j}}\right)\\[6pt]&={\frac {1}{\sqrt[{60}]{q^{11}\left(j-1728\right)^{11}}}}\,_{2}F_{1}\left({\tfrac {11}{60}},{\tfrac {41}{60}};{\tfrac {6}{5}};-{\tfrac {1728}{j-1728}}\right)\\[6pt]&=1+q^{2}+q^{3}+q^{4}+q^{5}+2q^{6}+2q^{7}+\cdots \end{aligned}}}
with the coefficients of the q-expansion being OEIS: A003114 and OEIS: A003106, respectively, where ( a ; q ) ∞ {\displaystyle (a;q)_{\infty }} denotes the infinite q-Pochhammer symbol, j is the j-function, and 2F1 is the hypergeometric function. The Rogers–Ramanujan continued fraction is then
R ( q ) = q 11 60 H ( q ) q − 1 60 G ( q ) = q 1 5 ∏ n = 1 ∞ ( 1 − q 5 n − 1 ) ( 1 − q 5 n − 4 ) ( 1 − q 5 n − 2 ) ( 1 − q 5 n − 3 ) = q 1 / 5 ∏ n = 1 ∞ ( 1 − q n ) ( n | 5 ) = q 1 / 5 1 + q 1 + q 2 1 + q 3 1 + ⋱ {\displaystyle {\begin{aligned}R(q)&={\frac {q^{\frac {11}{60}}H(q)}{q^{-{\frac {1}{60}}}G(q)}}=q^{\frac {1}{5}}\prod _{n=1}^{\infty }{\frac {(1-q^{5n-1})(1-q^{5n-4})}{(1-q^{5n-2})(1-q^{5n-3})}}=q^{1/5}\prod _{n=1}^{\infty }(1-q^{n})^{(n|5)}\\[8pt]&={\cfrac {q^{1/5}}{1+{\cfrac {q}{1+{\cfrac {q^{2}}{1+{\cfrac {q^{3}}{1+\ddots }}}}}}}}\end{aligned}}}
( n ∣ m ) {\displaystyle (n\mid m)} is the Jacobi symbol. One should be careful with notation since the formulas employing the j-function j {\displaystyle j} will be consistent with the other formulas only if q = e 2 π i τ {\displaystyle q=e^{2\pi i\tau }} (the square of the nome) is used throughout this section since the q-expansion of the j-function (as well as the well-known Dedekind eta function) uses q = e 2 π i τ {\displaystyle q=e^{2\pi i\tau }} . However, Ramanujan, in his examples to Hardy and given below, used the nome q = e π i τ {\displaystyle q=e^{\pi i\tau }} instead.
Special values If q is the nome or its square, then q − 1 60 G ( q ) {\displaystyle q^{-{\frac {1}{60}}}G(q)} and q 11 60 H ( q ) {\displaystyle q^{\frac {11}{60}}H(q)} , as well as their quotient R ( q ) {\displaystyle R(q)} , are related to modular functions of τ {\displaystyle \tau } . Since they have integral coefficients, the theory of complex multiplication implies that their values for τ {\displaystyle \tau } involving an imaginary quadratic field are algebraic numbers that can be evaluated explicitly.
Examples of R(q) Given the general form where Ramanujan used the nome q = e π i τ {\displaystyle q=e^{\pi i\tau }} ,
R ( q ) = q 1 / 5 1 + q 1 + q 2 1 + q 3 1 + ⋱ {\displaystyle R(q)={\cfrac {q^{1/5}}{1+{\cfrac {q}{1+{\cfrac {q^{2}}{1+{\cfrac {q^{3}}{1+\ddots }}}}}}}}}
f when τ = i {\displaystyle \tau =i} ,
R ( e − π ) = e − π 5 1 + e − π 1 + e − 2 π 1 + ⋱ = 1 2 φ ( 5 − φ 3 / 2 )