Continued fraction closely related to the Rogers–Ramanujan identities
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.
Domain coloring representation of the convergent
A
400
(
q
)
/
B
400
(
q
)
{\displaystyle A_{400}(q)/B_{400}(q)}
of the function
q
−
1
/
5
R
(
q
)
{\displaystyle q^{-1/5}R(q)}
, where
R
(
q
)
{\displaystyle R(q)}
is the Rogers–Ramanujan continued fraction.
Definition
Representation of the approximation
q
1
/
5
A
400
(
q
)
/
B
400
(
q
)
{\displaystyle q^{1/5}A_{400}(q)/B_{400}(q)}
of the Rogers–Ramanujan continued fraction.
Given the functions G (q ) and H (q ) appearing in the Rogers–Ramanujan identities,
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 }}}\\&=\prod _{n=1}^{\infty }{\frac {1}{(1-q^{5n-1})(1-q^{5n-4})}}\\&={\sqrt[{60}]{qj}}\,_{2}F_{1}\left(-{\tfrac {1}{60}},{\tfrac {19}{60}};{\tfrac {4}{5}};{\tfrac {1728}{j}}\right)\\&={\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)\\&=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 }}}\\&=\prod _{n=1}^{\infty }{\frac {1}{(1-q^{5n-2})(1-q^{5n-3})}}\\&={\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)\\&={\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)\\&=1+q^{2}+q^{3}+q^{4}+q^{5}+2q^{6}+2q^{7}+\cdots \end{aligned}}}
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 2 F1 is the hypergeometric function , then the Rogers–Ramanujan continued fraction is,
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
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})}}\\&={\cfrac {q^{1/5}}{1+{\cfrac {q}{1+{\cfrac {q^{2}}{1+{\cfrac {q^{3}}{1+\ddots }}}}}}}}\end{aligned}}}
Modular functions
If
q
=
e
2
π
i
τ
{\displaystyle q=e^{2\pi {\rm {i}}\tau }}
, 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 modular functions of
τ
{\displaystyle \tau }
. Since they have integral coefficients, the theory of complex multiplication implies that their values for
τ
{\displaystyle \tau }
an imaginary quadratic irrational are algebraic numbers that can be evaluated explicitly.
Examples
R
(
e
−
2
π
)
=
e
−
2
π
5
1
+
e
−
2
π
1
+
e
−
4
π
1
+
⋱
=
5
+
5
2
−
ϕ
{\displaystyle R{\big (}e^{-2\pi }{\big )}={\cfrac {e^{-{\frac {2\pi }{5}}}}{1+{\cfrac {e^{-2\pi }}{1+{\cfrac {e^{-4\pi }}{1+\ddots }}}}}}={{\sqrt {5+{\sqrt {5}} \over 2}}-\phi }}
R
(
e
−
2
5
π
)
=
e
−
2
π
5
1
+
e
−
2
π
5
1
+
e
−
4
π
5
1
+
⋱
=
5
1
+
(
5
3
/
4
(
ϕ
−
1
)
5
/
2
−
1
)
1
/
5
−
ϕ
{\displaystyle R{\big (}e^{-2{\sqrt {5}}\pi }{\big )}={\cfrac {e^{-{\frac {2\pi }{\sqrt {5}}}}}{1+{\cfrac {e^{-2\pi {\sqrt {5}}}}{1+{\cfrac {e^{-4\pi {\sqrt {5}}}}{1+\ddots }}}}}}={\frac {\sqrt {5}}{1+{\big (}5^{3/4}(\phi -1)^{5/2}-1{\big )}^{1/5}}}-{\phi }}
where
ϕ
=
1
+
5
2
{\displaystyle \phi ={\frac {1+{\sqrt {5}}}{2}}}
is the golden ratio .
Relation to modular forms
It can be related to the Dedekind eta function , a modular form of weight 1/2, as,[1]
1
R
(
q
)
−
R
(
q
)
=
η
(
τ
5
)
η
(
5
τ
)
+
1
{\displaystyle {\frac {1}{R(q)}}-R(q)={\frac {\eta ({\frac {\tau }{5}})}{\eta (5\tau )}}+1}
1
R
5
(
q
)
−
R
5
(
q
)
=
[
η
(
τ
)
η
(
5
τ
)
]
6
+
11
{\displaystyle {\frac {1}{R^{5}(q)}}-R^{5}(q)=\left[{\frac {\eta (\tau )}{\eta (5\tau )}}\right]^{6}+11}
Relation to j-function
Among the many formulas of the j-function , one is,
j
(
τ
)
=
(
x
2
+
10
x
+
5
)
3
x
{\displaystyle j(\tau )={\frac {(x^{2}+10x+5)^{3}}{x}}}
where
x
=
[
5
η
(
5
τ
)
η
(
τ
)
]
6
{\displaystyle x=\left[{\frac {{\sqrt {5}}\,\eta (5\tau )}{\eta (\tau )}}\right]^{6}}
Eliminating the eta quotient, one can then express j (τ ) in terms of
r
=
R
(
q
)
{\displaystyle r=R(q)}
as,
j
(
τ
)
=
−
(
r
20
−
228
r
15
+
494
r
10
+
228
r
5
+
1
)
3
r
5
(
r
10
+
11
r
5
−
1
)
5
j
(
τ
)
−
1728
=
−
(
r
30
+
522
r
25
−
10005
r
20
−
10005
r
10
−
522
r
5
+
1
)
2
r
5
(
r
10
+
11
r
5
−
1
)
5
{\displaystyle {\begin{aligned}&j(\tau )=-{\frac {(r^{20}-228r^{15}+494r^{10}+228r^{5}+1)^{3}}{r^{5}(r^{10}+11r^{5}-1)^{5}}}\\[6pt]&j(\tau )-1728=-{\frac {(r^{30}+522r^{25}-10005r^{20}-10005r^{10}-522r^{5}+1)^{2}}{r^{5}(r^{10}+11r^{5}-1)^{5}}}\end{aligned}}}
where the numerator and denominator are polynomial invariants of the icosahedron . Using the modular equation between
R
(
q
)
{\displaystyle R(q)}
and
R
(
q
5
)
{\displaystyle R(q^{5})}
, one finds that,
j
(
5
τ
)
=
−
(
r
20
+
12
r
15
+
14
r
10
−
12
r
5
+
1
)
3
r
25
(
r
10
+
11
r
5
−
1
)
{\displaystyle j(5\tau )=-{\frac {(r^{20}+12r^{15}+14r^{10}-12r^{5}+1)^{3}}{r^{25}(r^{10}+11r^{5}-1)}}}
let
z
=
r
5
−
1
r
5
{\displaystyle z=r^{5}-{\frac {1}{r^{5}}}}
,then
j
(
5
τ
)
=
−
(
z
2
+
12
z
+
16
)
3
z
+
11
{\displaystyle j(5\tau )=-{\frac {\left(z^{2}+12z+16\right)^{3}}{z+11}}}
where
z
∞
=
−
[
5
η
(
25
τ
)
η
(
5
τ
)
]
6
−
11
,
z
0
=
−
[
η
(
τ
)
η
(
5
τ
)
]
6
−
11
,
z
1
=
[
η
(
5
τ
+
2
5
)
η
(
5
τ
)
]
6
−
11
,
z
2
=
−
[
η
(
5
τ
+
4
5
)
η
(
5
τ
)
]
6
−
11
,
z
3
=
[
η
(
5
τ
+
6
5
)
η
(
5
τ
)
]
6
−
11
,
z
4
=
−
[
η
(
5
τ
+
8
5
)
η
(
5
τ
)
]
6
−
11
{\displaystyle {\begin{aligned}&z_{\infty }=-\left[{\frac {{\sqrt {5}}\,\eta (25\tau )}{\eta (5\tau )}}\right]^{6}-11,\ z_{0}=-\left[{\frac {\eta (\tau )}{\eta (5\tau )}}\right]^{6}-11,\ z_{1}=\left[{\frac {\eta ({\frac {5\tau +2}{5}})}{\eta (5\tau )}}\right]^{6}-11,\\[6pt]&z_{2}=-\left[{\frac {\eta ({\frac {5\tau +4}{5}})}{\eta (5\tau )}}\right]^{6}-11,\ z_{3}=\left[{\frac {\eta ({\frac {5\tau +6}{5}})}{\eta (5\tau )}}\right]^{6}-11,\ z_{4}=-\left[{\frac {\eta ({\frac {5\tau +8}{5}})}{\eta (5\tau )}}\right]^{6}-11\end{aligned}}}
which in fact is the j-invariant of the elliptic curve ,
y
2
+
(
1
+
r
5
)
x
y
+
r
5
y
=
x
3
+
r
5
x
2
{\displaystyle y^{2}+(1+r^{5})xy+r^{5}y=x^{3}+r^{5}x^{2}}
parameterized by the non-cusp points of the modular curve
X
1
(
5
)
{\displaystyle X_{1}(5)}
.
Functional equation
For convenience, one can also use the notation
r
(
τ
)
=
R
(
q
)
{\displaystyle r(\tau )=R(q)}
when q = e2πiτ . While other modular functions like the j-invariant satisfies,
j
(
−
1
τ
)
=
j
(
τ
)
{\displaystyle j(-{\tfrac {1}{\tau }})=j(\tau )}
and the Dedekind eta function has,
η
(
−
1
τ
)
=
−
i
τ
η
(
τ
)
{\displaystyle \eta (-{\tfrac {1}{\tau }})={\sqrt {-i\tau }}\,\eta (\tau )}
the functional equation of the Rogers–Ramanujan continued fraction involves[2] the golden ratio
ϕ
{\displaystyle \phi }
,
r
(
−
1
τ
)
=
1
−
ϕ
r
(
τ
)
ϕ
+
r
(
τ
)
{\displaystyle r(-{\tfrac {1}{\tau }})={\frac {1-\phi \,r(\tau )}{\phi +r(\tau )}}}
Incidentally,
r
(
7
+
i
10
)
=
i
{\displaystyle r({\tfrac {7+i}{10}})=i}
Modular equations
There are modular equations between
R
(
q
)
{\displaystyle R(q)}
and
R
(
q
n
)
{\displaystyle R(q^{n})}
. Elegant ones for small prime n are as follows.[3]
For
n
=
2
{\displaystyle n=2}
, let
u
=
R
(
q
)
{\displaystyle u=R(q)}
and
v
=
R
(
q
2
)
{\displaystyle v=R(q^{2})}
, then
v
−
u
2
=
(
v
+
u
2
)
u
v
2
.
{\displaystyle v-u^{2}=(v+u^{2})uv^{2}.}
For
n
=
3
{\displaystyle n=3}
, let
u
=
R
(
q
)
{\displaystyle u=R(q)}
and
v
=
R
(
q
3
)
{\displaystyle v=R(q^{3})}
, then
(
v
−
u
3
)
(
1
+
u
v
3
)
=
3
u
2
v
2
.
{\displaystyle (v-u^{3})(1+uv^{3})=3u^{2}v^{2}.}
For
n
=
5
{\displaystyle n=5}
, let
u
=
R
(
q
)
{\displaystyle u=R(q)}
and
v
=
R
(
q
5
)
{\displaystyle v=R(q^{5})}
, then
(
v
4
−
3
v
3
+
4
v
2
−
2
v
+
1
)
v
=
(
v
4
+
2
v
3
+
4
v
2
+
3
v
+
1
)
u
5
.
{\displaystyle (v^{4}-3v^{3}+4v^{2}-2v+1)v=(v^{4}+2v^{3}+4v^{2}+3v+1)u^{5}.}
For
n
=
11
{\displaystyle n=11}
, let
u
=
R
(
q
)
{\displaystyle u=R(q)}
and
v
=
R
(
q
11
)
{\displaystyle v=R(q^{11})}
, then
u
v
(
u
10
+
11
u
5
−
1
)
(
v
10
+
11
v
5
−
1
)
=
(
u
−
v
)
12
.
{\displaystyle uv(u^{10}+11u^{5}-1)(v^{10}+11v^{5}-1)=(u-v)^{12}.}
Regarding
n
=
5
{\displaystyle n=5}
, note that
v
10
+
11
v
5
−
1
=
(
v
2
+
v
−
1
)
(
v
4
−
3
v
3
+
4
v
2
−
2
v
+
1
)
(
v
4
+
2
v
3
+
4
v
2
+
3
v
+
1
)
.
{\displaystyle v^{10}+11v^{5}-1=(v^{2}+v-1)(v^{4}-3v^{3}+4v^{2}-2v+1)(v^{4}+2v^{3}+4v^{2}+3v+1).}
Other results
Ramanujan found many other interesting results regarding R (q ).[4] Let
u
=
R
(
q
a
)
{\displaystyle u=R(q^{a})}
,
v
=
R
(
q
b
)
{\displaystyle v=R(q^{b})}
, and
ϕ
{\displaystyle \phi }
as the golden ratio .
If
a
b
=
4
π
2
{\displaystyle ab=4\pi ^{2}}
, then
(
u
+
ϕ
)
(
v
+
ϕ
)
=
5
ϕ
.
{\displaystyle (u+\phi )(v+\phi )={\sqrt {5}}\,\phi .}
If
5
a
b
=
4
π
2
{\displaystyle 5ab=4\pi ^{2}}
, then
(
u
5
+
ϕ
5
)
(
v
5
+
ϕ
5
)
=
5
5
ϕ
5
.
{\displaystyle (u^{5}+\phi ^{5})(v^{5}+\phi ^{5})=5{\sqrt {5}}\,\phi ^{5}.}
The powers of R (q ) also can be expressed in unusual ways. For its cube ,
R
3
(
q
)
=
α
β
{\displaystyle R^{3}(q)={\frac {\alpha }{\beta }}}
where,
α
=
∑
n
=
0
∞
q
2
n
1
−
q
5
n
+
2
−
∑
n
=
0
∞
q
3
n
+
1
1
−
q
5
n
+
3
{\displaystyle \alpha =\sum _{n=0}^{\infty }{\frac {q^{2n}}{1-q^{5n+2}}}-\sum _{n=0}^{\infty }{\frac {q^{3n+1}}{1-q^{5n+3}}}}
β
=
∑
n
=
0
∞
q
n
1
−
q
5
n
+
1
−
∑
n
=
0
∞
q
4
n
+
3
1
−
q
5
n
+
4
{\displaystyle \beta =\sum _{n=0}^{\infty }{\frac {q^{n}}{1-q^{5n+1}}}-\sum _{n=0}^{\infty }{\frac {q^{4n+3}}{1-q^{5n+4}}}}
For its fifth power, let
w
=
R
(
q
)
R
2
(
q
2
)
{\displaystyle w=R(q)R^{2}(q^{2})}
, then,
R
5
(
q
)
=
w
(
1
−
w
1
+
w
)
2
,
R
5
(
q
2
)
=
w
2
(
1
+
w
1
−
w
)
{\displaystyle R^{5}(q)=w\left({\frac {1-w}{1+w}}\right)^{2},\;\;R^{5}(q^{2})=w^{2}\left({\frac {1+w}{1-w}}\right)}
References
Rogers, L. J. (1894), "Second Memoir on the Expansion of certain Infinite Products" , Proc. London Math. Soc. , s1-25 (1): 318–343, doi :10.1112/plms/s1-25.1.318
Berndt, B. C.; Chan, H. H.; Huang, S. S.; Kang, S. Y.; Sohn, J.; Son, S. H. (1999), "The Rogers–Ramanujan continued fraction" (PDF) , Journal of Computational and Applied Mathematics , 105 (1–2): 9–24, doi :10.1016/S0377-0427(99)00033-3
External links