Étant données les fonctions G (q ) et H (q ) apparaissant dans les identités de Rogers-Ramanujan ,
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}}}
et
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}}}
où
(
a
;
q
)
∞
{\displaystyle (a;q)_{\infty }}
q-symbole de Pochhammer infini, j est le j-invariant , et 2 F1 est la fonction hypergéométrique (les coefficients des développements en séries entières forment les suites de l'OEIS A003114 A003106
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}}}
Fonctions modulaires
modifier
Liens avec les formes modulaires
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Liens avec le j-invariant
modifier
Parmi les nombreuses relations vérifiées par le j-invariant , on a
j
(
τ
)
=
(
x
2
+
10
x
+
5
)
3
x
{\displaystyle j(\tau )={\frac {(x^{2}+10x+5)^{3}}{x}}}
où
x
=
[
5
η
(
5
τ
)
η
(
τ
)
]
6
{\displaystyle x=\left[{\frac {{\sqrt {5}}\,\eta (5\tau )}{\eta (\tau )}}\right]^{6}}
Éliminant le quotient, on peut exprimer j (τ ) en termes de
r
=
R
(
q
)
{\displaystyle r=R(q)}
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}}}
où le numérateur et le dénominateur sont des invariants polynomiaux de l'icosaèdre . La relation modulaire entre
R
(
q
)
{\displaystyle R(q)}
R
(
q
5
)
{\displaystyle R(q^{5})}
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)}}}
Soit
z
=
r
5
−
1
r
5
{\displaystyle z=r^{5}-{\frac {1}{r^{5}}}}
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}}}
où
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}}}
ce qui est le j-invariant de la courbe elliptique
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}}
courbe modulaire
X
1
(
5
)
{\displaystyle X_{1}(5)}
Équation fonctionnelle
modifier
On pose désormais systématiquement
r
(
τ
)
=
R
(
q
)
{\displaystyle r(\tau )=R(q)}
q = e2πiτ . Là où d'autres fonctions modulaires, par exemple le j-invariant, vérifient :
j
(
−
1
τ
)
=
j
(
τ
)
{\displaystyle j(-{\tfrac {1}{\tau }})=j(\tau )}
et qu'on a pour la fonction êta de Dedekind :
η
(
−
1
τ
)
=
−
i
τ
η
(
τ
)
{\displaystyle \eta (-{\tfrac {1}{\tau }})={\sqrt {-i\tau }}\,\eta (\tau )}
l'équation fonctionnelle de la fraction continue de Rogers–Ramanujan met en jeu[ 3] nombre d'or
ϕ
=
1
+
5
2
{\displaystyle \phi ={\frac {1+{\sqrt {5}}}{2}}}
r
(
−
1
τ
)
=
1
−
ϕ
r
(
τ
)
ϕ
+
r
(
τ
)
{\displaystyle r(-{\tfrac {1}{\tau }})={\frac {1-\phi \,r(\tau )}{\phi +r(\tau )}}}
On a d'autre part
r
(
7
+
i
10
)
=
i
{\displaystyle r({\tfrac {7+i}{10}})=i}
Équations modulaires
modifier
Autres résultats
modifier
Ramanujan a découvert beaucoup d'autres propriétés intéressantes de R (q )[ 5]
u
=
R
(
q
a
)
{\displaystyle u=R(q^{a})}
v
=
R
(
q
b
)
{\displaystyle v=R(q^{b})}
ϕ
{\displaystyle \phi }
nombre d'or ,
si
a
b
=
4
π
2
{\displaystyle ab=4\pi ^{2}}
(
u
+
ϕ
)
(
v
+
ϕ
)
=
5
ϕ
.
{\displaystyle (u+\phi )(v+\phi )={\sqrt {5}}\,\phi .}
si
5
a
b
=
4
π
2
{\displaystyle 5ab=4\pi ^{2}}
(
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}.}
Les puissances de R (q ) vérifient également des relations inattendues. Ainsi,
R
3
(
q
)
=
α
β
{\displaystyle R^{3}(q)={\frac {\alpha }{\beta }}}
où
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}}}}
Posant
w
=
R
(
q
)
R
2
(
q
2
)
{\displaystyle w=R(q)R^{2}(q^{2})}
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)}
Notes et références
modifier
↑ (en) Godfrey Harold Hardy , « The Indian Mathematician Ramanujan The American Mathematical Monthly vol. 44, no 3, mars 1937 , p. 137-155 (lire en ligne ) ↑ (en) Duke, W. "Continued Fractions and Modular Functions", http://www.math.ucla.edu/~wdduke/preprints/bams4.pdf ↑ (en) Duke, W. "Continued Fractions and Modular Functions" (p.9)↑ (en) Berndt, B. et al. "The Rogers–Ramanujan Continued Fraction" [lire en ligne ] .↑ (en) Berndt, B. et al. "The Rogers–Ramanujan Continued Fraction"
(en) L. J. Rogers , « Second Memoir on the Expansion of certain Infinite Products », Proc. London Math. Soc. vol. s1-25, no 1, 1894 , p. 318–343 (DOI 10.1112/plms/s1-25.1.318 (en) B. C. Berndt , H. H. Chan , S. S. Huang , S. Y. Kang , J. Sohn et S. H. Son , « The Rogers–Ramanujan continued fraction », Journal of Computational and Applied Mathematics vol. 105, 1999 , p. 9 (DOI 10.1016/S0377-0427(99)00033-3 lire en ligne ) 
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e74f98d4b21f51c75b6da28b9eff82591f1b1726)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed5aa7dbd4c48804f6de43f7bdbebc2d8ceadfa9)
A003114 et 
![{\displaystyle {\frac {1}{R^{5}(q)}}-R^{5}(q)=\left[{\frac {\eta (\tau )}{\eta (5\tau )}}\right]^{6}+11}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd161c8522f2d0446ee11792d6f37d8f8aeb7d5)
![{\displaystyle x=\left[{\frac {{\sqrt {5}}\,\eta (5\tau )}{\eta (\tau )}}\right]^{6}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/595729ac7ecb75f7ab2f3bdf7f3189dc3f74c279)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/883827756d7f47131c78695dfb009752642db072)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc22fc990829510a6e4dc7225adb6abe4ad177f)
