Rogers–Ramanujan continued fraction - Alchetron, the free social encyclopedia


$Rogers–Ramanujan continued fraction$

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) 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 + ⋯

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 + ⋯

A003114 and A003106, respectively, where ( a ; q ) ∞ denotes the infinite q-Pochhammer symbol, j is the j-function, and ₂F₁ 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 + ⋱

Modular functions

If q = e 2 π i τ , then q − 1 60 G ( q ) and q 11 60 H ( q ) , as well as their quotient R ( q ) , are modular functions of τ . Since they have integral coefficients, the theory of complex multiplication implies that their values for τ 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 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

where ϕ = 1 + 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 R ( q ) − R ( q ) = η ( τ 5 ) η ( 5 τ ) + 1 1 R 5 ( q ) − R 5 ( q ) = [ η ( τ ) η ( 5 τ ) ] 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

where

x = [ 5 η ( 5 τ ) η ( τ ) ] 6

Eliminating the eta quotient, one can then express j(τ) in terms of 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

where the numerator and denominator are polynomial invariants of the icosahedron. Using the modular equation between R ( q ) and 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 )

let z = r 5 − 1 r 5 ,then j ( 5 τ ) = − ( z 2 + 12 z + 16 ) 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

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

parameterized by the non-cusp points of the modular curve X 1 ( 5 ) .

Functional equation

For convenience, one can also use the notation r ( τ ) = R ( q ) when q = e^2πiτ. While other modular functions like the j-invariant satisfies,

j ( − 1 τ ) = j ( τ )

and the Dedekind eta function has,

η ( − 1 τ ) = − i τ η ( τ )

the functional equation of the Rogers–Ramanujan continued fraction involves the golden ratio ϕ ,

r ( − 1 τ ) = 1 − ϕ r ( τ ) ϕ + r ( τ )

Incidentally,

r ( 7 + i 10 ) = i

Modular equations

There are modular equations between R ( q ) and R ( q n ) . Elegant ones for small prime n are as follows.

For n = 2 , let u = R ( q ) and v = R ( q 2 ) , then v − u 2 = ( v + u 2 ) u v 2 .

For n = 3 , let u = R ( q ) and v = R ( q 3 ) , then ( v − u 3 ) ( 1 + u v 3 ) = 3 u 2 v 2 .

For n = 5 , let u = R ( q ) and 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 .

For n = 11 , let u = R ( q ) and v = R ( q 11 ) , then u v ( u 10 + 11 u 5 − 1 ) ( v 10 + 11 v 5 − 1 ) = ( u − v ) 12 .

Regarding 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 ) .

Other results

Ramanujan found many other interesting results regarding R(q). Let u = R ( q a ) , v = R ( q b ) , and ϕ as the golden ratio.

If a b = 4 π 2 , then ( u + ϕ ) ( v + ϕ ) = 5 ϕ . If 5 a b = 4 π 2 , then ( u 5 + ϕ 5 ) ( v 5 + ϕ 5 ) = 5 5 ϕ 5 .

The powers of R(q) also can be expressed in unusual ways. For its cube,

R 3 ( q ) = α β

where,

α = ∑ n = 0 ∞ q 2 n 1 − q 5 n + 2 − ∑ n = 0 ∞ q 3 n + 1 1 − q 5 n + 3 β = ∑ n = 0 ∞ q n 1 − q 5 n + 1 − ∑ n = 0 ∞ q 4 n + 3 1 − q 5 n + 4

For its fifth power, let 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 )

References

Rogers–Ramanujan continued fraction Wikipedia

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