Rogers-Ramanujan Continued Fraction


The Rogers-Ramanujan continued fraction is a generalized continued fraction defined by


(Rogers 1894, Ramanujan 1957, Berndt et al. 1996, 1999, 2000). It was discovered by Rogers (1894), independently by Ramanujan around 1913, and again independently by Schur in 1917. Modulo the factor of q^(1/5) added for convenience, it provides a geometric series q-analog of the golden ratio


The convergents A_n(q)/B_n(q) of q^(-1/5)R(q) are given by


(OEIS A128915 and A127836; Sills 2003, p. 25, identity 3-14).

The fraction can be expressed in closed form in terms of q-series by


and in terms of the Ramanujan theta function




In the upper half-plane and modulo branch cuts, it can also be expressed exactly in terms of the Dedekind eta function eta(tau) by




(Trott 2004).

The coefficients of q^n in the Maclaurin series of R(q)/q^(1/5) for n=0, 1, 2, ... are 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 2, -3, ... (OEIS A007325).


The fraction converges quickly for points sufficiently far from the unit circle in the complex plane. For values |q|<1, the series converges to a unique value, while for |q|>1, it converges to two possible values. The value of the nth convergent of the continued fraction R_n(q) can be expressed in terms of the unique value inside the unit disk as

 R_n(q)={q^(1/5)(-q^(-1))^(-1/5)[R_n(-q^(-1))]^(-1)   for n even; q^(-4/5)(q^(-4))^(-1/5)R(q^(-4))   for n odd

(Andrews et al. 1992, Trott 2004).

Amazingly, Ramanujan showed that R(e^(-pisqrt(n))) is an algebraic number for all positive rational r. Special cases include


where r_1 is the root of x^8+14x^7+22x^6+22x^5+30x^4-22x^3+22x^2-14x+1 near 0.51142.... r_1 can be written as

 r_1=1/8(3+sqrt(5))(RadicalBox[5, 4]-1)(sqrt(10+2sqrt(5))-(3+RadicalBox[5, 4])(RadicalBox[5, 4]-1))

(Yi 2001, Trott 2004). The values of r_n have been computed by Trott for all values of n>=10, and the algebraic degrees of r_1, r_2, ... are 8, 4, 32, 8, 40, 16, 64, 16, 96, 20, ... (OEIS A082682; Trott 2004).

R(q) satisfies the amazing equalities


where (q)_infty=(q;q)_infty is a q-Pochhammer symbol. It also satisfies


(Watson 1929ab; Berndt 1991, pp. 265-267; Berndt et al. 1996, 2000; Son 1998).



these quantities satisfy the modular equations


(Berndt et al. 1996, 2000). Trott (2004) gives modular equations of orders 2 to 15 and the primes 17, 19, and 23.

As discussed by Hardy (Ramanujan 1962, pp. xxvii and xxviii), Berndt and Rankin (1995), and Berndt et al. (1996, 2000), Ramanujan also defined the generalized continued fraction


Ramanujan also considered the continued fraction


(Berndt 1991, p. 30; Berndt et al. 1996, 2000), of which the special case F(q)=F(1,q) is plotted above.

Terminating at a term aq^n gives


(Berndt et al. 1996, 2000).

The real roots of F(q) are 0.576149, 0.815600, 0.882493, 0.913806, 0.931949, 0.943785, 0.952125, ..., the smallest of which was found by Ramanujan (Berndt et al. ). F(q) and its smallest positive root are related to the enumeration of coins in a fountain (Berndt 1991, Berndt et al. 1996, 2000) and the study of birth and death processes (Berndt et al. 1996, 2000; Parthasarathy et al. 1998). In general, the least positive root q_0(a) of F(a,q) is given as a->infty by


(OEIS A050203; Berndt et al. 1996, 2000). Ramanujan gave the amazing approximations


For a=1, these approximations give

q_0^((1))(1)=1/3sqrt(3) approx 0.57735
q_0^((2))(1)=3/(110)(9+7sqrt(3)) approx 0.576119.

More generally, for the broad class of q^_ defined as q^_=e^(2piitau), R(q) can be evaluated in terms of the j-function j(tau) and the icosahedral equation as


with one of the r_i as r=R(q) (Duke 2004). As an example, R(e^(-2pi)) has tau=sqrt(-1)=i, so j(tau)=12^3. Substituting 12^3 into the equation, one of its factors will be a quartic with the root r=R(e^(-2pi)).

Furthermore, the numerator and the denominator (with a constant) can be combined to form a perfect square,


which are in fact polynomial invariants of the icosahedral group.

See also

Bauer-Muir Transformation, Fountain, Generalized Continued Fraction, Golden Ratio, q-Series, Ramanujan Theta Functions, Rogers-Ramanujan Identities

Portions of this entry contributed by Tito Piezas III

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Rogers-Ramanujan Continued Fraction

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Piezas, Tito III and Weisstein, Eric W. "Rogers-Ramanujan Continued Fraction." From MathWorld--A Wolfram Web Resource.

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