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Ramanujan Continued Fractions

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Ramanujan continued fractions are generalized continued fractions associated with the work of Ramanujan (Hardy 1999, p. 8). Examples with closed-form expressions found by Ramanujan include the almost integers

1/(1+)(e^(-2pi))/(1+)(e^(-4pi))/(1+...)=(sqrt((5+sqrt(5))/2)-(sqrt(5)+1)/2)e^(2pi/5)
(1)
=e^(2pi/5)(sqrt(phisqrt(5))-phi)
(2)
=0.9981360...
(3)

(OEIS A091667; Watson 1929ab; Ramanathan 1984, p. 215; Hardy 1999, p. 8), where phi is the golden ratio, its multiplicative inverse

1+(e^(-2pi))/(1+)(e^(-4pi))/(1+)(e^(-6pi))/(1+...)=1/2[1+sqrt(5)+sqrt(2(5+sqrt(5)))]e^(-2pi/5)
(4)
=(e^(-2pi/5))/(sqrt(phisqrt(5))-phi)
(5)
=1.0018674...
(6)

(OEIS A091899; Ramanathan 1984, p. 215), and

1/(1+)(e^(-2pisqrt(5)))/(1+)(e^(-4pisqrt(5)))/(1+...)={(sqrt(5))/(1+[5^(3/4)(phi-1)^(5/2)-1]^(1/5))-phi}e^(2pi/sqrt(5))
(7)
=0.99999920...
(8)

(OEIS A091668; Watson 1929ab; Ramanathan 1984, p. 215; Berndt and Rankin 1995, p. 57; Hardy 1999, p. 8) and its multiplicative inverse

1+(e^(-2pisqrt(5)))/(1+)(e^(-4pisqrt(5)))/(1+...)=(e^(-2pi/sqrt(5)))/((sqrt(5))/(1+[5^(3/4)(phi-1)^(5/2)-1]^(1/5))-phi)
(9)
=1.000000791267...
(10)

(OEIS A091900).

Other examples include the integrals

4int_0^infty(xe^(-xsqrt(5)))/(coshx)dx=1/2[zeta(2,1/4(1+sqrt(5)))-zeta(2,1/4(3+sqrt(5)))]
(11)
=1/2[psi_1(1/4(1+sqrt(5)))-psi_1(1/4(3+sqrt(5)))]
(12)
=1/(1+)(1^2)/(1+)(1^2)/(1+)(2^2)/(1+)(2^2)/(1+)(3^2)/(1+)(3^2)/(1+)...
(13)
=0.5683000...
(14)

(OEIS A091659; Preece 1931, Perron 1954; Berndt and Rankin 1995, pp. 57 and 65; Hardy 1999, p. 8; Berndt and Dixit 2020, p. 72), where zeta(a,z) is the Hurwitz zeta function and psi_1(z) is the trigamma function, and

2int_0^infty(x^2e^(-xsqrt(3)))/(sinhx)dx=-1/2psi_2(1/2(1+sqrt(3)))
(15)
=1/(1+)(1^3)/(1+)(1^3)/(3+)(2^3)/(1+)(2^3)/(5+)(3^3)/(1+)(3^3)/(7+)...
(16)
=0.5269391...
(17)

(OEIS A091660; Preece 1931, Perron 1954; Berndt and Rankin 1995, pp. 57 and 65; Berndt and Dixit 2020, p. 72), where psi_2(z) is a polygamma function.

Ramanujan also gave a generalized continued fraction whose convergents have three distinct subsequential limits. For |q|<1, define

T_n(q)=1/(1+K_(j=1)^(n)(-1)/(1+q^j)).
(18)

The subsequences with n in the three residue classes modulo 3 converge to distinct points of the Riemann sphere, although the complete sequence does not converge (Andrews et al. 2005). Set

Omega(q)=((omega^2q;q)_infty)/((omegaq;q)_infty),
(19)

where omega=e^(2pii/3) is a cube root of unity and (a;q)_infty is the q-Pochhammer symbol. The three limits are

L_r(q)=lim_(m->infty)T_(3m+r)(q)=-omega^2(Omega(q)-omega^(r+1))/(Omega(q)-omega^(r-1))((q^2;q^3)_infty)/((q;q^3)_infty)
(20)

for r=0, 1, 2, with an infinite value interpreted on the Riemann sphere. For example, at q=0 the convergents repeat the cycle infty, 0, 1, so L_0(0)=1, L_1(0)=infty, and L_2(0)=0. Bowman and Mc Laughlin (2007) give related constructions with k distinct subsequential limits for any integer k>=2.

See also

Continued Fraction Constants, Generalized Continued Fraction, Rogers-Ramanujan Continued Fraction, Stern-Stolz Theorem

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References

Andrews, G. E.; Berndt, B. C.; Sohn, J.; Yee, A. J.; and Zaharescu, A. "On Ramanujan's Continued Fraction for (q^2;q^3)_infty/(q;q^3)_infty." Trans. Amer. Math. Soc. 355, 2397-2411, 2003. https://doi.org/10.1090/S0002-9947-02-03155-0.Andrews, G. E.; Berndt, B. C.; Sohn, J.; Yee, A. J.; and Zaharescu, A. "Continued Fractions with Three Limit Points." Adv. Math. 192, 231-258, 2005. https://doi.org/10.1016/j.aim.2004.04.004.Berndt, B. C. and Dixit, A. "Ramanujan's Beautiful Integrals." Hardy-Ramanujan J. 43, 69-82, 2020. https://doi.org/10.46298/hrj.2021.7429.Berndt, B. C. and Rankin, R. A. Ramanujan: Letters and Commentary. Providence, RI: Amer. Math. Soc., 1995.Bowman, D. and Mc Laughlin, J. "Continued Fractions with Multiple Limits." Adv. Math. 210, 578-606, 2007. https://doi.org/10.1016/j.aim.2006.07.004. https://arxiv.org/abs/math/0403027.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Perron, O. "Über die Preece'schen Kettenbrüche." Sitz. Bayer. Akad. Wiss. München Math.-Naturwiss. Kl. 1953, 21-56, 1954. https://publikationen.badw.de/de/003383728/003383728.pdf.Preece, C. T. "Theorems Stated by Ramanujan (X)." J. London Math. Soc. 6, 22-32, 1931. https://doi.org/10.1112/jlms/s1-6.1.22.Ramanathan, K. G. "On Ramanujan's Continued Fraction." Acta Arith. 43, 209-226, 1984. https://doi.org/10.4064/aa-43-3-209-226.Ramanujan, S. The Lost Notebook and Other Unpublished Papers. New Delhi, India: Narosa, 1988.Sloane, N. J. A. Sequences A091659, A091660, A091667, A091668, A091899, and A091900 in "The On-Line Encyclopedia of Integer Sequences."Trott, M. "Series Expansions." §1.6.4 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. https://doi.org/10.1007/0-387-28815-5. https://www.mathematicaguidebooks.org/.Watson, G. N. "Theorems Stated by Ramanujan (VII): Theorems on Continued Fractions." J. London Math. Soc. 4, 39-48, 1929a. https://doi.org/10.1112/jlms/s1-4.1.39.Watson, G. N. "Theorems Stated by Ramanujan (IX): Two Continued Fractions." J. London Math. Soc. 4, 231-237, 1929b. https://doi.org/10.1112/jlms/s1-4.3.231.

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Ramanujan Continued Fractions

Cite this as:

Weisstein, Eric W. "Ramanujan Continued Fractions." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RamanujanContinuedFractions.html

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