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Ramanujan Continued Fractions

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Ramanujan developed a number of interesting closed-form expressions for generalized continued fractions. These 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 1929, 1931; 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), 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 1929, 1931; Ramanathan 1984; 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/5))/((sqrt(5))/(1+[5^(3/4)(phi-1)^(5/2)]^(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 1953; Berndt and Rankin 1995, pp. 57 and 65; Hardy 1999, p. 8), 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 1953; Berndt and Rankin 1995, pp. 57 and 65), where psi_2(z) is a polygamma function.

See also

Continued Fraction Constants, Generalized Continued Fraction, Rogers-Ramanujan Continued Fraction

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References

Berndt, B. C. and Rankin, R. A. Ramanujan: Letters and Commentary. Providence, RI: Amer. Math. Soc., 1995.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Perron, O. "Über die Preeceschen Kettenbrüche." Sitz. Bayer. Akad. Wiss. München Math. Phys. Kl., 21-56, 1953.Preece, C. T. "Theorems Stated by Ramanujan (X)." J. London Math. Soc. 6, 22-32, 1931.Ramanathan, K. G. "On Ramanujan's Continued Fraction." Acta. Arith. 43, 209-226, 1984.Sloane, N. J. A. Sequences A091659, A091660, A091667, A091668, A091899, and A091900 in "The On-Line Encyclopedia of Integer Sequences."Watson, G. N. "Theorems Stated by Ramanujan (VII): Theorems on a Continued Fraction." J. London Math. Soc. 4, 39-48, 1929.Watson, G. N. "Theorems Stated by Ramanujan (IX): Two Continued Fractions." J. London Math. Soc. 4, 231-237, 1929.

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Ramanujan Continued Fractions

Cite this as:

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

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