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Ramanujan Continued Fractions

المؤلف:  Berndt, B. C. and Rankin, R. A.

المصدر:  Ramanujan: Letters and Commentary. Providence, RI: Amer. Math. Soc., 1995.

الجزء والصفحة:  ...

12-5-2020

1243

Ramanujan Continued Fractions

Ramanujan developed a number of interesting closed-form expressions for generalized continued fractions. These include the almost integers

			(1)
			(2)
			(3)

(OEIS A091667; Watson 1929, 1931; Hardy 1999, p. 8), where  is the golden ratio, its multiplicative inverse

			(4)
			(5)
			(6)

(OEIS A091899; Ramanathan 1984), and

		{(sqrt(5))/(1+[5^(3/4)(phi-1)^(5/2)-1]^(1/5))-phi}e^(2pi/sqrt(5))" src="https://mathworld.wolfram.com/images/equations/RamanujanContinuedFractions/Inline22.gif" style="height:48px; width:233px" />	(7)
			(8)

(OEIS A091668; Watson 1929, 1931; Ramanathan 1984; Berndt and Rankin 1995, p. 57; Hardy 1999, p. 8) and its multiplicative inverse

			(9)
			(10)

(OEIS A091900).

Other examples include the integrals

			(11)
			(12)
			(13)
			(14)

(OEIS A091659; Preece 1931; Perron 1953; Berndt and Rankin 1995, pp. 57 and 65; Hardy 1999, p. 8), where  is the Hurwitz zeta function and  is the trigamma function, and

			(15)
			(16)
			(17)

(OEIS A091660; Preece 1931; Perron 1953; Berndt and Rankin 1995, pp. 57 and 65), where  is a polygamma function.

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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.