Khinchin Harmonic Mean

Let  be the simple continued fraction of a "generic" real number, where the numbers  are the partial quotients. Then the Khinchin (or Khintchine) harmonic mean

(1)

defined analogously to the Khinchin constant  but with the partial quotients taken to the  power, exists and has a unique common value (except for a set of real numbers with measure zero) given by

			(2)
			(3)
			(4)

(OEIS A087491; Bailey et al. 1997, Plouffe).

Khinchin's constant  and the Khinchin harmonic mean  are just two of an infinite family of such constants , the first few of which are summarized in the following table.

	OEIS	value
0	A002210	2.685452001065306445309714835481795693820382293994462
	A087491	1.745405662407346863494596309683661067294936618777984
	A087492	1.450340328495630406052983076680697881408299979605904
	A087493	1.313507078687985766717339447072786828158129861484792
	A087494	1.236961809423730052626227244453422567420241131548937
	A087495	1.189003926465513154062363732771403397386092512639671
	A087496	1.156552374421514423152605998743410046840213070718761
	A087497	1.133323363950865794910289694908868363599098282411797
	A087498	1.115964408978716690619156419345349695769491182230400
	A087499	1.102543136670728013836093402522568351022221284149318
	A087500	1.091877041209612678276110979477638256493272651429656

REFERENCES:

Bailey, D. H.; Borwein, J. M.; and Crandall, R. E. "On the Khintchine Constant." Math. Comput. 66, 417-431, 1997.

Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 161, 2003.

Khinchin, A. Ya. "Average Values." §16 in Continued Fractions. New York: Dover, pp. 86-94, 1997.

Plouffe, S. "The Khintchine Harmonic Mean." http://pi.lacim.uqam.ca/piDATA/khintchine1.txt.

Sloane, N. J. A. Sequence A087491 in "The On-Line Encyclopedia of Integer Sequences."