Khinchin Harmonic Mean
المؤلف:
Bailey, D. H.; Borwein, J. M.; and Crandall, R. E.
المصدر:
"On the Khintchine Constant." Math. Comput. 66
الجزء والصفحة:
...
31-1-2020
1004
Khinchin Harmonic Mean
Let ![[a_0;a_1,a_2,...]](http://mathworld.wolfram.com/images/equations/KhinchinHarmonicMean/Inline1.gif)
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
(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."
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