Somos,s Quadratic Recurrence Constant
المؤلف:
Finch, S. R.
المصدر:
Mathematical Constants. Cambridge, England: Cambridge University Press, 2003.
الجزء والصفحة:
...
29-3-2020
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Somos's Quadratic Recurrence Constant
Somos's quadratic recurrence constant is defined via the sequence
 |
(1)
|
with 
. This has closed-form solution
![g_n=exp[-2^n(partialLi_n(1/2))/(partialn)|_(n=0)+1/2(partialPhi(1/2,s,n+1))/(partials)|_(s=0)],](https://mathworld.wolfram.com/images/equations/SomossQuadraticRecurrenceConstant/NumberedEquation2.gif) |
(2)
|
where 
is a polylogarithm, 
is a Lerch transcendent. The first few terms are 1, 2, 12, 576, 1658880, 16511297126400, ... (OEIS A052129). The terms of this sequence have asymptotic growth as
 |
(3)
|
(OEIS A116603; Finch 2003, p. 446, 
term corrected), where 
is known as Somos's quadratic recurrence constant. Here, the generating function 
in 
satisfies the functional equation
 |
(4)
|
Expressions for 
include
(OEIS A112302; Ramanujan 2000, p. 348; Finch 2003, p. 446; Guillera and Sondow 2005).
Expressions for 
include
(OEIS A114124; Finch 2003, p. 446; Guillera and Sondow 2005; J. Borwein, pers. comm., Feb. 6, 2005), where 
is a polylogarithm.
is also given by the unit square integral
(Guillera and Sondow 2005).
Ramanujan (1911; 2000, p. 323) proposed finding the nested radical expression
 |
(17)
|
which converges to 3. Vijayaraghavan (in Ramanujan 2000, p. 348) gives the justification of his process both in general, and in the particular example of 
.
REFERENCES:
Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, 2003.
Guillera, J. and Sondow, J. "Double Integrals and Infinite Products for Some Classical Constants Via Analytic Continuations of Lerch's Transcendent." 16 June 2005 https://arxiv.org/abs/math.NT/0506319.
Ramanujan, S. Question No. 298. J. Indian Math. Soc. 1911.
Ramanujan, S. Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, P. V. S. Aiyar, and B. M. Wilson). Providence, RI: Amer. Math. Soc., 2000.
Sloane, N. J. A. Sequences A052129, A112302, A114124, and A116603 in "The On-Line Encyclopedia of Integer Sequences."
Somos, M. "Several Constants Related to Quadratic Recurrences." Unpublished note. 1999.
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