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Somos's Quadratic Recurrence Constant

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Somos's quadratic recurrence constant is sigma=lim_(n->infty)g_n^(1/2^n), where the sequence g_n is defined by

g_n=ng_(n-1)^2
(1)

for integers n>=1, with g_0=1. This has closed-form solution

g_n=exp[-2^n(partialLi_s(1/2))/(partials)|_(s=0)+1/2(partialPhi(1/2,s,n+1))/(partials)|_(s=0)],
(2)

where Li_s(z) is a polylogarithm and Phi(z,s,a) is a Lerch transcendent. For n=0, 1, ..., the first few terms are 1, 1, 2, 12, 576, 1658880, 16511297126400, ... (OEIS A052129). As n->infty, the terms of this sequence have asymptotic growth given by

g_n=sigma^(2^n)(n+2-n^(-1)+4n^(-2)-21n^(-3)+138n^(-4)-1091n^(-5)+O(n^(-6)))^(-1)
(3)

(OEIS A116603; Finch 2003, p. 446, n^(-4) term corrected). The formal generating function A(x) of these asymptotic coefficients satisfies the functional equation

(1+x)^2=(A^2(x))/(A(x/(1+x))).
(4)

Expressions for sigma include

sigma=sqrt(1sqrt(2sqrt(3sqrt(4...))))
(5)
=product_(k=1)^(infty)k^(1/2^k)
(6)
=product_(k=1)^(infty)((k+1)/k)^(1/2^k)
(7)
=product_(n=1)^(infty)product_(k=0)^(n)(k+1)^((-1)^(k+n)(n; k))
(8)
=1.661687949...
(9)

(OEIS A112302; Ramanujan 2000, p. 348; Finch 2003, p. 446; Guillera and Sondow 2008).

Expressions for lnsigma include

lnsigma=sum_(k=1)^(infty)(1/2)^klnk
(10)
=sum_(k=1)^(infty)((-1)^(k-1)Li_k(1/2))/k
(11)
=-(partialLi_s(z))/(partials)|_(s=0,z=1/2)
(12)
=sum_(n=1)^(infty)sum_(k=0)^(n)(-1)^(n+k)(n; k)ln(k+1)
(13)
=0.5078339...
(14)

(OEIS A114124; Finch 2003, p. 446; J. Borwein, pers. comm., Feb. 6, 2005; Guillera and Sondow 2008), where Li_s(z) is a polylogarithm.

lnsigma is also given by the unit square integral

lnsigma=int_0^1int_0^1(-x)/((2-xy)ln(xy))dxdy
(15)
=int_0^1(1-x)/((x-2)lnx)dx
(16)

(Guillera and Sondow 2008).

Ramanujan (1911; 2000, p. 323) proposed finding the nested radical expression

sqrt(1+2sqrt(1+3sqrt(1+4sqrt(1+5sqrt(...)))))
(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 lnsigma.

See also

Glaisher-Kinkelin Constant, Nested Radical, Nested Radical Constant, Unit Square Integral

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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." Ramanujan J. 16, 247-270, 2008. https://doi.org/10.1007/s11139-007-9102-0.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.Ramanujan, S. Question No. 298. J. Indian Math. Soc. 1911.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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Somos's Quadratic Recurrence Constant

Cite this as:

Weisstein, Eric W. "Somos's Quadratic Recurrence Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SomossQuadraticRecurrenceConstant.html

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