Somos's Quadratic Recurrence Constant

Somos's quadratic recurrence constant is defined via the sequence


with g_0=1. This has closed-form solution


where Li_n(z) is a polylogarithm, Phi(z,s,a) 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


(OEIS A116603; Finch 2003, p. 446, n^(-4) term corrected), where sigma is known as Somos's quadratic recurrence constant. Here, the generating function A(x) in x=1/n satisfies the functional equation


Expressions for sigma include

=product_(n=1)^(infty)product_(k=0)^(n)(k+1)^((-1)^(k+n)(n; k))

(OEIS A112302; Ramanujan 2000, p. 348; Finch 2003, p. 446; Guillera and Sondow 2005).

Expressions for lnsigma include

=sum_(n=1)^(infty)sum_(k=0)^(n)(-1)^(n+k)(n; k)ln(k+1)

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

lnsigma is also given by the unit square integral


(Guillera and Sondow 2005).

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


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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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, 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 Unpublished note. 1999.

Referenced on Wolfram|Alpha

Somos's Quadratic Recurrence Constant

Cite this as:

Weisstein, Eric W. "Somos's Quadratic Recurrence Constant." From MathWorld--A Wolfram Web Resource.

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