with first few terms 1, 1, 2, 12, 576, 1658880, ... (sequence A052129 in the OEIS). This sequence can be shown to have asymptotic behaviour as follows:[1]
Jesus Guillera and Jonathan Sondow, "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent", Ramanujan Journal 16 (2008), 247–270 (Provides an integral and a series representation). arXiv:math/0506319
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somos, quadratic, recurrence, constant, mathematics, named, after, michael, somos, number, displaystyle, sigma, sqrt, sqrt, sqrt, cdots, cdots, this, easily, written, into, more, quickly, converging, product, representation, displaystyle, sigma, sigma, sigma, . In mathematics Somos quadratic recurrence constant named after Michael Somos is the number s 1 2 3 1 1 2 2 1 4 3 1 8 displaystyle sigma sqrt 1 sqrt 2 sqrt 3 cdots 1 1 2 2 1 4 3 1 8 cdots This can be easily re written into the far more quickly converging product representation s s 2 s 2 1 1 2 3 2 1 4 4 3 1 8 5 4 1 16 displaystyle sigma sigma 2 sigma left frac 2 1 right 1 2 left frac 3 2 right 1 4 left frac 4 3 right 1 8 left frac 5 4 right 1 16 cdots which can then be compactly represented in infinite product form by s k 1 1 1 k 1 2 k displaystyle sigma prod k 1 infty left 1 frac 1 k right frac 1 2 k The constant s arises when studying the asymptotic behaviour of the sequence g 0 1 g n n g n 1 2 n gt 1 displaystyle g 0 1 g n ng n 1 2 qquad n gt 1 with first few terms 1 1 2 12 576 1658880 sequence A052129 in the OEIS This sequence can be shown to have asymptotic behaviour as follows 1 g n s 2 n n 2 O 1 n displaystyle g n sim frac sigma 2 n n 2 O frac 1 n Guillera and Sondow give a representation in terms of the derivative of the Lerch transcendent ln s 1 2 F s 1 2 0 1 displaystyle ln sigma frac 1 2 frac partial Phi partial s left frac 1 2 0 1 right where ln is the natural logarithm and F displaystyle Phi z s q is the Lerch transcendent Finally s 1 661687949633594121296 displaystyle sigma 1 661687949633594121296 dots sequence A112302 in the OEIS Notes Edit Weisstein Eric W Somos s Quadratic Recurrence Constant MathWorld References EditSteven R Finch Mathematical Constants 2003 Cambridge University Press p 446 ISBN 0 521 81805 2 Jesus Guillera and Jonathan Sondow Double integrals and infinite products for some classical constants via analytic continuations of Lerch s transcendent Ramanujan Journal 16 2008 247 270 Provides an integral and a series representation arXiv math 0506319 This mathematics related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Somos 27 quadratic recurrence constant amp oldid 1135928227, wikipedia, wiki, book, books, library,