Somos' quadratic recurrence constant

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In mathematics, Somos' quadratic recurrence constant, named after Michael Somos, is the number

This can be easily re-written into the far more quickly converging product representation

The constant σ arises when studying the asymptotic behaviour of the sequence

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]

Guillera and Sondow give a representation in terms of the derivative of the Lerch transcendent:

where ln is the natural logarithm and (zsq) is the Lerch transcendent.

Using series acceleration it is the sum of the n-th differences of ln(k) at k=1 as given by:

Finally,

(sequence A112302 in the OEIS).

Notes[edit]

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