A new family of almost identities (Q951271)

The authors present a new family of almost identities. These are based on series that sum to elements close to either rationals or rational multiples of \(\pi\). They study the sequence of real numbers \(u_ n\), defined by \[ u_ n=\log 2\sum_ {k=-\infty}^ {+\infty}\frac{1}{(2^ {k/2}+ 2^ {-k/2})^ n} \] for \(n\in {\mathbb N}^ +\). Here they treat the cases \(n=1,2\). The explanation of the phenomenon takes its roots in the theory of Mellin transforms. More details, proof of the recurrence relation for \(u_n\), and explanation for the phenomena for \(u_3\) and \(u_4\), are given in an extended version of this paper [see \url{arXiv:math/0409014}]. For related work, see also \textit{J. M. Borwein} and \textit{P. B. Borwein}, Am. Math. Mon. 99, No. 7, 622--640 (1992; Zbl 0762.40003).