On the Sums ∞ K=−∞ (4k + 1)
نویسنده
چکیده
The sum in the title is a rational multiple of π for all integers n = 2, 3, 4, . . . for which the sum converges absolutely. This is equivalent to a celebrated theorem of Euler. Of the many proofs that have appeared since Euler, a simple one was discovered only recently by Calabi [4]: the sum is written as a definite integral over the unit n-cube, then transformed into the volume of a polytope Πn ⊂ R whose vertices’ coordinates are rational multiples of π. We review Calabi’s proof, and give two further interpretations. First we define a simple linear operator T on L(0, π/2), and show that T is self-adjoint and compact, and that Vol(Πn) is the trace of T . We find that the spectrum of T is {1/(4k + 1) : k ∈ Z}, with each eigenvalue 1/(4k + 1) occurring with multiplicity 1; thus Vol(Πn) is the sum of the n-th powers of these eigenvalues. We also interpret Vol(Πn) combinatorially in terms of the number of alternating permutations of n + 1 letters, and if n is even also in terms of the number of cyclically alternating permutations of n letters. We thus relate these numbers with S(n) without the intervention of Bernoulli and Euler numbers or their generating functions.
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تاریخ انتشار 2008