N ov 2 00 1 SOME NEW FORMULAS FOR π

m at h . N T / 0 11 02 38 v 3 1 0 O ct 2 00 2 SOME NEW FORMULAS FOR π

We show how to find series expansions for π of the form π = ∞ n=0 S(n) mn pn a n , where S(n) is some polynomial in n (depending on m, p, a). We prove that there exist such expansions for m = 8k, p = 4k, a = (−4) k , for any k, and give explicit examples for such expansions for small values of m, p and a.

O ct 2 00 1 SOME NEW FORMULAS FOR π

Matematiska Institutionen, Lunds Universitet, SE-221 00 Lund, Sweden. E-mail: [email protected], [email protected] Institut für Mathematik der Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria. E-mail: [email protected] WWW: http://www.mat.univie.ac.at/~kratt Abstract. We show how to find arbitrarily fast convergent series expansions for π of the form π = ∑∞ n=0 S(n)...

0 v 1 3 0 N ov 2 00 5 New result of the measurement of the direct photon emission in K + → π + π 0 γ decay

N ov 2 00 6 Measurements of K e 4 and K ± → π 0 π 0 π ± decays

1 9 N ov 2 00 4 Power Sums Related to Semigroups S ( d 1 , d 2 , d 3 )

The explicit formulas for the sums of positive powers of the integers si unrepresentable by the triple of integers d1, d2, d3 ∈ N, gcd(d1, d2, d3) = 1, are derived.