Abstract. We prove two conjectures of Brändén on the real-rootedness of polynomials Q n (x) and R n (x) which are related to the Boros-Moll polynomials P n (x). In fact, we show that both Q n (x) and R n (x) form Sturm sequences. The first conjecture implies the 2-log-concavity of P n (x), and the second conjecture implies the 3-log-concavity of P n (x).
In this note we investigate the function B k, (n), which counts the number of (k, )-regular bipartitions of n. We shall prove an infinite family of congruences modulo 11: for α ≥ 2 and n ≥ 0, B 3,11 3 α n + 5 · 3 α−1 − 1 2 ≡ 0 (mod 11).
Let p(n) denote the number of overpartitions of n. In this paper, we shall show that for n ≥ 0,where r = 8, 52, 68, and 72. In addition, we present a short alternative proof of the congruence p(40n + 35) ≡ 0 (mod 5), which is conjectured by Hirschhorn and Sellers.
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