Skew log-concavity of the Boros-Moll sequences

Boros and Moll [1, 2] explored a special class of Jacobi polynomials in their study of a quartic integral. They have shown that for any \(a>-1\) and any nonnegative integer m, where Using Ramanujan’s master theorem, Boros and Moll [2] derived the following formula for \(P_{m}(a)\): which implies that the coefficient of \(a^{i}\) in \(P_{m}(a)\) is positive for \(0\leq i \leq m\). Let \(d_{i}(m)\) be given by The polynomial \(P_{m}(a)\) is called the Boros-Moll polynomial, and the sequence \(\{d_{i}(m)\}_{0 \leq i \leq m}\) of the coefficients is called a Boros-Moll sequence. From (1.3), we know that \(d_{i}(m)\) can be given by

Some combinatorial properties of \(\{d_{i}(m)\}_{0 \leq i \leq m}\) have been proved. Boros and Moll [1] proved that the sequence \(\{d_{i}(m)\}_{0 \leq i \leq m}\) is unimodal, and the maximum element appears in the middle. Recall that a sequence \(\{a_{i}\}_{0 \leq i \leq m}\) of real numbers is said to be unimodal if there exists an index \(0 \leq j \leq m\) such that and \(\{a_{i}\}_{0 \leq i \leq m}\) is said to be log-concave if where \(a_{-1}=a_{m+1}=0\). Moll [2] conjectured that the sequence \(\{d_{i}(m)\}_{0 \leq i \leq m}\) is log-concave. Kauers and Paule [3] proved this conjecture based on recurrence relations found using a computer algebra approach. Recently, Chen and Xia [4] showed that the sequence \(\{d_{i}(m)\}_{0 \leq i \leq m}\) satisfies the strongly ratio monotone property which implies the log-concavity and the spiral property. They [5] also confirmed a conjecture of Moll which says that \(\{i(i+1) (d_{i}^{2}(m) -d_{i-1}(m)d_{i+1}(m) )\}_{1\leq i \leq m}\) attains its minimum at \(i=m\). Chen et al. [6] proved that the Boros-Moll sequences are interlacing log-concave. Chen and Gu [7] showed that the sequence \(\{d_{i}(m)\}_{0 \leq i \leq m}\) satisfies the reverse ultra log-concavity. Chen and Xia [8] proved that the Boros-Moll sequences are 2-log-concave, and Xia [9] studied the concavity and convexity of the Boros-Moll sequences.

By (1.4), which yields Therefore, (1.6) holds when \(m=1\).

Hence, in the following, we always assume that \(m\geq2\) and \(i\geq1\). We first recall the following three recurrence relations derived by Kauers and Paule [3]: and

Now we represent the difference \(d_{i}^{2}(m+i)-d_{i-1}(m+i-1)d_{i+1}(m+i+1)\) in terms of \(d_{i}(m+i)\) and \(d_{i}(m+i+1)\). Thanks to (2.1), (2.2) and (2.3), where with It is easy to check that where Note that A is positive. Hence, in order to prove that the right-hand side of (2.4) is positive, it suffices to prove that when Δ is nonnegative, Therefore, in the following, we assume that \(\Delta\geq0\).

Recall that Kauers and Paule [3] proved the following inequality: Replacing m by \(m+i\), we see that It is a routine to verify that where Note that when \(m\geq2\) and \(i \geq1\), G is positive. Thus the right-hand side of (2.12) is positive. On the other hand, which is positive. Therefore, from (2.12), we have which can be rewritten as From (2.11) and (2.13), we obtain (2.10) and this completes the proof.