The ratio monotonicity of the Boros-Moll polynomials
نویسندگان
چکیده
منابع مشابه
The ratio monotonicity of the Boros-Moll polynomials
In their study of a quartic integral, Boros and Moll discovered a special class of Jacobi polynomials, which we call the Boros-Moll polynomials. Kauers and Paule proved the conjecture of Moll that these polynomials are logconcave. In this paper, we show that the Boros-Moll polynomials possess the ratio monotone property which implies the log-concavity and the spiral property. We conclude with a...
متن کاملBrändén’s Conjectures on the Boros-Moll Polynomials
We prove two conjectures of Brändén on the real-rootedness of the polynomials Qn(x) and Rn(x) which are related to the Boros-Moll polynomials Pn(x). In fact, we show that both Qn(x) and Rn(x) form Sturm sequences. The first conjecture implies the 2-log-concavity of Pn(x), and the second conjecture implies the 3-log-concavity of Pn(x). AMS Classification 2010: Primary 26C10; Secondary 05A20, 30C15.
متن کاملThe Reverse Ultra Log-Concavity of the Boros-Moll Polynomials
Based on the recurrence relations on the coefficients of the Boros-Moll polynomials Pm(a) = ∑ i di(m)a i derived independently by Kauers and Paule, and Moll, we are led to the discovery of the reverse ultra log-concavity of the sequence {di(m)}. We also show that the sequence {i!di(m)} is log-concave for m ≥ 1. Two conjectures are proposed.
متن کاملOn the Combinatorics of the Boros-Moll Polynomials
The Boros-Moll polynomials arise in the evaluation of a quartic integral. The original double summation formula does not imply the fact that the coefficients of these polynomials are positive. Boros and Moll proved the positivity by using Ramanujan’s Master Theorem to reduce the double sum to a single sum. Based on the structure of reluctant functions introduced by Mullin and Rota along with an...
متن کاملThe Concavity and Convexity of the Boros-Moll Sequences
In their study of a quartic integral, Boros and Moll discovered a special class of sequences, which is called the Boros–Moll sequences. In this paper, we consider the concavity and convexity of the Boros–Moll sequences {di(m)}i=0. We show that for any integer m > 6, there exist two positive integers t0(m) and t1(m) such that di(m)+di+2(m) > 2di+1(m) for i ∈ [0, t0(m)] ⋃ [t1(m),m−2] and di(m)+di...
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 2009
ISSN: 0025-5718
DOI: 10.1090/s0025-5718-09-02223-6