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We find a combinatorial setting for the coefficients of the Boros–Moll polynomials Pm(a) in terms of partially 2-colored permutations. Using this model, we give a combinatorial proof of a recurrence relation on the coefficients of Pm(a). This approach enables us to give a combinatorial interpretation of the log-concavity of Pm(a) which was conjectured by Moll and confirmed by Kauers and Paule.
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 extension...
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