Abstract
The Nekrasov–Okounkov hook length formula provides a fundamental link between the theory of partitions and the coefficients of powers of the Dedekind eta function. In this paper we examine three conjectures presented by Amdeberhan. The first conjecture is a refined Nekrasov–Okounkov formula involving hooks with trivial legs. We give a proof of the conjecture. The second conjecture is on properties of the roots of the underlying D’Arcais polynomials. We give a counterexample and present a new conjecture. The third conjecture is on the unimodality of the coefficients of the involved polynomials. We confirm the conjecture up to the polynomial degree 1000.
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Notes
The reviewer kindly put our attention on the paper of William Keith.
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Acknowledgements
The reviewer of the paper kindly informed us on the paper of W. Keith. We thank the reviewer for this important observation related to Conjecture 1. We also thank W. Keith for very useful comments which are incorporated into the paper. Most of the results had been obtained in July and August 2018 at the RWTH Aachen. The authors benefited from an invitation and excellent working atmosphere. We thank Prof. Dr. Rabe and Prof. Dr. Krieg for useful comments. The paper was finalized at the German University of Technology in Oman.
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Heim, B., Neuhauser, M. On conjectures regarding the Nekrasov–Okounkov hook length formula. Arch. Math. 113, 355–366 (2019). https://doi.org/10.1007/s00013-019-01335-4
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DOI: https://doi.org/10.1007/s00013-019-01335-4