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New proofs of two $ q$-analogues of Koshy's formula


Authors: Emma Yu Jin and Markus E. Nebel
Journal: Proc. Amer. Math. Soc. 143 (2015), 5027-5042
MSC (2010): Primary 05A19
Published electronically: April 21, 2015
MathSciNet review: 3411124
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Abstract: In this paper we prove a $ q$-analogue of Koshy's formula in terms of the Narayana polynomial due to Lassalle, and a $ q$-analogue of Koshy's formula in terms of $ q$-hypergeometric series due to Andrews, by applying the inclusion-exclusion principle on Dyck paths and on partitions. We generalize these two $ q$-analogues of Koshy's formula for $ q$-Catalan numbers to that for $ q$-Ballot numbers. This work also answers an open question by Lassalle and two questions raised by Andrews in 2010. We conjecture that if $ n$ is odd, then for $ m\ge n\ge 1$, the polynomial $ (1+q^n){m\brack n-1}_q$ is unimodal. If $ n$ is even, for any even $ j\ne 0$ and $ m\ge n\ge 1$, the polynomial $ (1+q^n)[j]_q{m\brack n-1}_q$ is unimodal. This implies the answer to the second problem posed by Andrews.


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Additional Information

Emma Yu Jin
Affiliation: Department of Computer Science, University of Kaiserslautern, Kaiserslautern, Germany
Email: jin@cs.uni-kl.de

Markus E. Nebel
Affiliation: Department of Computer Science, University of Kaiserslautern, Kaiserslautern, Germany – and – Department of Mathematics and Computer Science, University of Southern Denmark, Denmark
Email: nebel@cs.uni-kl.de

DOI: https://doi.org/10.1090/proc/12627
Received by editor(s): September 4, 2013
Received by editor(s) in revised form: March 19, 2014, and September 10, 2014
Published electronically: April 21, 2015
Additional Notes: The work of the first author was supported by research grants from DFG (Deutsche Forschungsgemeinschaft), JI 207/1-1.
Communicated by: Jim Haglund
Article copyright: © Copyright 2015 American Mathematical Society