New proofs of two $q$-analogues of Koshy’s formula
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- by Emma Yu Jin and Markus E. Nebel PDF
- Proc. Amer. Math. Soc. 143 (2015), 5027-5042 Request permission
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.References
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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
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 5027-5042
- MSC (2010): Primary 05A19
- DOI: https://doi.org/10.1090/proc/12627
- MathSciNet review: 3411124