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Overpartition analogues of q-bi\(^{s}\)nomial coefficients: basic properties and log-concavity

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Abstract

In this paper, we define overpartition analogues of q-bi\(^{s}\)nomial coefficients as generating functions for the number of overpartitions fitting inside the \((N-1)\times M\) rectangle in which no part appears more than s times, which we call over q-bi\(^{s}\)nomial (resp. over (q, t)-bi\(^{s}\)nomial) coefficients. We study basic properties and we prove the (q, t)-log-concavity (resp. the q-log-concavity) of over (q, t)-bi\(^{s}\)nomial (resp. over q-bi\(^{s}\)nomial) coefficients. We also extend the Dousse and Kim’s results on the (q, t)-log-concavity of over (q, t)-binomial coefficients.

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Acknowledgements

The authors would like to thank the referees for many valuable remarks and suggestions to improve the original manuscript.

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  1. LMAM laboratory, University of Mohamed Seddik Ben Yahia, Jijel, BP 98 Ouled Aissa, 18000, Jijel, Algeria

    Yousra Ghemit & Moussa Ahmia

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  1. Yousra Ghemit
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  2. Moussa Ahmia
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Correspondence to Moussa Ahmia.

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This work was supported by DG-RSDT (Algeria), PRFU Project, No. C00L03UN180120220002.

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Ghemit, Y., Ahmia, M. Overpartition analogues of q-bi\(^{s}\)nomial coefficients: basic properties and log-concavity. Ramanujan J 62, 431–455 (2023). https://doi.org/10.1007/s11139-023-00706-4

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  • DOI: https://doi.org/10.1007/s11139-023-00706-4

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