Abstract
We study the q-bracket operator of Bloch and Okounkov, recently examined by Zagier and other authors, when applied to functions defined by two classes of sums over the parts of an integer partition. We derive convolution identities for these functions and link both classes of q-brackets through divisor sums. As a result, we generalize Euler’s classic convolution identity for the partition function and obtain an analogous identity for the totient function. As corollaries, we generalize Stanley’s Theorem on the number of ones in all partitions of n, and provide several new combinatorial results.
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Acknowledgements
I would like to thank Christophe Vignat for guiding me through my first forays into mathematics, and Armin Straub for first introducing me to Stanley’s Theorem and the current literature on partitions. I would also like to thank George Andrews, Robert Schneider, and the anonymous referee for helpful comments. Last but not least, I would like to thank Ellicott 3 for always providing me with (questionable) inspiration.
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Wakhare, T. Special classes of q-bracket operators. Ramanujan J 47, 309–316 (2018). https://doi.org/10.1007/s11139-017-9956-8
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DOI: https://doi.org/10.1007/s11139-017-9956-8