Articles written in Proceedings – Mathematical Sciences

    • On 3-way combinatorial identities


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      In this paper, we provide combinatorial meanings to two generalized basic series with the aid of associated lattice paths. These results produce two new classes of infinite 3-way combinatorial identities. Five particular cases are also discussed. These particular cases provide new combinatorial versions of Göllnitz–Gordon identities and Göllnitz identity. Seven $q$-identities of Slater and five $q$-identities of Rogers are further explored using the same combinatorial object. These results are an extension of the work of Goyal and Agarwal (Utilitas Math. 95 (2014) 141–148), Agarwal and Rana (Utilitas Math. 79 (2009) 145–155), and Agarwal (J. Number Theory 28 (1988) 299–305).

    • Combinatorial identities for tenth order mock theta functions


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      In this paper, the open problem posed by Sareen and Rana (Proc. Indian Acad. Sci. (Math. Sci.) 126 (2016) 549–556) is addressed. Here, we interpret two tenth order mock theta functions combinatorially in terms of lattice paths. Then we extend enumeration of one of these with Bender–Knuth matrices; the other by using Frobenius partitions. The combinatorial interpretation of one of these mock theta functions in terms of Frobenius partitions gives an answer to the open problem. Finally, we establish bijections between different classes of combinatorial objects which lead us to one 4-way and one 3-way combinatorial identity.

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