On combinatorial extensions of some Ramanujan’s mock theta functions

M. Goyal

M. Goyal IK Gujral Punjab Techn. Univ., Jalandhar, India

Keywords: Ramanujan’s mock theta functions, mock theta functions, $(n t)$--color partitions, weighted lattice paths, associated lattice paths, anti-hook differences

Abstract

Five mock theta functions of S. Ramanujan are combinatorially interpreted by means of certain associated lattice path functions and antihook differences. These results provide new combinatorial interpretations of five mock theta functions of Ramanujan.
Using a bijection between the associated lattice path functions and the $(n+t)$-color partitions and then between the associated lattice path functions and the weighted lattice path functions, we extend the works by Agarwal and Agarwal and Rana to five new 3-way combinatorial identities. These results are further extended to 4-way combinatorial identities by using bijection between the $(n+t)$-color partitions and the partitions with certain antihook differences. These interesting results present elegant combinatorial links between Ramanujan's mock theta functions, $(n+t)$-color partitions, weighted lattice paths, associated lattice paths, and antihook differences.

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