2018/12/04 by Atul Dixit, Pramod Eyyunni, Dixit, Atul +5
Mathematics · #11P84 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary 11P81 #Secondary 05A17
paper · pdf · doi:10.48550/arxiv.1812.01424
openalex publication_date 2018/12/04 · openalex created_date 2022/08/01 · openalex updated_date 2026/07/28
We obtain a finite analogue of a recent generalization of an identity in\nRamanujan's Notebooks. Differentiating it with respect to one of the parameters\nleads to a result whose limiting case gives a finite analogue of Andrews'\nfamous identity for textupspt(n). The latter motivates us to extend the\ntheory of the restricted partition function p(n, N), namely, the number of\npartitions of n with largest parts less than or equal to N, by obtaining\nthe finite analogues of rank and crank for vector partitions as well as of the\nrank and crank moments. As an application of the identity for our finite\nanalogue of the spt-function, namely textupspt(n, N), we prove an\ninequality between the finite second rank and crank moments. The other results\nobtained include finite analogues of a recent identity of Garvan, an identity\nrelating d(n, N) and lpt(n, N), namely the finite analogues of the divisor\nand largest parts functions respectively, and a finite analogue of the\nBeck-Chern theorem. We also conjecture an inequality between the finite\nanalogues of k^ textupth rank and crank moments.\n