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Articles

The partition-frequency enumeration matrix

Published in Springer

2022

We develop a calculus that gives an elementary approach to enumerate partition-like objects using an infinite number-theoretic matrix. We call this matrix the partition-frequency enumeration (PFE) matrix. This matrix unifies a large number of results connecting number-theoretic functions to partition-theoretic functions. The calculus is extended to arbitrary generating functions, and functions with Weierstrass products. As a by-product, we recover (and extend) some well-known recurrence relations for many number-theoretic functions, including the sum of divisors function, Ramanujan’s 𝜏$\tau $ function, sums of squares and triangular numbers, and for 𝜁(2𝑛)$\zeta (2n)$, where *n* is a positive integer. These include classical results due to Euler, Ewell, Ramanujan, Lehmer, and others. As one application, we embed Ramanujan’s famous congruences 𝑝(5𝑛+4)≡0(mod5)$p(5n+4)\equiv 0\phantom{\rule{thickmathspace}{0ex}}(\mathrm{m}\mathrm{o}\mathrm{d}\phantom{\rule{thinmathspace}{0ex}}5)$ and 𝜏(5𝑛+5)≡0(mod5)$\tau (5n+5)\equiv 0\phantom{\rule{thickmathspace}{0ex}}(\mathrm{m}\mathrm{o}\mathrm{d}\phantom{\rule{thinmathspace}{0ex}}5)$ into an infinite family of such congruences.

Postprint Version

Content may be subject to copyright.This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Spr... ...This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s11139-022-00573-5

About the journal

Journal | Data powered by TypesetThe Ramanujan Journal |
---|---|

Publisher | Data powered by TypesetSpringer |

Open Access | No |