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• W3127975105 abstract "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 $$tau $$ function, sums of squares and triangular numbers, and for $$zeta (2n)$$ , 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 $$p(5n+4)equiv 0; (mathrm{mod},5)$$ and $$tau (5n+5)equiv 0; (mathrm{mod},5)$$ into an infinite family of such congruences." @default.
• W3127975105 created "2021-02-15" @default.
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• W3127975105 date "2022-04-23" @default.
• W3127975105 modified "2023-10-16" @default.
• W3127975105 title "The partition-frequency enumeration matrix" @default.
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• W3127975105 doi "https://doi.org/10.1007/s11139-022-00573-5" @default.
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