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The partition-frequency enumeration matrix

, Hartosh Singh Bal
Published in Springer

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 𝜏τ function, sums of squares and triangular numbers, and for 𝜁(2𝑛)ζ(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)0 (mod 5) and 𝜏(5𝑛+5)≡0(mod5)τ(5n+5)0 (mod 5) into an infinite family of such congruences.

About the journal
JournalData powered by TypesetThe Ramanujan Journal
PublisherData powered by TypesetSpringer
Open AccessNo