Math @ Duke

Publications [#302442] of Heekyoung Hahn
Papers Published
 Hahn, H, Convolution sums of some functions on divisors,
Rocky Mountain Journal of Mathematics, vol. 37 no. 5
(December, 2007),
pp. 15931622, Rocky Mountain Mathematics Consortium, ISSN 00357596 [doi]
(last updated on 2021/06/16)
Abstract: One of the main goals in this paper is to establish convolution sums of functions for the divisor sums σ̃ (n) = Σ (1) d and σ̂ = Σ (l) d , for certain s, which were first defined by Glaisher. We first introduce three functions P(q), E(q), and Q(q) related to σ̃(n), σ̂(n), and σ̃ (n), respectively, and then we evaluate them in terms of two parameters x and z in Ramanujan's theory of elliptic functions. Using these formulas, we derive some identities from which we can deduce convolution sum identities. We discuss some formulae for determining r (n) and r (n), s = 4, 8, in terms of σ̃(n), σ̂(n), and σ̃ (n), where r (n) denotes the number of representations of n as a sum of s squares and δ (n) denotes the number of representations of n as a sum of s triangular numbers. Finally, we find some partition congruences by using the notion of colored partitions. Copyright ©2007 Rocky Mountain Mathematics Consortium. s d/n s d/n 3 s s 3 s s d1 s (n/d)1 s


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