Divisor functions and Weierstrass functions arising from $q$-series
Bull. Korean Math. Soc. 2012 Vol. 49, No. 4, 693-704
Printed July 1, 2012
Daeyeoul Kim and Min-Soo Kim
National Institute for Mathematical Sciences, Kyungnam University
Abstract : We consider Weierstrass functions and divisor functions arising from $q$-series. Using these we can obtain new identities for divisor functions. Farkas [3] provided a relation between the sums of divisors satisfying congruence conditions and the sums of numbers of divisors satisfying congruence conditions. In the proof he took logarithmic derivative to theta functions and used the heat equation. In this note, however, we obtain a similar result by differentiating further. For any $n \geq 1$, we have $$ k \cdot \tau_{2; k, l} (n) = 2 n \cdot E_{\frac{k-l}{2}}(n;k) + l \cdot \tau_{1; k, l} (n) + 2 k \cdot \sum_{j=1}^{n-1} E_{\frac{k-l}{2}}(j;k) \tau_{1; k, l}(n-j).$$ Finally, we shall give a table for $E_1 (N;3),\sigma(N),\tau_{1;3,1}(N)$ and $ \tau_{2;3,1}(N)$ $(1\leq N\leq 50)$ and state simulation results for them.
Keywords : divisor function, $q$-series
MSC numbers : 11P83, 05A17
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