DIVISOR FUNCTIONS AND WEIERSTRASS FUNCTIONS ARISING FROM q-SERIES

Title & Authors
DIVISOR FUNCTIONS AND WEIERSTRASS FUNCTIONS ARISING FROM q-SERIES
Kim, Dae-Yeoul; Kim, Min-Soo;

Abstract
We consider Weierstrass functions and divisor functions arising from $\small{q}$-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 $\small{n{\geq}1}$, we have $\small{k{\cdot}{\tau}_{2;k,l}(n)=2n{\cdot}E_{\frac{k-l}{2}}(n;k)+l{\cdot}{\tau}_{1;k,l}(n)+2k{\cdot}{\sum_{j=1}^{n-1}}E_{\frac{k-1}{2}(j;k){\tau}_{1;k,l}(n-j)}$. Finally, we shall give a table for $\small{E_1(N;3)}$, $\small{{\sigma}(N)}$, $\small{{\tau}_{1;3,1}(N)}$ and $\small{{\tau}_{2;3,1}(N)}$ ($\small{1{\leq}N{\leq}50}$) and state simulation results for them.
Keywords
divisor function;q-serie;
Language
English
Cited by
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2.
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3.
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4.
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5.
CONGRUENCES OF THE WEIERSTRASS ${\wp}(x)$ AND ${\wp}^{{\prime}{\prime}}(x)$($x=\frac{1}{2}$, $\frac{\tau}{2}$, $\frac{\tau+1}{2}$)-FUNCTIONS ON DIVISORS, Bulletin of the Korean Mathematical Society, 2013, 50, 1, 241
References
1.
B. Cho, D. Kim, and J. K. Koo, Divisor functions arising from q-series, Publ. Math. Debrecen 76 (2010), no. 3-4, 495-508.

2.
B. Cho, D. Kim, and J. K. Koo, Modular forms arising from divisor functions, J. Math. Anal. Appl. 356 (2009), no. 2, 537-547.

3.
H. M. Farkas, On an arithmetical function, Ramanujan J. 8 (2004), no. 3, 309-315.

4.
H. M. Farkas, On an arithmetical function. II, Complex analysis and dynamical systems II, 121-130, Contemp. Math., 382, Amer. Math. Soc., Providence, RI, 2005.

5.
N. J. Fine, Basic Hypergeometric Series and Applications, American Mathematical Society, Providence, RI, 1988.

6.
D. Kim and J. K. Koo, Algebraic integer as values of elliptic functions, Acta Arith. 100 (2001), no. 2, 105-116.

7.
D. Kim and J. K. Koo, Algebraic numbers, transcendental numbers and elliptic curves derived from infinite products, J. Korean Math. Soc. 40 (2003), no. 6, 977-998.

8.
D. Kim and J. K. Koo, On the infinite products derived from theta series I, J. Korean Math. Soc. 44 (2007), no. 1, 55-107.

9.
S. Lang, Elliptic Functions, Addison-Wesley, 1973.

10.
J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer-Verlag, New York, 1994.