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학술연구/단체지원/교육 등 연구자 활동을 지속하도록 DBpia가 지원하고 있어요.
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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 $$k{\cdot}{\tau}_{2;k,l}(n)=2n{\cdot}E_{\frac{k-l}{2}}(n;k)+l{\cdot}{\tau}_{1.k,l}(n)+2k{\cdot}{\sum_{j=1}^{n-1}}E_{\frac{k-1}{2}(j;k){\tau}_{1.k,l}(n-j)$$. Finally, we shall give a table for $E_1(N;3)$, ${\sigma}(N)$, ${\tau}_{1.3,1}(N)$ and ${\tau}_{2;3,1}(N)$ ($1{\leq}N{\leq}50$) and state simulation results for them.
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참고문헌 (10)
참고문헌 신청
B. Cho / 2009 / Modular forms arising from divisor functions / J. Math. Anal. Appl 356 (2) : 537 ~ 547
B. Cho / 2010 / Divisor functions arising from q-series / Publ. Math. Debrecen 76 (3) : 495 ~ 508
D. Kim / 2001 / Algebraic integer as values of elliptic functions / Acta Arith 100 (2) : 105 ~ 116
H. M. Farkas / 2004 / On an arithmetical function / Ramanujan J 8 (3) : 309 ~ 315
H. M. Farkas / 2005 / On an arithmetical function. II, Complex analysis and dynamical systems II / Contemp. Math. 382 : 121 ~ 130
B. Cho / 2010 / Divisor functions arising from q-series / Publ. Math. Debrecen 76 (3) : 495 ~ 508
D. Kim / 2001 / Algebraic integer as values of elliptic functions / Acta Arith 100 (2) : 105 ~ 116
H. M. Farkas / 2004 / On an arithmetical function / Ramanujan J 8 (3) : 309 ~ 315
H. M. Farkas / 2005 / On an arithmetical function. II, Complex analysis and dynamical systems II / Contemp. Math. 382 : 121 ~ 130
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