Browse > Article
http://dx.doi.org/10.4134/BKMS.2013.50.1.241

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  

Kim, Daeyeoul (National Institute for Mathematical Sciences)
Kim, Aeran (Department of Mathematics and Institute of Pure and Applied Mathematics Chonbuk National University)
Park, Hwasin (Department of Mathematics and Institute of Pure and Applied Mathematics Chonbuk National University)
Publication Information
Bulletin of the Korean Mathematical Society / v.50, no.1, 2013 , pp. 241-261 More about this Journal
Abstract
In this paper, we find the coefficients for the Weierstrass ${\wp}(x)$ and ${\wp}^{{\prime}{\prime}}(x)$($x=\frac{1}{2}$, $\frac{\tau}{2}$, $\frac{\tau+1}{2}$)-functions in terms of the arithmetic identities appearing in divisor functions which are proved by Ramanujan ([23]). Finally, we reprove congruences for the functions ${\mu}(n)$ and ${\nu}(n)$ in Hahn's article [11, Theorems 6.1 and 6.2].
Keywords
Weierstrass ${\wp}(x)$ functions; convolution sums;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 B. C. Berndt, Ramanujan's Notebooks. Part II, Springer-Verlag, New York, 1989.
2 M. Besgue, Extrait d'une lettre de M. Besgue a M Liouville, J. Math. Pures Appl. 7 (1862), 256.
3 B. Cho, D. Kim, and J. K. Koo, Divisor functions arising from q-series, Publ. Math. Debrecen 76 (2010), no. 3-4, 495-508.
4 B. Cho, D. Kim, and J. K. Koo, Modular forms arising from divisor functions, J. Math. Anal. Appl. 356 (2009), no. 2, 537-547.   DOI   ScienceOn
5 L. E. Dickson, History of the Theory of Numbers. Vol. I: Divisibility and Primality, Chelsea Publishing Co., New York New York, 1966.
6 L. E. Dickson, History of the Theory of Numbers. Vol. II: Diophantine analysis, Chelsea Publishing Co., New York New York, 1966.
7 N. J. Fine, Basic hypergeometric series and applications, American Mathematical Society, Providence, RI, 1988.
8 J. W. L. Glaisher, Expressions for the five powers of the series in which the coefficients are the sums of the divisors of the exponents, Mess. Math. 15 (1885), 33-36.
9 J. W. L. Glaisher, On certain sums of products of quantities depending upon the divisors of a number, Mess. Math. 15 (1885), 1-20.
10 J. W. L. Glaisher, On the square of the series in which the coefficients are the sums of the divisors of the exponents, Mess. Math. 14 (1884).
11 H. Hahn, Convolution sums of some functions on divisors, Rocky Mountain J. Math. 37 (2007), no. 5, 1593-1622.   DOI   ScienceOn
12 J. G. Huard, Z. M. Ou, B. K. Spearman, and K. S. Willians, Elementary evaluation of certain convolution sums involving divisor functions, Number theory for the millennium, II (Urbana, IL, 2000), 229274, A K Peters, Natick, MA, 2002.
13 D. Kim and M. Kim, Divisor functions and Weierstrass functions arising from q series, Bull. Korean Math. Soc. 49 (2012), no. 4, 693-704.   과학기술학회마을   DOI   ScienceOn
14 D. Kim, H. Park, and A. Kim, Arithmetic identities arising from the basic hypergeometric series and elliptic curve, Submitted.
15 S. Lang, Elliptic Functions, Addison-Wesly, 1973.
16 D. H. Lehmer, Selected papers, Vol. II, Charles Babbage Research Centre, St. Pierre, Manitoba, 1981.
17 D. H. Lehmer, Some functions of Ramanujan, Math. Student 27 (1959), 105-116.
18 J. Liouville, Sur quelques formules generales qui peuvent etre utiles dans la theorie des nombres, Jour. de Math. (2) (1864), 389-400.
19 J. Lutzen, Joseph Liouville 1809-1882, Master of Pure and Applied Mathematics, Springer-Verlag, Berlin/Heidelberg, 1998.
20 P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc. (2) 19 (1920), no. 1, 75-113.
21 R. A. Rankin, Elementary proofs of relations between Eisenstein series, Proc. Roy. Soc. Edinburgh Sect. A 76 (1976/77), no. 2, 107-117.   DOI
22 G. Melfi, On some modular identities, Number theory (Eger, 1996), 371-382, de Gruyter, Berlin, 1998.
23 S. Ramanujan, Collected papers Srinivasa Ramanujan, AMS Chelsea Publishing, Providence, RI, 2000.
24 S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 (1916), no. 9, 159-184.
25 J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer-Verlag, 1994.
26 J. H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, 1986.
27 N. P. Skoruppa, A quick combinatorial proof of Eisenstein series identities, J. Number Theory 43 (1993), no. 1, 68-73.   DOI   ScienceOn