[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/JKMS.2015.52.2.333

DERIVATIVE FORMULAE FOR MODULAR FORMS AND THEIR PROPERTIES

Aygunes, Aykut Ahmet (Department of Mathematics Faculty of Art and Science University of Akdeniz)

Publication Information

Journal of the Korean Mathematical Society / v.52, no.2, 2015 , pp. 333-347 More about this Journal

Abstract

In this paper, by using the modular forms of weight nk ( $2{\leq}n{\in}\mathbb{N}$ and $k{\in}\mathbb{Z}$ ), we construct a formula which generates modular forms of weight 2nk+4. This formula consist of some known results in [14] and [4]. Moreover, we obtain Fourier expansion of these modular forms. We also give some properties of an operator related to the derivative formula. Finally, by using the function $j_4$ , we obtain the Fourier coefficients of modular forms with weight 4.

Keywords

Eisenstein series; modular forms; cusp forms; Fourier series; operators; derivative formula; theta function; Jacobi theta function;

Citations & Related Records

Reference

1	T. M. Apostol, Modular functions and Dirichlet series in Number Theory, Berlin, Heidelberg and New York, Springer-Verlag, 1976.
2	A. A. Aygunes, A formula for generating modular forms with weight 4, to appear.
3	A. A. Aygunes and Y. Simsek, Hecke Operators Related to the Generalized Dedekind Eta Functions and Applications, Numer. Anal. Appl. Math. Vol. I-III, Book Series: AIP Conference Proceedings Volume: 1281 (2010), 1098-1101.
4	A. A. Aygunes, Y. Simsek, and H.M. Srivastava, A sequence of modular forms associated with higher order derivative Weierstrass-type functions, to appear.
5	E. Hecke, Mathematische Werke, Vandenhoeck & Ruprecht in Gottingen, 1983.
6	A. Hurwitz, Ueber die Differentialgleichungen dritter Ordnung, welchen die Formen mit linearen Transformationen in sich genu gen, Math. Ann. 33 (1889), no. 3, 345-352. DOI
7	L. J. P. Kilford, Modular Forms: a classical and computational introduction, Imperial College Press, 2008.
8	C. H. Kim and J. K. Koo, On the modular function j4 of level 4, J. Korean Math. Soc. 35 (1998), no. 4, 903-931.
9	C. H. Kim and J. K. Koo, Arithmetic of the modular function j4, J. Korean Math. Soc. 36 (1999), no. 4, 707-723.
10	F. Klein, Ueber Multiplicatorgleichungen, Math. Ann. 15 (1879), no. 1, 86-88. DOI
11	N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, New York, 1993.
12	S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 (1916), no. 9, 159-184.
13	A. Sebbar and A. Sebbar, Eisenstein series and modular differential equations, Canad. Math. Bull. 55 (2012), no. 2, 400-409. DOI
14	J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, New York, Heidelberg and Berlin, Springer-Verlag, 1994.
15	Y. Simsek, Relations between theta-functions Hardy sums Eisenstein series and Lambert series in the transformation formula of log $η_g,_h_(g,h)$ (z), J. Number Theory 99 (2003), no. 2, 338-360. DOI ScienceOn
16	B. Van der Pol, On a non-linear partial differential equation satisfied by the logarithm of the Jacobian theta-functions, with arithmetical applications. I, II, Nederl. Akad. Wetensch. Proc. Ser. A. 54, Indagationes Math. 13 (1951), 261-271, 272-284.
17	Y. Simsek, On normalized Eisenstein series and new theta functions, Proc. Jangjeon Math. Soc. 8 (2005), no. 1, 25-34.