×

zbMATH — the first resource for mathematics

On the Schröter formula for theta functions. (English) Zbl 1185.33026
Suppose that \(\tau\) is a complex number with positive imaginary part, and let \(q=e^{\pi i\tau}\). The Schröter formula in terms of the Jacobi theta function \(\theta_3(z|\tau)=1+2\sum_{n=1}^{\infty}q^{n^2}\cos(2nz)\) asserts that \[ \begin{split} \theta_3(rax+by|r\tau)\theta_3(-sbx+ay|s\tau)\\ =\sum_{k=0}^{m-1}q^{rk^2}e^{2ki(rax+by)}\theta_3(mx+rak\pi\tau|m\tau)\theta_3(my+rsbk\pi\tau|rsm\tau), \end{split} \] where \(a,b,r\) and \(s\) are positive integers, \(m=ra^2+sb^2\), and \(\gcd(b,m)=1\). In this paper, the authors study various identities of theta functions using Schröter formula. More precisely, they prove some identities concerning products of two theta functions and their sums, which two of them reprove two well-known Ramanujan identities related to the modular equation of degree 5. Also, the Hirschhorn septuple product identity is obtained as a special case of Schröter formula.

MSC:
33E05 Elliptic functions and integrals
11F27 Theta series; Weil representation; theta correspondences
11P84 Partition identities; identities of Rogers-Ramanujan type
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1093/qmath/3.1.29 · Zbl 0046.04203
[2] DOI: 10.1093/qmath/3.1.158 · Zbl 0046.27202
[3] DOI: 10.1007/978-1-4612-0965-2
[4] Chapman R., Sém. Lothar. Combin. 42 pp 4–
[5] DOI: 10.1006/jnth.1995.1083 · Zbl 0844.11064
[6] DOI: 10.1142/S1793042105000017 · Zbl 1081.33034
[7] Chu W., Electron. J. Combin. 14 pp 10–
[8] Enneper A., Elliptische Functionen: Theorie und Geschichte (1890)
[9] DOI: 10.1216/rmjm/1181072146 · Zbl 0853.11029
[10] DOI: 10.1090/S0002-9939-99-04791-7 · Zbl 0932.11029
[11] Hirschhorn M. D., J. Reine Angew. Math. 326 pp 1–
[12] DOI: 10.1017/S1446788700019728
[13] DOI: 10.1007/BF03322989 · Zbl 1053.11015
[14] Liu Z.-G., Integers 1 pp 14–
[15] DOI: 10.1016/j.aim.2004.07.006 · Zbl 1138.11014
[16] DOI: 10.1016/j.aim.2006.10.005 · Zbl 1162.11025
[17] Liu Z.-G., J. Ramanujan Math. Soc. 22 pp 283–
[18] DOI: 10.2140/pjm.2009.240.135 · Zbl 1169.33005
[19] Ramanujan S., Notebooks 1 (1957)
[20] Ramanujan S., Notebooks 2 (1957)
[21] Shen L.-C., Trans. Amer. Math. Soc. 345 pp 323–
[22] Shen L.-C., Proc. Amer. Math. Soc. 123 pp 1521–
[23] Watson G. N., J. London Math. Soc. 4 pp 39–
[24] Whittaker E. T., A Course of Modern Analysis (1990)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.