A MODULAR TRANSFORMATION FOR A GENERALIZED THETA FUNCTION WITH MULTIPLE PARAMETERS S. BHARGAVA, M. S. MAHADEVA NAIKA AND M. C. MAHESHKUMAR
Abstract. We obtain a modular transformation for the theta function ∞ ∞ X X
q a(m
2
+mn)+cn2 +λm+µn+ν
ζ Am+Bn z Cm+Dn .
−∞ −∞
We are thus able to unify and extend several modular transformations in literature.
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S. BHARGAVA, M. S. MAHADEVA NAIKA AND M. C. MAHESHKUMAR
[12] M. D. Hirschhorn, F. G. Garvan and J. M. Borwein, Cubic Analogues of the Jacobian Theta Function θ(z, q), Canadian J. 45(1993), 673−694. ¨ ber die analytische Theorie der quadratischen Formen, Ann. Math., 36 (1935), [13] C. L. Siegel, U 527. [14] C. L. Siegel, Indefinite quadratische Formen und Funktionentheorie, I, Math. Ann., 124 (1951), 17; II, 364.
(S. BHARGAVA) Department of studies in Mathematics, University of Mysore, Manasagangotri, Mysore-570 006, INDIA. E-mail address:
[email protected] (M. S. MAHADEVA NAIKA AND M. C. MAHESHKUMAR) Department of Mathematics, Bangalore University, Central College Campus, Bangalore-560 001, INDIA E-mail address:
[email protected],
[email protected]
