Schröter's Formula

Let a general Theta Function be defined as

$$T(x,q)\equiv \sum_{n=-\infty}^\infty x^nq^{n^2},$$

then

$$T(x,q^a)T(x,q^b) = \sum_{k=0}^{a+b-1} y^kq^{bk^2}T(xyq^{2bk},q^{a+b}) T(y^qx^{-b}q^{2abk},q^{ab(1+b)}).$$

See also Blecksmith-Brillhart-Gerst Theorem, Jacobi Triple Product, Ramanujan Theta Functions

References

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 111, 1987.

Tannery, J. and Molk, J. Elements de la Théorie des Fonctions Elliptiques, 4 vols. Paris: Gauthier-Villars et fils, 1893-1902.