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Schröter's Formula

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Schröter's formula is a product identity for theta functions. Let a general theta function be defined as

T(x,q)=sum_(n=-infty)^inftyx^nq^(n^2),

then

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

See also

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

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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, France: Gauthier-Villars et fils, 1893-1902.

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Schröter's Formula

Cite this as:

Weisstein, Eric W. "Schröter's Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SchroetersFormula.html

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