Jacobi triple product

The Jacobi triple product is an identity equating an infinite product to a bilateral series in powers of a variable, with a parameter controlling the coefficients.

Jacobi theta functions: A family of complex functions defined by rapidly convergent series or infinite products and characterized by periodicity properties. The triple product gives a product representation for the standard theta functions.
q-series: A power series whose coefficients or exponents depend on powers of a parameter q. The triple product is a central transformation between q-series and q-products.
Carl Gustav Jacob Jacobi: A German mathematician whose work shaped elliptic functions, number theory, and classical mechanics. Jacobi established the identity within his work on elliptic functions.
Integer partition: A way of expressing an integer as a sum of positive integers, where order does not matter. Generating functions derived from the identity encode restricted integer partitions.
Infinite product: A product with infinitely many factors, defined through a limit of finite partial products. The identity’s product side is an infinite product whose convergence depends on its parameters.
Laurent series: A power series that permits both positive and negative integer powers of its variable. The bilateral expansion is naturally a Laurent series in the variable.
Elliptic functions: Meromorphic functions with two independent periods in the complex plane. The triple product arose from formulas for elliptic functions and their associated theta functions.
Generating function: A formal power series whose coefficients record a sequence or combinatorial structure. The series expansion translates product factors into combinatorial counting data.
Bilateral series: A series whose index ranges over both positive and negative integers. The identity’s series side sums terms indexed across all integers.
Complex analysis: The study of functions of complex variables, including their differentiability, integrals, and singularities. Analytic convergence conditions give the identity its standard domain of validity.

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