Q-theta Function
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the ''q''-theta function (or modified Jacobi theta function) is a type of ''q''-series which is used to define
elliptic hypergeometric series In mathematics, an elliptic hypergeometric series is a series Σ''c'n'' such that the ratio ''c'n''/''c'n''−1 is an elliptic function of ''n'', analogous to generalized hypergeometric series where the ratio is a rational function o ...
. It is given by :\theta(z;q):=\prod_^\infty (1-q^nz)\left(1-q^/z\right) where one takes 0 ≤ , ''q'',  < 1. It obeys the identities :\theta(z;q)=\theta\left(\frac;q\right)=-z\theta\left(\frac;q\right). It may also be expressed as: :\theta(z;q)=(z;q)_\infty (q/z;q)_\infty where (\cdot \cdot )_\infty is the
q-Pochhammer symbol In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer symb ...
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See also

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elliptic hypergeometric series In mathematics, an elliptic hypergeometric series is a series Σ''c'n'' such that the ratio ''c'n''/''c'n''−1 is an elliptic function of ''n'', analogous to generalized hypergeometric series where the ratio is a rational function o ...
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Jacobi theta function In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theo ...
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Ramanujan theta function In mathematics, particularly -analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi triple product takes on a particularly elegant for ...


References

Q-analogs Theta functions {{combin-stub