20 Theta Functions Properties 20.10 Integrals 20.12 Mathematical Applications

§20.11 Generalizations and Analogs

Keywords:: theta functions
Permalink:: http://dlmf.nist.gov/20.11

§20.11(i) Gauss Sum
§20.11(ii) Ramanujan’s Theta Function and -Series
§20.11(iii) Ramanujan’s Change of Base
§20.11(iv) Theta Functions with Characteristics
§20.11(v) Permutation Symmetry

§20.11(i) Gauss Sum

Notes:: See Bellman (1961, pp. 38–39), Walker (1996, pp. 181–182), and McKean and Moll (1999, pp. 140–147 and 151–152).
Keywords:: Gauss sums, Poisson identity, theta functions
Permalink:: http://dlmf.nist.gov/20.11.i

For relatively prime integers with and even, the Gauss sum is defined by

20.11.1 $G(m,n)=\sum\limits_{{k=0}}^{{n-1}}e^{{-\pi ik^{2}m/n}};$

Symbols:: : base of exponential function, : integer, : integer and : Gauss sum
Permalink:: http://dlmf.nist.gov/20.11.E1
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see Lerch (1903). It is a discrete analog of theta functions. If both are positive, then allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)):

20.11.2 $\frac{1}{\sqrt{n}}G(m,n)=\frac{1}{\sqrt{n}}\sum\limits_{{k=0}}^{{n-1}}e^{{-\pi ik% ^{2}m/n}}=\frac{e^{{-\pi i/4}}}{\sqrt{m}}\sum\limits_{{j=0}}^{{m-1}}e^{{\pi ij% ^{2}n/m}}=\frac{e^{{-\pi i/4}}}{\sqrt{m}}G(-n,m).$

Symbols:: : base of exponential function, : integer, : integer and : Gauss sum
Permalink:: http://dlmf.nist.gov/20.11.E2
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This is the discrete analog of the Poisson identity (§1.8(iv)).

§20.11(ii) Ramanujan’s Theta Function and -Series

Keywords:: theta functions
Referenced by:: §20.11(iii)
Permalink:: http://dlmf.nist.gov/20.11.ii

Ramanujan’s theta function is defined by

20.11.3 $f(a,b)=\sum_{{n=-\infty}}^{\infty}a^{{n(n+1)/2}}b^{{n(n-1)/2}},$

Symbols:: : integer
Permalink:: http://dlmf.nist.gov/20.11.E3
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where $a,b\in\Complex$ and $\left|ab\right|<1$ . With the substitutions $a=qe^{{2iz}}$ , $b=qe^{{-2iz}}$ , with $q=e^{{i\pi\tau}}$ , we have

20.11.4 $f(a,b)=\mathop{\theta_{{3}}\/}\nolimits\!\left(z\middle|\tau\right).$

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, : complex and $\tau$ : lattice parameter
Permalink:: http://dlmf.nist.gov/20.11.E4
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In the case identities for theta functions become identities in the complex variable , with $\left|q\right|<1$ , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7).

§20.11(iii) Ramanujan’s Change of Base

Keywords:: theta functions
Permalink:: http://dlmf.nist.gov/20.11.iii

As in §20.11(ii), the modulus of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in -series via (20.9.1). However, in this case is no longer regarded as an independent complex variable within the unit circle, because is related to the variable $\tau=\tau(k)$ of the theta functions via (20.9.2). This is Jacobi’s inversion problem of §20.9(ii).

The first of equations (20.9.2) can also be written

20.11.5 $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{1}{2};1;k^{2}% \right)={\mathop{\theta_{{3}}\/}\nolimits^{{2}}}\!\left(0\middle|\tau\right);$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\tau$ : lattice parameter and : modulus
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see §19.5. Similar identities can be constructed for $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(\tfrac{1}{3},\tfrac{2}{3};1;k^{2}\right)$ , $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(\tfrac{1}{4},\tfrac{3}{4};1;k^{2}\right)$ , and $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(\tfrac{1}{6},\tfrac{5}{6};1;k^{2}\right)$ . These results are called Ramanujan’s changes of base. Each provides an extension of Jacobi’s inversion problem. See Berndt et al. (1995) and Shen (1998). For applications to rapidly convergent expansions for $\pi$ see Chudnovsky and Chudnovsky (1988), and for applications in the construction of elliptic-hypergeometric series see Rosengren (2004).

§20.11(iv) Theta Functions with Characteristics

Keywords:: theta functions
Permalink:: http://dlmf.nist.gov/20.11.iv

Multidimensional theta functions with characteristics are defined in §21.2(ii) and their properties are described in §§21.3(ii), 21.5(ii), and 21.6. For specialization to the one-dimensional theta functions treated in the present chapter, see Rauch and Lebowitz (1973) and §21.7(iii).

§20.11(v) Permutation Symmetry

Defines:: $\mathop{\varphi_{{n,m}}\/}\nolimits\!\left(z,q\right)$ : combined theta function
Keywords:: theta functions
Referenced by:: Other Changes
Permalink:: http://dlmf.nist.gov/20.11.v
Addition (effective with 1.0.5):: The material in this subsection has been added. It will be incorporated into the next printed edition.
Reported 2011-09-27

A further development on the lines of Neville’s notation (§20.1) is as follows.

For , , and $m\neq n$ , define twelve combined theta functions $\mathop{\varphi_{{m,n}}\/}\nolimits\!\left(z,q\right)$ by

20.11.6 $\mathop{\varphi_{{m,1}}\/}\nolimits\!\left(z,q\right)=\frac{{\mathop{\theta_{{% 1}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)\mathop{\theta_{{m}}\/}\nolimits% \!\left(z,q\right)}{\mathop{\theta_{{m}}\/}\nolimits\!\left(0,q\right)\mathop{% \theta_{{1}}\/}\nolimits\!\left(z,q\right)},$

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\varphi_{{n,m}}\/}\nolimits\!\left(z,q\right)$ : combined theta function, : integer, : complex and : nome
Permalink:: http://dlmf.nist.gov/20.11.E6
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20.11.7 $\mathop{\varphi_{{1,n}}\/}\nolimits\!\left(z,q\right)=\frac{\mathop{\theta_{{n% }}\/}\nolimits\!\left(0,q\right)\mathop{\theta_{{1}}\/}\nolimits\!\left(z,q% \right)}{{\mathop{\theta_{{1}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)% \mathop{\theta_{{n}}\/}\nolimits\!\left(z,q\right)},$

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\varphi_{{n,m}}\/}\nolimits\!\left(z,q\right)$ : combined theta function, : integer, : complex and : nome
Permalink:: http://dlmf.nist.gov/20.11.E7
Encodings:: TeX, pMML, png

20.11.8 $\mathop{\varphi_{{m,n}}\/}\nolimits\!\left(z,q\right)=\frac{\mathop{\theta_{{n% }}\/}\nolimits\!\left(0,q\right)\mathop{\theta_{{m}}\/}\nolimits\!\left(z,q% \right)}{\mathop{\theta_{{m}}\/}\nolimits\!\left(0,q\right)\mathop{\theta_{{n}% }\/}\nolimits\!\left(z,q\right)},$

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\varphi_{{n,m}}\/}\nolimits\!\left(z,q\right)$ : combined theta function, : integer, : integer, : complex and : nome
Permalink:: http://dlmf.nist.gov/20.11.E8
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Then

20.11.9 $\mathop{\varphi_{{m,n}}\/}\nolimits\!\left(z,q\right)=\mathop{\varphi_{{m,1}}% \/}\nolimits\!\left(z,q\right)\mathop{\varphi_{{1,n}}\/}\nolimits\!\left(z,q% \right)=\frac{1}{\mathop{\varphi_{{n,m}}\/}\nolimits\!\left(z,q\right)}=\frac{% \mathop{\varphi_{{m,1}}\/}\nolimits\!\left(z,q\right)}{\mathop{\varphi_{{n,1}}% \/}\nolimits\!\left(z,q\right)}=\frac{\mathop{\varphi_{{1,n}}\/}\nolimits\!% \left(z,q\right)}{\mathop{\varphi_{{1,m}}\/}\nolimits\!\left(z,q\right)}.$

Symbols:: $\mathop{\varphi_{{n,m}}\/}\nolimits\!\left(z,q\right)$ : combined theta function, : integer, : integer, : complex and : nome
Permalink:: http://dlmf.nist.gov/20.11.E9
Encodings:: TeX, pMML, png

The importance of these combined theta functions is that sets of twelve equations for the theta functions often can be replaced by corresponding sets of three equations of the combined theta functions, plus permutation symmetry. Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas.

For further information, see Carlson (2011).

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