# §20.11 Generalizations and Analogs

## §20.11(i) Gauss Sum

For relatively prime integers $m,n$ with $n>0$ and $mn$ even, the Gauss sum $G(m,n)$ is defined by

 20.11.1 $G(m,n)=\sum\limits_{k=0}^{n-1}e^{-\pi ik^{2}m/n};$ ⓘ Defines: $G(m,n)$: Gauss sum (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer and $n$: integer Permalink: http://dlmf.nist.gov/20.11.E1 Encodings: TeX, pMML, png See also: Annotations for §20.11(i), §20.11 and Ch.20

see Lerch (1903). It is a discrete analog of theta functions. If both $m,n$ are positive, then $G(m,n)$ allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)):

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

This is the discrete analog of the Poisson identity (§1.8(iv)).

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

Ramanujan’s theta function $f(a,b)$ is defined by

 20.11.3 $f(a,b)=\sum_{n=-\infty}^{\infty}a^{n(n+1)/2}b^{n(n-1)/2},$ ⓘ Symbols: $n$: integer Permalink: http://dlmf.nist.gov/20.11.E3 Encodings: TeX, pMML, png See also: Annotations for §20.11(ii), §20.11 and Ch.20

where $a,b\in\mathbb{C}$ 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 $f(a,b)=\theta_{3}\left(z\middle|\tau\right).$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$: theta function, $z$: complex and $\tau$: lattice parameter Permalink: http://dlmf.nist.gov/20.11.E4 Encodings: TeX, pMML, png See also: Annotations for §20.11(ii), §20.11 and Ch.20

In the case $z=0$ identities for theta functions become identities in the complex variable $q$, 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

As in §20.11(ii), the modulus $k$ of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in $q$-series via (20.9.1). However, in this case $q$ is no longer regarded as an independent complex variable within the unit circle, because $k$ 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 ${{}_{2}F_{1}}\left(\tfrac{1}{2},\tfrac{1}{2};1;k^{2}\right)={\theta_{3}}^{2}% \left(0\middle|\tau\right);$

see §19.5. Similar identities can be constructed for ${{}_{2}F_{1}}\left(\tfrac{1}{3},\tfrac{2}{3};1;k^{2}\right)$, ${{}_{2}F_{1}}\left(\tfrac{1}{4},\tfrac{3}{4};1;k^{2}\right)$, and ${{}_{2}F_{1}}\left(\tfrac{1}{6},\tfrac{5}{6};1;k^{2}\right)$. 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

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

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

For $m=1,2,3,4$, $n=1,2,3,4$, and $m\neq n$, define twelve combined theta functions $\varphi_{m,n}\left(z,q\right)$ by

 20.11.6 $\displaystyle\varphi_{m,1}\left(z,q\right)$ $\displaystyle=\frac{\theta_{1}'\left(0,q\right)\theta_{m}\left(z,q\right)}{% \theta_{m}\left(0,q\right)\theta_{1}\left(z,q\right)},$ $m=2,3,4$, 20.11.7 $\displaystyle\varphi_{1,n}\left(z,q\right)$ $\displaystyle=\frac{\theta_{n}\left(0,q\right)\theta_{1}\left(z,q\right)}{% \theta_{1}'\left(0,q\right)\theta_{n}\left(z,q\right)},$ $n=2,3,4$, 20.11.8 $\displaystyle\varphi_{m,n}\left(z,q\right)$ $\displaystyle=\frac{\theta_{n}\left(0,q\right)\theta_{m}\left(z,q\right)}{% \theta_{m}\left(0,q\right)\theta_{n}\left(z,q\right)},$ $m,n=2,3,4$.

Then

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

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).