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

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

20.11.1 | $$G(m,n)=\sum _{k=0}^{n-1}{\mathrm{e}}^{-\pi \mathrm{i}{k}^{2}m/n};$$ | ||

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 | $$\frac{1}{\sqrt{n}}G(m,n)=\frac{1}{\sqrt{n}}\sum _{k=0}^{n-1}{\mathrm{e}}^{-\pi \mathrm{i}{k}^{2}m/n}=\frac{{\mathrm{e}}^{-\pi \mathrm{i}/4}}{\sqrt{m}}\sum _{j=0}^{m-1}{\mathrm{e}}^{\pi \mathrm{i}{j}^{2}n/m}=\frac{{\mathrm{e}}^{-\pi \mathrm{i}/4}}{\sqrt{m}}G(-n,m).$$ | ||

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

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

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

where $a,b\in \u2102$ and $$. With the substitutions $a=q{\mathrm{e}}^{2\mathrm{i}z}$, $b=q{\mathrm{e}}^{-2\mathrm{i}z}$, with $q={\mathrm{e}}^{\mathrm{i}\pi \tau}$, we have

20.11.4 | $$f(a,b)={\theta}_{3}\left(z\right|\tau ).$$ | ||

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 | $${}_{2}F_{1}(\frac{1}{2},\frac{1}{2};1;{k}^{2})={\theta}_{3}^{2}\left(0\right|\tau );$$ | ||

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

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 $m\ne n$, define twelve combined theta functions ${\phi}_{m,n}(z,q)$ by

20.11.6 | ${\phi}_{m,1}(z,q)$ | $={\displaystyle \frac{{\theta}_{1}^{\prime}(0,q){\theta}_{m}(z,q)}{{\theta}_{m}(0,q){\theta}_{1}(z,q)}},$ | ||

$m=2,3,4$, | ||||

20.11.7 | ${\phi}_{1,n}(z,q)$ | $={\displaystyle \frac{{\theta}_{n}(0,q){\theta}_{1}(z,q)}{{\theta}_{1}^{\prime}(0,q){\theta}_{n}(z,q)}},$ | ||

$n=2,3,4$, | ||||

20.11.8 | ${\phi}_{m,n}(z,q)$ | $={\displaystyle \frac{{\theta}_{n}(0,q){\theta}_{m}(z,q)}{{\theta}_{m}(0,q){\theta}_{n}(z,q)}},$ | ||

$m,n=2,3,4$. | ||||

Then

20.11.9 | $${\phi}_{m,n}(z,q)={\phi}_{m,1}(z,q){\phi}_{1,n}(z,q)=\frac{1}{{\phi}_{n,m}(z,q)}=\frac{{\phi}_{m,1}(z,q)}{{\phi}_{n,1}(z,q)}=\frac{{\phi}_{1,n}(z,q)}{{\phi}_{1,m}(z,q)}.$$ | ||

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).