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§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)=k=0n-1-πk2m/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 1nG(m,n)=1nk=0n-1-πk2m/n=-π/4mj=0m-1πj2n/m=-π/4mG(-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)=n=-an(n+1)/2bn(n-1)/2,

where a,b and |ab|<1. With the substitutions a=q2z, b=q-2z, with q=πτ, we have

20.11.4 f(a,b)=θ3(z|τ).

In the case z=0 identities for theta functions become identities in the complex variable q, with |q|<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 τ=τ(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 F12(12,12;1;k2)=θ32(0|τ);

see §19.5. Similar identities can be constructed for F12(13,23;1;k2), F12(14,34;1;k2), and F12(16,56;1;k2). 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 π 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 mn, define twelve combined theta functions φm,n(z,q) by

20.11.6 φm,1(z,q) =θ1(0,q)θm(z,q)θm(0,q)θ1(z,q),
20.11.7 φ1,n(z,q) =θn(0,q)θ1(z,q)θ1(0,q)θn(z,q),
20.11.8 φm,n(z,q) =θn(0,q)θm(z,q)θm(0,q)θn(z,q),


20.11.9 φm,n(z,q)=φm,1(z,q)φ1,n(z,q)=1φn,m(z,q)=φm,1(z,q)φn,1(z,q)=φ1,n(z,q)φ1,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).