Nome

Given a Jacobi theta function, the nome is defined as

			(1)
			(2)
			(3)

(Borwein and Borwein 1987, pp. 41, 109 and 114), where is the half-period ratio, is the complete elliptic integral of the first kind, and is the elliptic modulus. The nome is implemented in the Wolfram Language as EllipticNomeQ[m].

Extreme care is needed when consulting the literature, as it is common in the theory of modular functions (and in particular the Dedekind eta function) to use the symbol to denote , i.e., the square of the usual nome (e.g., Berndt 1993, p. 139). In this work, the modular version of is denoted

(4)

(e.g., Borwein and Borwein 1987, p. 118).

The nome in plotted above in the complex -plane.

Various notations for Jacobi theta functions involving the nome include

(5)

where is the half-period ratio (Whittaker and Watson 1990, p. 464) and

(6)

Special values include

			(7)
			(8)
			(9)

The nome has Maclaurin series in parameter given by

(10)

(OEIS A002639 and A119349).

The nome has derivative

(11)

where is a complete elliptic integral of the first kind and is the elliptic modulus.

There exists a nonlinear third-order differential equation

(12)

for (Bertrand and Zudilin 2000; Trott 2006, pp. 29-31).