Ramanujan Theta Functions

The Ramanujan theta functions include the two-variable function defined by

(1)

for (Berndt 1991, p. 34; Berndt et al. 2000). It satisfies

(2)

and

			(3)
			(4)

(Berndt 1991, pp. 34-35; Berndt et al. 2000), where is an infinite q-Pochhammer symbol. The identity expressing as an infinite product is the Jacobi triple product.

For , a one-argument form of is also defined by

			(5)
			(6)
			(7)

(OEIS A010815; Berndt 1991, pp. 36-37; Berndt et al. 2000), where is a q-Pochhammer symbol. The identities above are equivalent to the pentagonal number theorem.

The function also satisfies

			(8)
			(9)

Ramanujan's -function is defined by

			(10)
			(11)
			(12)
			(13)
			(14)

(OEIS A000122), where is a Jacobi theta function (Berndt 1991, pp. 36-37). Special values of include

			(15)
			(16)

where is a gamma function (Berndt and Rebák 2021, p. 44).

Ramanujan's -function is defined by

			(17)
			(18)
			(19)
			(20)
			(21)
			(22)

(OEIS A010054; Berndt 1991, p. 37). Fractional powers are taken at their principal values, and the expression involving at is interpreted by its limit.

The Ramanujan chi-function is defined for by

(23)

(Berndt 1991, p. 37), where is a q-Pochhammer symbol.

A different function is sometimes denoted by , with the prime being part of the name rather than a derivative, and is defined by

(24)

where is again a Jacobi theta function. With the same principal value convention, it has the special value

(25)