Ramanujan Theta Functions

Ramanujan's two-variable theta function is defined by

(1)

for (Berndt 1985, p. 34; Berndt et al. 2000). It satisfies

(2)

and

			(3)
			(4)

(Berndt 1985, pp. 34-35; Berndt et al. 2000), where is a q-Pochhammer symbol, i.e., a q-series.

A one-argument form of is also defined by

			(5)
			(6)
			(7)

(OEIS A010815; Berndt 1985, 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 1985, pp. 36-37). is a generalization of , with the two being connected by

(15)

Special values of include

			(16)
			(17)

where is a gamma function.

Ramanujan's -function is defined by

			(18)
			(19)
			(20)
			(21)
			(22)
			(23)

(OEIS A010054; Berndt 1985, p. 37).

Ramanujan's -function is defined by

			(24)
			(25)
			(26)

(OEIS A000700; Berndt 1985, p. 37).

A different function is sometimes defined as

(27)

where is again a Jacobi theta function, which has special value

(28)