Mock Theta Function

Ramanujan was the first to extensively study these Theta Function-like functions

$$f(q) = \sum_{n=0}^\infty {q^{n^2}\over (1+q)^2(1+q^2)^2\cdots(1+q^n)^2}$$

$$\phi(q) = \sum_{n=0}^\infty {q^{n^2}\over (1+q^2)(1+q^4)\cdots(1+q^{2n})}.$$

See also q-Series, Theta Function

References

Bellman, R. E. A Brief Introduction to Theta Functions. New York: Holt, Rinehart, and Winston, 1961.