It is well known that we have the following pentagon number theorem by Euler:
$\prod_{n=1}^\infty\left(1-q^{n}\right)=\sum_{-\infty}^{\infty}(-1)^nq^{\large \frac{3n^2-n}2}$.
However, how to calculate the following for $|q|<1$ : $$(1) \prod_{n=1}^\infty\left(1-q^{2n-1}\right)$$
$$(2) \prod_{n=1}^\infty\left(1+q^{2n-1}\right)$$
$$(3) \prod_{n=1}^\infty\left(1-q^{2n}\right)$$
$$(4) \prod_{n=1}^\infty\left(1+q^{2n}\right)$$
Can you give some suggestions about them?
Four related infinite q-products can be expressed in terms of the Ramanujan theta function $\,f(-q) := (1-q^1)(1-q^2)(1-q^3)\cdots\,$ as follows:
(A081362) $$Q_3(q) := (1-q^1)(1-q^3)(1-q^5)\cdots = \frac{f(-q)}{f(-q^2)} .\tag{1}$$ (A000700)
$$Q_2(q) := (1+q^1)(1+q^3)(1+q^5)\cdots = \frac{f^2(-q^2)}{f(-q)f(-q^4)} .\tag{2}$$ (A274719) $$ Q_0(q) := (1-q^2)(1-q^4)(1-q^6)\cdots = f(-q^2) .\tag{3}$$ (A035457)
$$Q_1(q) := (1+q^2)(1+q^4)(1+q^6)\cdots = \frac{f(-q^4)}{f(-q^2)} .\tag{4}$$
Consult the linked OEIS entries for some details on their computation. They are all special cases of a $q$-Pochhammer symbol. Consult the linked Wikipedia article for some details. The OEIS sequence A115977 "Expansion of elliptic modular function lambda in powers of the nome q". has some details and references.