On a continued fraction of Ramanujan, two and four-square theorems and bibasic \(q\)-Appell functions (Q6188410)

The Rogers-Ramanujan continued fraction is defined as \[ R(q)=\frac{q^{1/5}}{1}_{+}\frac{q}{1}_{+}\frac{q^2}{1}_{+} \frac{q^3}{1}_{+}\cdots, \qquad |q|<1. \] In the present paper, the author considers its analogue \[ C_1(q)=\frac{q^{1/8}}{1}_{+}\frac{1+q}{1}_{+}\frac{q^2}{1}_{+} \frac{q+q^3}{1}_{+}\frac{q^4}{1}_{+}\cdots, \qquad |q|<1, \] and gives some identities analogous to the identities for \(R(q)\). In particular, he provides a product-sum identity for \(C_1(q)\) and connect two and four-square theorems with \(C_1(q)\).