Your transformation (*) is related to the Mellin transform (with $\epsilon=s$). In particular, your result is obtain as the limit $s\to0^+$ of Ramanujan's master theorem $$ \frac{\sin(\pi s)}{\pi}\int_0^\infty x^{s-1}\left(\,\lambda(0) - x\,\lambda(1) + x^2\,\lambda(2) -\,\cdots\,\right) dx = \,\lambda(-s)\,.$$

Your transformation (*) is related to the Mellin transform (with $\epsilon=s$). In particular, your result is obtained as the limit $s\to0^+$ of Ramanujan's master theorem $$ \frac{\sin(\pi s)}{\pi}\int_0^\infty x^{s-1}\left(\,\lambda(0) - x\,\lambda(1) + x^2\,\lambda(2) -\,\cdots\,\right) dx = \,\lambda(-s)\,.$$