One (inessential) mistake in your reasoning is that $b_0=(-1)^K$, rather than $b_0=-1$.

The only other mistake in your reasoning can be that the series defining your $f(x)$ actually has the zero radius of convergence. In fact, this is the single essential mistake in your reasoning. Indeed, the $a_j$'s must all be real, according to their recursive definition, which would contradict (for $K\ge2$) the simple comments by Gerry Myerson and me if the series defining your $f(x)$ had a nonzero radius of convergence.

A mistake in your reasoning is that $b_0=(-1)^K$, rather than $b_0=-1$.

One (inessential) mistake in your reasoning is that $b_0=(-1)^K$, rather than $b_0=-1$.

The only other mistake in your reasoning can be that the series defining your $f(x)$ actually has the zero radius of convergence. In fact, this is the single essential mistake in your reasoning. Indeed, the $a_j$'s must all be real, according to their recursive definition, which would contradict the simple comments by Gerry Myerson and me if the series defining your $f(x)$ had a nonzero radius of convergence.

One (inessential) mistake in your reasoning is that $b_0=(-1)^K$, rather than $b_0=-1$.

The only other mistake in your reasoning can be that the series defining your $f(x)$ actually has the zero radius of convergence. In fact, this is the single essential mistake in your reasoning. Indeed, the $a_j$'s must all be real, according to their recursive definition, which would contradict (for $K\ge2$) the simple comments by Gerry Myerson and me if the series defining your $f(x)$ had a nonzero radius of convergence.