  We construct the existence theory for generalized fractional Bessel
differential equations and find the solutions in the form of fractional or
logarithmic fractional power series. We figure out the cases when the series
solution is unique, non-unique, or does not exist. The uniqueness theorem in
space $C^p$ is proved for the corresponding initial value problem. We are
concerned with the following homogeneous generalized fractional Bessel equation
$$
  \sum_{i=1}^{m}d_i\, x^{\alpha_i}D^{\alpha_i} u(x) + (x^\beta - \nu^2)u(x)=0,
\ \ \alpha_i > 0, \beta > 0, $$ which includes the standard fractional and
classical Bessel equations as particular cases. Mostly, we consider fractional
derivatives in Caputo sense and construct the theory for positive coefficients
$d_i$. Our theory leads to a threshold admissible value for $\nu^2$, which
perfectly fits to the known results. Our findings are supported by several
numerical examples and counterexamples that justify the necessity of the
imposed conditions. The key point in the investigation is forming proper
fractional power series leading to an algebraic characteristic equation.
Depending on its roots and their multiplicity/complexity, we find the system of
linearly independent solutions.
