A further generalization of the modified Möbius inversion formula (Q1202820)

The following general inversion formula of Möbius type is given: \[ A(x)=\sum_{n=1}^ \infty B(a_ n x)\iff B(x)=A(x/a_ 1)+\sum_{n=1}^ \infty A_ n(x/a_ 1^{n+1}), \] where \(\{a_ n\}\) is an arbitrary sequence of distinct non-zero real numbers and \[ A_ n(x)=(-1)^ n \sum_{m_ 1=2}^ \infty \dots \sum_{m_ n=2}^ \infty A(m_{m_ 1}\dots a_{m_ n} x). \] The convergence problem of the series is also analyzed and a higher dimensional extension of the inversion formula is carried out. It has previously been discovered that such inversion formulas have applications in physics.