Multifractal analysis of some inhomogeneous multinomial measures with distinct analytic Olsen's $b$ and $B$ functions

Inhomogeneous multinomial measures on the mixed symbolic spaces and the real line are given. By counting the zeros of the corresponding generalized Dirichlet polynomials, one obtains a probability measure whose Olsen's functions $b$ and $B$ are analytic and their graphs differ except at two points where they are tangent. Also, interpretations of the Legendre transform of $b$ and $B$ are given in terms of dimensions.