Analytic Representations in the 3-dim Frobenius Problem

We consider the Diophantine problem of Frobenius for semigroup ${\sf S}({\bf d}^3)$where${\bf d}^3$denotesthetuple$(d_1,d_2,d_3)$,$\gcd(d_1,d_2,d_3)=1$. Based on the Hadamard product of analytic functions we have found the analytic representation for the diagonal elements $a_{kk}({\bf d}^3)$oftheJohnso{n}^{\mathrm{\prime }}smatrixofminimalrelationsintermsof$d_1,d_2,d_3$. Bearing in mind the results of the recent paper this gives the analytic representation for the Frobenius number $F({\mathbf{d}}^3)$, genus $G({\mathbf{d}}^3)$ and the Hilbert series $H({\mathbf{d}}^3;z)$ for the semigroups ${\sf S}({\bf d}^3)$. This representation does complement the Curtis’ theorem on the non-algebraic representation of the Frobenius number $F({\mathbf{d}}^3)$. We also give a procedure to calculate the diagonal and off-diagonal elements of the Johnson’s matrix.