Closed-form solutions to irreducible Newton-Puiseux equations by Lagrange inversion formula and diagonalization on polynomial sequences of binomial-type

In a recent article published in 2017, Barroso, Pérez, and Popescu-Pampu employ the Lagrange inversion formula to solve certain Newton-Puiseux equations when the solutions to the inverse problems are given. More precisely, for an irreducible f ( x , y ) ∈ K [ [ x , y ] ] f(x,y)\in K[[x,y]] over an algebraically closed field K K of characteristic zero, they calculate the coefficients of η ( x 1 / n ) \eta (x^{1/n}) which would meet f ( x , η ( x 1 / n ) ) = 0 f(x,\eta (x^{1/n}))=0 in terms of the coefficients of ξ ( y 1 / m ) \xi (y^{1/m}) that satisfy f ( ξ ( y 1 / m ) , y ) = 0 f(\xi (y^{1/m}),y)=0 . This article will present an alternative approach to solving the problem using diagonalizations on polynomial sequences of binomial-type. Along the way, a close relationship between binomial-type sequences and the Lagrange inversion formula will be observed. In addition, it will extend the result to give the coefficients of η ( x 1 / n ) \eta (x^{1/n}) directly in terms of the coefficients of f ( x , y ) f(x,y) . As an application, an infinite series formula for the roots of complex polynomials will be obtained together with a sufficient condition for its convergence.