Closed-Form Rational Approximations of Fractional, Analog and Digital Differentiators/Integrators

This paper provides closed-form formulas for coefficients of convergents of some popular continued fraction expansions (CFEs) approximating
$s^{\nu}$
, with
$-1<\nu<1$
, and
$(2/T)^{\nu}((z-1)/(z+1))^{\nu}$
. The expressions of the coefficients are given in terms of
$\nu$
and of the degree
$n$
of the polynomials defining the convergents. The formulas greatly reduce the effort for approximating fractional operators and show the equivalence between two well-known CFEs in a given condition.