  While Pad\'e approximation is a general method for improving convergence of
series expansions, Gell-Mann--Low renormalization group normally relies on the
presence of special symmetries. We show that in the single-variable case, the
latter becomes an integral Hermite-Pad\'e approximation, needing no special
symmetries. It is especially useful for interpolating between expansions for
small values of a variable and a scaling law of known exponent for large
values. As an example, we extract the scaling-law prefactor for the one-body
density matrix of the Lieb-Liniger gas. Using a new result for the 4th-order
term in the short-distance expansion, we find a remarkable agreement with known
ab initio numerical results.
