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Asymptotic normalization coefficients (ANCs) are fundamental nuclearconstants playing an important role in nuclear physics and astrophysics. Wederive a new useful relationship between ANC of the Gamow radial wave functionand the renormalized (due to the Coulomb interaction) Coulomb-nuclear partialscattering amplitude. We use an analytical approximation in the form of aseries for the nonresonant part of the phase shift which can be analyticallycontinued to the point of an isolated resonance pole in the complex plane ofthe momentum. Earlier, this method which we call the $S$-matrix pole method wasused by us to find the resonance pole energy. We find the corresponding fittingparameters for the $^5\rm{He},\,^5\rm{Li}$, and $^{16}\rm{O}$ concreteresonance states. Additionally, based on the theory of the effective range, wecalculate the parameters of the $p_{3/2}$ and $p_{1/2}$ resonance states of thenuclei $^5\rm{He}$ and $^5\rm{Li}$ and compare them with the results obtainedby the $S$-matrix pole method. ANC values are found which can be used tocalculate the reaction rate through the $^{16}\rm{O}$ resonances which lieslightly above the threshold for the $\alpha^{12}\rm{C}$ channel.

Resonance-state properties from a phase shift analysis with the $S$-matrix pole method and the effective-range method