Pascal's triangle: An origin of Daubechies polynomials and an analytic expression for associated filter coefficients

After showing that Daubechies polynomial coefficients can be simply obtained from Pascal's triangle by some elementary additions, we propose a derivation of the spectral factorization by using the elementary symmetric functions. This derivation leads us to present an analytic expression, able to compute Daubechies wavelet filter coefficients from the roots of the associated Daubechies polynomial. Thus, these coefficients are directly obtained and without recurrence. At last, we measure the quality of the coefficient sets generated by this expression and we compare it with two well-known methods.

► Daubechies polynomial coefficients can be obtained from Pascal's triangle.
► These coefficients are computed by a very fast algorithm.
► An analytic expression able to compute Daubechies filter coefficients is given.
► Coefficient set is better than the one of well-known methods for small orders.