Generalized Pascal Matrices, Inverses, Computations and Properties Using One-to-One Rational Polynomial s-z Transformations

This paper proposes a one-to-one mapping between the coefficients of continuous-time ($s$-domain) and discrete-time ($z$-domain) IIR transfer functions such that the $s$ -domain numerator/denominator coefficients can be uniquely mapped to the $z$-domain numerator/denominator coefficients. The one-to-one mapping provides a firm basis for proving the inverses of the so-called generalized Pascal matrices from various first-order $s$- $z$ transformations. We also derive recurrence formulas for recursively determining the inner elements of the generalized Pascal matrices from their boundary ones. Consequently, all the elements of the whole generalized Pascal matrix can be easily generated through utilizing their neighbourhood, which can be exploited for further simplifying the Pascal matrix generations. Finally, we reveal and prove some interesting properties of the generalized Pascal matrices.