Network properties of a pair of generalized polynomials
In this article, it is shown that there exists an intimate relationship between the network functions of certain ladder one-port and two-port networks, and a set of generalized two-variable polynomials defined by U/sub n/(x,y)=xU/sub n-1/(x,y)+yU/sub n-2/(x,y), n/spl ges/2, U/sub 0/(x,y)=0, U/sub 1/(x,y)=1, and V/sub n/(x,y)=xV/sub n-1/(x,y)+yV/sub n-2/(x,y), n/spl ges/2, V/sub 0/(x,y)=2, V/sub 1/(x,y)=x. Observing that well-known polynomials such as Fibonacci, Chebyshev, Jacobsthal, Pell and Morgan-Voyce polynomials are special cases of these generalized polynomials, it is shown how using these polynomials we can derive elegant relations amongst these various polynomials. Also, using the well-established properties of two-element-kind one-and two-port networks, we then obtain a number of interesting results regarding the location of the zeros of these polynomials, as well as their derivatives.