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_n(x,y)=xU_n-1(x,y)+yU_n-2(x,y), n⩾2, U₀(x,y)=0, U₁(x,y)=1, and V_n(x,y)=xV _n-1(x,y)+yV_n-2(x,y), n⩾2, V₀(x,y)=2, V₁(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