Numerical triangle, Fibonacci sequence and ladder networks: Some further results

Fibonacci Polynomials their Properties and Applications

“…Applying (32) in circuit theory we get a simple expression for a parrallel connection of infinite number of resistors with resistances determined by successive Fibonacci numbers. Such type of electric circuits can be considered as an alternative structure for ladder networks composed of identical resistors (see [9,16]).…”

“…The Fibonacci polynomials can be used to represent a given physical quantity, for example, voltage in a voltage divider (see [11,13,16)), or a fixed number as a sum of suitable components (see [4,20]). Our particular interest here is the decomposition of a given value g into elements of a Fibonacci polynomial of n-th degree, i.e.…”

“…in which the first two values fo and f are known. It is usualy assumed (see [2,15,16,20]) that the absolute values of the subscripts in (3) are . I k I =0,1,2,... and the initial values are fo = 1 and f, = 1.…”

“…Solving this equality gives k = 0. Thus, using (16) we can state that there exists any polynomial (1) with index of concentration S which has some zeros located in the open unit disk centered at 0. It means that the location of the zeros of the polynomial (1) in the open unit disk corresponds to values of the index of concentration greater than From (20) and (37) we obtain…”

Transfer function of multi-stage active filters: a solution based on Pascal's triangle and a general expression

“…In general, a polynomial characterizes the electrical network over a frequency range. New methods were developed to find the coefficients of these characteristic polynomials using recurrence relations [4], which generate the correct coefficients for a ladder network of any given number of stages. These methods are less accessible to third or fourth year undergraduate students as they involve the matrix formalism or knowledge of theorems, such as Cagney-Hamilton [3] theorems applied to ladder networks, from which the modified numerical triangle is being generated.…”