1. Introduction and notation

As the reader will easily see, the results of this paper amount to a transfer—from continuous to discrete time, via the approach introduced in [1]—of the findings reported in [2] and [3] (see also Chapters 3 and 7 of [4]).

Throughout this paper the following notation is used: N and L are two arbitrary positive integers N ≥ 2, L ≥ 2), , the indices n and m run from 1 to N, the discrete-time variable ℓ = 0, 1, 2, ... takes all nonnegative integer values, the N dependent variables z_n (ℓ) are generally complex numbers and. being generally defined (see below) as the N zeros of a polynomial of degree N in its (complex) argument z, they are the elements of an unordered set of N elements identified hereafter with the notation z˜(ℓ); likewise in the following the notation f˜ denotes the unordered set of N elements f_n.

2. Results

3. Proof

The starting point of the proof of Proposition 2.1 is the definition (2.3) of the polynomial ψ_N (z; ℓ). The consistency of this definition with the assignment of the initial data z˜(0) and z˜(1) has already been noted above. What remains to be proven is that the formula

—which, with ψ_N (z; ℓ) defined by (2.3), clearly coincides with the statement of Proposition 2.1—implies that the N zeros z_n (ℓ) satisfy the evolution equation ( 2.1a). To this end we note that since by definition (see (2.3)) the dependence of ψ_N (z; ℓ) on the discrete-time variable ℓ is linear, ψ_N (z; ℓ) satisfies identically the linear second-order difference equation For z = z_n (ℓ + 2), via (3.1), this formula implies (2.1a). Q. E. D.

4. Envoy

The result reported in the above Proposition 2.1 is likely to look, at least at first sight, somewhat remarkable, especially in view of the arbitrariness of the assignment of the 2N numbers z_n(0) and f_n (or, equivalently, z_n(0) and z_n(1); see the above Remark 2.1). But of course, after its validity has been proven, it shall be considered obvious—as all valid mathematical results in some sense are. A potential application of this finding is as a tool to test the accuracy of numerical routines to compute the zeros of polynomials of arbitrary degree N: by comparing, with the simple explicit outcome detailed in the above Corollary 2.1, the results yielded by the application of such routines in order to solve numerically—from the initial data detailed in Proposition 2.1, up to ℓ = L—the discrete-time evolution (2.1); which indeed requires finding the zeros of appropriate polynomials of degree N at every step of this discrete-time evolution. In this context the flexibility implied by the possibility to assign arbitrarily the two integers N and L and the 2N, generally complex, numbers z_n(0) and f_n might be quite useful. Specialists in numerical analysis might be interested to explore in detail the vistas implied by such possibilities: note for instance that, for N = 20 and f_m = m, Corollary 2.1—for any arbitrary assignment of the parameters L and x_n(0)—yields the 20 zeros of the perfidious Wilkinson polynomial [5].

An extension of the findings reported in this paper to the case in which the finite positive integer N is replaced by ∞ is of course possible, see [2].

A (perhaps less elegant) variant of the approach described in this paper—characterized by the replacement of the system of second-order discrete-time evolution equations (2.1a) by systems of first-order discrete-time evolution equations—is of course possible, in analogy to the treatments of the continuous-time cases, see [1],[2] and Chapter 3 of [4].