In this paper, we investigate the generalized Fibonacci (Horadam) polynomials and deal with, in detail, two special cases which call them $(r,s)$-Fibonacci $(r,s)$-Lucas polynomials. We present Binet's formulas, generating functions, Simson's summation formulas for these polynomial sequences. Moreover, give some identities matrices associated with Finally, several expressions combinatorial resu...

Pollution has become a very serious threat to our environment. Monitoring pollution is the rst step toward planning to save the environment. The use of dierential equations, monitoring pollution has become possible. In this paper, a Ritz-collocation method is introduced to solve non-linear oscillatory systems such as modelling the pollution of a system of lakes. The method is based upon Bernoul...

As two-dimensional coupled system of nonlinear partial differential equations does not give enough smooth solutions, when approximated by linear, quadratic and cubic polynomials and gives poor convergence or no convergence. In such cases, approximation by zero degree polynomials like Haar wavelets (continuous functions with finite jumps) are most suitable and reliable. Therefore, modified numer...

Littlewood polynomials are polynomials with each of their coefficients in the set {−1, 1}. We compute asymptotic formulas for the arithmetic mean values of the Mahler’s measure and the Lp norms of Littlewood polynomials of degree n − 1. We show that the arithmetic means of the Mahler’s measure and the Lp norms of Littlewood polynomials of degree n − 1 are asymptotically e−γ/2 √ n and Γ(1+ p/2)1...

We give new sufficient conditions for a sequence of polynomials to have only real zeros based on the method of interlacing zeros. As applications we derive several well-known facts, including the reality of zeros of orthogonal polynomials, matching polynomials, Narayana polynomials and Eulerian polynomials. We also settle certain conjectures of Stahl on genus polynomials by proving them for cer...