Let p be an odd prime and q = pm. Let l be an odd positive integer. Let p ≡ −1 (mod l) or p ≡ 1 (mod l) and l | m. By employing the integer sequence an = l−1 2 ∑ t=1 ( 2 cos π(2t− 1) l )n , which can be considered as a generalized Lucas sequence, we construct all the permutation binomials P (x) = xr + xu of the finite field Fq .

In this paper, we define and investigate the generalized Friedrich sequences deal with, in detail, two special cases, namely, Friedrich-Lucas sequences. We present Binet's formulas, generating functions, Simson summation formulas for these Moreover, give some identities matrices related with Furthermore, show that there are close relationsbetween Friedrich, third order Jacobsthal, modified thir...

In this paper we consider the general sequences U„ and Vn satisfying the recurrences Un+2=mUn+l + Un, Vn+2=mV„+l+V„, (1.1) where m is a given positive integer, and UQ = 0, Ux = 1, V0 = 2, V1 = m. We shall occasionally refer to these sequences as U(m) and V(m) to emphasize their dependence on the parameter m. They can be represented by the Binet forms Un = {a-ni{a-P\ Vn = a+f3\ (1.2) where a+j3 ...

If p(z) is univariate polynomial with complex coefficients having all its zeros inside the closed unit disk, then the Gauss-Lucas theorem states that all zeros of p′(z) lie in the same disk. We study the following question: what is the maximum distance from the arithmetic mean of all zeros of p(z) to a nearest zero of p′(z)? We obtain bounds for this distance depending on degree. We also show t...

Using $ (p, q) $-Lucas polynomials and bi-Bazilevic type functions of order $\rho +i\xi,$ we defined a new subclass biunivalent functions. We obtained coefficient inequalities for belonging to the subclass. In addition these results, upper bound Fekete-Szegö functional was obtained. Finally, some special values parameters, several corollaries were presented.

In this paper, we prove several identities involving linear combinations of convolutions the generalized Fibonacci and Lucas sequences. Our results apply more generally to broader classes second-order linearly recurrent sequences with constant coefficients. As a consequence, obtain as special cases many relating exactly four amongst Fibonacci, Lucas, Pell, Pell–Lucas, Jacobsthal, Jacobsthal–Luc...

An operational approach introduced by Gould and Hopper to the construction of generalized Hermite polynomials is followed in the hypercomplex context to build multidimensional generalized Hermite polynomials by the consideration of an appropriate basic set of monogenic polynomials. Directly related functions, like Chebyshev polynomials of first and second kind are constructed.

The hypercube is one of the best models for network topology a distributed system. Recently, Padovan cubes and Lucas–Padovan have been introduced as new interconnection topologies. Despite their asymmetric relatively sparse interconnections, are shown to possess attractive recurrent structures. In this paper, we determine cube polynomial cubes, well generating functions sequences these cubes. S...

This paper studies a suitably normalized set of generalized Hermite polynomials and sets down a relevant Mehler formula, Rodrigues formula, and generalized translation operator. Weighted generalized Hermite polynomials are the eigenfunctions of a generalized Fourier transform which satisfies an F. and M. Riesz theorem on the absolute continuity of analytic measures. The Bose-like oscillator cal...