This is a study of Favard interpolation—in which the nth derivatives of the interpolant are bounded above by a constant times the nth divided differences of the data—in the case the data is given on some subset of a rectangular lattice in R k. In some instances, depending on the geometry of this subset, we construct a Favard interpolant, and in other instances, we prove that none exists.

In the current article we present a fast and reliable algorithm for evaluating nth order pentadiagonal determinants in linear time. It is a natural generalization of the DETGTRI algorithm [M. El-Mikkawy, A fast algorithm for evaluating nth order tri-diagonal determinants, J. Comput. Appl. Math. 166 (2004) 581–584]. The algorithm is suited for implementation using computer algebra systems (CAS) ...

Let Ψn(x) be the monic polynomial having precisely all non-primitive nth roots of unity as its simple zeros. One has Ψn(x) = (x n − 1)/Φn(x), with Φn(x) the nth cyclotomic polynomial. The coefficients of Ψn(x) are integers that like the coefficients of Φn(x) tend to be surprisingly small in absolute value, e.g. for n < 561 all coefficients of Ψn(x) are ≤ 1 in absolute value. We establish variou...

The irreducibility of a polynomial in a prime modulus implies irreducibility over the rationals. However, the converse is certainly not true. In fact, there are some polynomials that are irreducible over the rationals yet reducible in every prime modulus. We show that the nth cyclotomic polynomial reduces modulo all primes if and only if the discriminant of the nth cyclotomic polynomial is a sq...

The latter is an especially attractive characterization of the nth central binomial coefficient. Contrast this with the nth central trinomial coefficient, B(n), defined to be the largest coefficient of the polynomial (1 + x + x). There is no simple closed-form expression for B(n) [2]. It possesses recursion (n+ 1)B(n+ 1) = (2n+ 1)B(n) + 3nB(n− 1), B(0) = B(1) = 1 and asymptotics B(n) ∼ r 3 4π n...