Abstract. Let p be a prime. A famous theorem of Lucas states that ( mp+s) ≡ ( np+t m) ( s) (mod p) ifm, n, s, t are nonnegative integers with s, t < p. Inthispaper n t we aim to prove a similar result for generalized binomial coefficients defined in terms of second order recurrent sequences

To recover a version of Barro's (1979) `random walk' tax smoothing outcome, we modify Lucas and Stokey's (1983) economy to permit only risk-free debt. This imparts near unit root like behavior to government debt, independently of the government expenditure process, a realistic

The Lucas triangle is an infinite triangular array of natural numbers that is a variant of Pascal’s triangle. In this note, we prove a property of the Lucas triangle that has been merely stated by prior researchers; we also present some apparently new properties of the

We are interested in a general equilibrium economy under leverage constraints. In the classical representative-agent Lucas economy, there is a unique stock price derived from the unique state-price density. In our economy, agents have diverse beliefs about future performance so (following Brown

Abstract. The celebrated Gauss-Lucas theorem states that all the roots of the derivative of a complex non-constant polynomial p lie in the convex hull of the roots of p, called the Lucas polygon of p. We improve the Gauss-Lucas theorem by proving that all the nontrivial roots of p0 lie in a smaller

In Matijasevic's paper [1] on Hilbert's Tenth Problem, Lemma 17 states that F 2 m divides Fmr if and only if Fm divides r. Here, we extend Lemma 17 to its counterpart in Lucas numbers and generalized Fibonacci numbers and explore divisibility by higher powers. In [2], Matijasevic

A proof of the Lucas–Lehmer test can be difficult to find, for most textbooks that state the result do not prove it. Over the past two decades, there have been some efforts to produce elementary versions of this famous result. However, the two that we acknowledge in this note did so by using

Abstract. The Lucas-Kanade (LK) method is a classic tracking algorithm ex-ploiting target structural constraints thorough template matching. Extended Lucas Kanade or ELK casts the original LK algorithm as a maximum likelihood opti-mization and then extends it by considering pixel object

− bc) , β = ca/(2b − ca) , γ = ab/(2c − ab) thus we have characterized the Lucas Circles of a triangle ABC by stating their radii. We proceed now to a geometric explanation. Let AD = h be the height from A in triangle ABC, then from the area-form we have bc = 2h, so we may write for the radius of OA, α

The classical Gauss - Lucas Theorem states that all the critical points (zeros of the derivative) of a nonconstant polynomial p lie in the convex hull \Xi of the zeros of p. It is proved that, actually, a subdomain of \Xi contains the critical points of p. 1.