Goldman, Joichi, and White proved a beautiful theorem showing that the falling factorial generating function for the rook numbers of a Ferrers board factors over the integers. Briggs and Remmel studied an analogue of rook placements where rows are replaced by sets of m rows called levels. They proved a version of the factorization theorem in that setting, but only for certain Ferrers boards. We...

Rook theory has been investigated by many people since its introduction by Kaplansky and Riordan in 1946. Goldman, Joichi and White in 1975 showed that the sum over k of the product of the (n− k)-th rook numbers multiplied by the k-th falling factorial polynomials factorize into a product. In the sequel, different types of generalizations and analogues of this product formula have been derived ...

We investigate the coefficients generated by expressing the falling factorial (xy) as a linear combination of falling factorial products xy for l, m = 1, . . . , k. Algebraic and combinatoric properties are discussed, some in relation to Stirling numbers.

Let n, p, k be three positive integers. We prove that the numbers n k 3 F 2 (1 − k, −p, p−n ; 1, 1−n ; 1) are positive integers which generalize the classical binomial coefficients. We give two generating functions for these integers, and a straightforward application. z k k! , where for an indeterminate a and some positive integer k, the raising factorial is defined by (a) k = a(a + 1). .. (a ...

We investigate non-homogeneous linear differential equations of the form x′′(t) + x′(t) − x (t) = p (t) where p (t) is either a polynomial or a factorial polynomial in t. We express the solution of these differential equations in terms of the coefficients of p (t), in the initial conditions, and in the solution of the corresponding homogeneous differential equation y′′(t) + y′(t) − y (t) = 0 wi...

Insomnia disorder is characterized by chronic dissatisfaction with sleep quantity or quality that is associated with difficulty falling asleep, frequent nighttime awakenings with difficulty returning to sleep, and/or awakening earlier in the morning than desired. Although progress has been made in our understanding of the nature, etiology, and pathophysiology of insomnia, there is still no univ...

Falls are serious health problems among older adults, and are the leading cause of fatal and nonfatal injuries treated by emergency medical services (EMS). Although considerable research has examined the risk factors of falls at the individual level, relatively few studies have addressed the risk factors at the neighborhood level. This study examines the characteristics of neighborhood environm...

In this article, improved bounds for (kn)! and the factorial polynomial are obtained using Kober’s inequality and Lagrange’s identity.