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Let s and t be variables. Define polynomials { n} in s, t by {0} = 0, {1} = 1, and { n} = s { n - 1} + t { n - 2} for n ≥ 2. If s, t are integers then the corresponding sequence of integers is called a Lucas sequence. Define an analogue of the binomial coefficients by where { n}! = {1} {2} ⋯ { n}. It is easy to see that is a polynomial in s and t. The purpose of this note is to give two combinatorial interpretations for this polynomial in terms of statistics on integer partitions inside a k × ( n - k) rectangle. When s = t = 1 we obtain combinatorial interpretations of the fibonomial coefficients which are simpler than any that have previously appeared in the literature. [ABSTRACT FROM AUTHOR]

In this short paper we establish identities involving sums of products of binomial coefficients and coefficients that satisfy the general second-order linear recurrence. We obtain generalizations of identities of Carlitz, Prodinger and Haukkanen. [ABSTRACT FROM AUTHOR]

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k · 2n + 1 is composite for all integers n ≥ 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) ∈ ℤ[x], with f(1) ≠ -d, such that f(x)xn + d is reducible over the rationals for all integers n ≥ 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erdös and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d ≡ 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d ≡ 6 (mod 12). We also consider the corresponding problem over 픽p, and in that situation, we show 1 ∈ S for all primes p. [ABSTRACT FROM AUTHOR]

We describe exercises in which students use PascGaloisJE to formulate conjectures about certain binomial identities which hold when the binomial coefficients are interpreted as elements in the cyclic group [image omitted] of integers modulo a prime integer p. In addition to having an appealing visual component, these exercises are open-ended and expose students to the investigative nature of mathematical discovery. [ABSTRACT FROM AUTHOR]

Abstract: Making use a linear operator, we introduce two novel subclasses and of meromorphically -valent functions with positive and negative coefficients, respectively. The main object of this paper is to investigate the various important properties and characteristics of these subclasses of meromorphically -valent functions. We also derive many interesting results for the Hadamard products of functions belonging to the classes and . [Copyright &y& Elsevier]

Abstract: For sums-of-squares formulas of the form where the are bilinear functions of the and . Let denote the smallest possible value of t allowing such a formula to hold. We have two well-known lower bounds on the size of . One was obtained independently by Hopf and Stiefel, and another by Atiyah. These bounds are given by requiring certain binomial coefficients be divisible by certain powers of 2. Although the behavior of the Hopf–Stiefel bound is fairly well understood, the Atiyah bound is not. In this paper we provide an efficient algorithm for computing the Atiyah bound and some results on which of the lower bounds is larger. [Copyright &y& Elsevier]

The article presents a mathematical activity which relates the coefficients in the formula to finding a parking spot. First, the students discover Pascal's triangle using a parking spot simulation. Then the activity relates Pascal's triangle to the coefficients in the binomial theorem. Graphing calculators and combinations are used to verify the binomial expansion. This activity supports a variety of learning styles in an enjoyable application. The relationship between the parking spot simulation and Pascal's triangle is explained.

Abstract: In this paper we establish a q-analogue of a congruence of Sun concerning the products of binomial coefficients modulo the square of a prime. [Copyright &y& Elsevier]

We give necessary and sufficient conditions that a polynomial f ( x, y) gives an ovoid of PG(3, q). Furthermore by considering Rédei polynomials associated to ovoids of PG(3, q) we obtain sets of equations relating the coefficients of f ( x, y). One such set of equations implies that if xiyj occurs as a term in f ( x, y) + xy then i and j must have disjoint binary expansions. [ABSTRACT FROM AUTHOR]

Abstract: We give a simple proof of Jackson''s terminating q-analogue of Dixon''s identity. [Copyright &y& Elsevier]

Provides a simple proof of Descartes' rule of signs in high school algebra. Proof of theorems; Variations in the signs of the sequence of coefficients; Number of positive zeros of p counting multiplicities by z; Nonnegative integers; Induction hypothesis; Arbitrary real numbers.