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Motivated by a problem proposed by Seiffert a quarter of century ago, we explicitly evaluate binomial sums with Pell and Lucas polynomials as weight functions. Their special cases result in several interesting identities concerning Fibonacci and Lucas numbers. [ABSTRACT FROM AUTHOR]

Abstract Following recent paper of the author about polynomial expressions with respect to binomial ideals, the current paper is devoted to the non-binomial case. Beside the established facts in the underlying theory, the correctness and termination of the algorithms are addressed. From the collection of examples, the main large-scale non-binomial one is of particular importance. Application of the new algorithms for the cases of moderately large binomial like cyclic 16-roots and cyclic 18-roots is discussed. A combinatorial analysis for complexity of the proposed algorithms is shown to be at least O (log ⁡ (log ⁡ n) n 2 (log ⁡ n) 2). [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions for a given polynomial to be a plucking polynomial of a rooted tree. We discuss the fact that different rooted trees can have the same polynomial. [ABSTRACT FROM AUTHOR]

Let F be a finite extension of ℚp. We show that every Schwartz function on F, with values in ℚp, is the p-adic uniform limit of a sequence of Schwartz functions, whose Fourier transforms tend uniformly to 0. The proof uses the notion of q-binomial coefficients and some classical identities between these polynomials. [ABSTRACT FROM AUTHOR]

We consider the problem of computing the coefficients of the reliability polynomial (RP) for a random graph with reliable vertices and unreliable edges. To speed up the computation, we use the meaning of RP coefficients in one of its representations and prove vector relations over vectors of binomial coefficients. [ABSTRACT FROM AUTHOR]

We introduce new q-tangent numbers based on the Carlitz q-analog of the Eulerian polynomial and the so-called doubloon combinatorial set-up. Those new q-tangent numbers are polynomials with positive integral coefficients. They are divisible by products of binomials of the form 1 + qi, the quotients being q-analogs of the reduced tangent numbers having an explicit combinatorial interpretation. [ABSTRACT FROM PUBLISHER]

In this paper, we consider translation and multiplication operators acting on the rings of symmetric and nonsymmetric polynomials and study their matrix coefficients with respect to the bases of Jack polynomials and interpolation polynomials. The main new insight is that the symmetric and nonsymmetric cases share a key combinatorial feature, that of a locally finite graded poset with a minimum element. This allows us to treat both cases in a simple and unified manner. [ABSTRACT FROM PUBLISHER]

Abstract: We confirm several conjectures of Guo, Jouhet and Zeng concerning the factors of alternating binomials sums. [Copyright &y& Elsevier]

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]

Abstract: In this paper we investigate some algebraic and geometric consequences which arise from an extremal bound on the Hilbert function of the general hyperplane section of a variety (Green''s Hyperplane Restriction Theorem). These geometric consequences improve some results in this direction first given by Green and extend others by Bigatti, Geramita, and Migliore. Other applications of our detailed investigation of how the Hilbert polynomial is written as a sum of binomials, are to conditions that must be satisfied by a polynomial if it is to be the Hilbert polynomial of a non-degenerate integral subscheme of (a problem posed by R.P. Stanley). We also give some new restrictions on the Hilbert function of a zero-dimensional reduced scheme with the Uniform Position Property. [Copyright &y& Elsevier]

Abstract: We estimate character sums with Catalan numbers and middle binomial coefficients modulo a prime p. We use this bound to show that the first at most elements of each sequence already fall in all residue classes modulo every sufficiently large p, which improves the previously known result requiring elements. We also study, using a different technique, similar questions for sequences satisfying polynomial recurrence relations like the Apéry numbers. We show that such sequences form a finite additive basis modulo p for every sufficiently large prime p. [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]

The main result of the paper establishes the strong log-concavity of certain sequences arising from representation of positive integers with respect to some integer basis. More precisely, given an integer basis B=(bi)i⩾0, for instance bi≔bi with b⩾2, and a positive integer m, let fℓ be the number of integers between 0 and m having exactly ℓ nonzero digits in their B-representation. It is shown that (fℓ)ℓ⩾0 is log-concave and some estimates for the peaks of these sequences are given. This theorem is indeed an inequality for elementary symmetric polynomials. It can be specialized to give the log-concavity of sequences of sums of special numbers, such as binomial coefficients, Stirling numbers of the first kind or their q-analogs. These sequences (fℓ)ℓ⩾0 can also be seen as f-vectors of compressed subsets in direct (poset) product of stars, where the compression is relative to the reverse-lexicographic order. [Copyright &y& Elsevier]

We provide a probabilistic approach using renewal theory to derive some novel identities involving Eulerian numbers and uniform B-splines. The renewal perspective leads to a unified treatment for the normalized binomial coefficients and the normalized Eulerian numbers when studying their limits of sums, as well as their associated distributions — the binomial distributions (Bernstein polynomials) and the Irwin–Hall distributions (uniform B-splines). We further extend the probabilistic perspective to h -Bernstein polynomials (Pólya–Eggenberger distributions) through conditional renewal processes, and derive new limits of various ways of summing for the two special numbers and associated distributions. The proposed probabilistic unification dramatically simplifies the proofs of some identities, which are far from obvious (such as for h -Bernstein polynomials) or do not otherwise even appear promising (such as for Eulerian numbers). [ABSTRACT FROM AUTHOR]