Valerio De Angelis

A pretty binomial identity

Elementary proofs abound: the first identity results from choosing x = y = 1 in the binomial expansion of (x+y). The second one may be obtained by comparing the coefficient of x in the identity (1 + x)(1 + x) = (1 + x). The reader is surely aware of many other proofs, including some combinatorial in nature. At the end of the previous century, the evaluation of these sums was trivialized by the work of H. Wilf and D. Zeilberger [8]. In the preface to the charming book [8], the authors begin with the phrase You’ve been up all night working on your new theory, you found the answer, and it is in the form that involves factorials, binomial coefficients, and so on, ... and then proceed to introduce the method of creative telescoping discussed in Section 3. This technique provides an automatic tool for the verification of these type of identities. The points of view presented in [3] and [10] provide an entertaining comparison of what is admissible as a proof. In this short note we present a variety of proofs of the identity

Complementary Bell Numbers: Arithmetical Properties and Wilf’s Conjecture

The 2-adic valuations of Bell and complementary Bell numbers are determined. The complementary Bell numbers are known to be zero at n = 2 and H. S. Wilf conjectured that this is the only case where vanishing occurs. N. C. Alexander and J. An proved (independently) that there are at most two indices where this happens. This paper presents yet an alternative proof of the latter.

Stirling's Series Revisited

1. INTRODUCTION. We present a concise and elementary derivation of the complete asymptotic expansion for the factorial function n!, which we will refer to as Stirling’s series. While there have been numerous published proofs of Stirling’s series and of its classical dominant term given by Stirling’s formula lim n→∞ n!e n n n √ 2πn = 1

From sequences to polynomials and back, via operator orderings

Bender and Dunne [“Polynomials and operator orderings,” J. Math. Phys. 29, 1727–1731 (1988)] showed that linear combinations of words qkpnqn−k, where p and q are subject to the relation qp − pq = i, may be expressed as a polynomial in the symbol z=12(qp+pq). Relations between such polynomials and linear combinations of the transformed coefficients are explored. In particular, examples yielding orthogonal polynomials are provided.

The Stern Diatomic Sequence via Generalized Chebyshev Polynomials

Abstract Let a(n) be the Stern diatomic sequence, and let x1,…, xr be the distances between successive 1s in the binary expansion of the (odd) positive integer n. We show that a(n) is obtained by evaluating generalized Chebyshev polynomials when the variables are given the values x1 + 1,…, xr + 1. We also derive a formula expressing the same polynomials in terms of sets of increasing integers of alternating parity and derive a determinant representation for a(n).

Wilf′s Conjecture

Abstract In a Note in this MONTHLY, Klazar raised the question of whether the alternating sum of the Stirling numbers of the second kind is ever zero for n ≠ 2. In this article, we present the history of this problem, and an economical account of a recent proof that there is at most one n ≠ 2 for which B± (n) =0.

Euler polynomials and identities for non-commutative operators

Three kinds of identities involving non-commutating operators and Euler and Bernoulli polynomials are studied. The first identity, as given by Bender and Bettencourt [Phys. Rev. D 54(12), 7710-7723 (1996)], expresses the nested commutator of the Hamiltonian and momentum operators as the commutator of the momentum and the shifted Euler polynomial of the Hamiltonian. The second one, by Pain [J. Phys. A: Math. Theor. 46, 035304 (2013)], links the commutators and anti-commutators of the monomials of the position and momentum operators. The third appears in a work by Figuieira de Morisson and Fring [J. Phys. A: Math. Gen. 39, 9269 (2006)] in the context of non-Hermitian Hamiltonian systems. In each case, we provide several proofs and extensions of these identities that highlight the role of Euler and Bernoulli polynomials.

Small Denominators: No Small Problem

It is always assumed that when either a or b is a fraction, it is in reduced form. The special case a= 9 and b = 7 appeared as a problem in [2]. As often happens in mathematics, the simplicity with which the problem is stated belies the complexity of solving it. We will observe interesting connections among the solution of (P), Farey sequences, and continued fractions. Many of these connections lead to good classroom problems in elementary number theory and computer science. Diligent readers will uncover some unanswered problems of their own. Is F(a, b) a function? The existence of a minimal denominator is ensured by the Well-Ordering Property of the natural numbers. We establish uniqueness of F(a, b) in the following proposition.