Some Remarkable Identities Involving Numbers

The article focuses on simple identities found for binomials, their divisibility, and basic inequalities. A general formula allowing factorization of the sum of like powers is introduced and used to prove elementary theorems for natural numbers. Formulas for short multiplication are sometimes referred in English or French as remarkable identities. The same formulas could be found in works concerning polynomial factorization, where there exists no single term for various identities. Their usability is not questionable, and they have been successfully utilized since for ages. For example, in his books published in 1731 (p. 385), Edward Hatton [3] wrote: “Note, that the differences of any two like powers of two quantities, will always be divided by the difference of the quantities without any remainer...”. Despite of its conceptual simplicity, the problem of factorization of sums/differences of two like powers could still be analyzed [7], giving new and possibly interesting results [6].Department of Carbohydrate Technology University of Agriculture Krakow, PolandGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.E. Hatton. An intire system of Arithmetic: or, Arithmetic in all its parts. Number 6. Printed for G. Strahan, 1731. http://books.google.pl/books?id=urZJAAAAMAAJRafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.M.I. Mostafa. A new approach to polynomial identities. The Ramanujan Journal, 8(4): 423-457, 2005. ISSN 1382-4090. doi:10.1007/s11139-005-0272-3.Werner Georg Nowak. On differences of two k-th powers of integers. The Ramanujan Journal, 2(4):421-440, 1998. ISSN 1382-4090. doi:10.1023/A:1009791425210.Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990

Lattice walk area combinatorics, some remarkable trigonometric sums and Ap\'ery-like numbers

Explicit algebraic area enumeration formulae are derived for various lattice walks generalizing the canonical square lattice walk, and in particular for the triangular lattice chiral walk recently introduced by the authors. A key element in the enumeration is the derivation of some remarkable identities involving trigonometric sums --which are also important building blocks of non trivial quantum models such as the Hofstadter model-- and their explicit rewriting in terms of multiple binomial sums. An intriguing connection is also made with number theory and some classes of Ap\'ery-like numbers, the cousins of the Ap\'ery numbers which play a central role in irrationality considerations for {\zeta}(2) and {\zeta}(3).Comment: 31 pages, 4 figure

Polynomial Moments with a weighted Zeta Square measure on the critical line

We prove closed-form identities for the sequence of moments $\int t^{2n}|\Gamma(s)\zeta(s)|^2dt$ on the whole critical line $s=1/2+it$. They are finite sums involving binomial coefficients, Bernoulli numbers, Stirling numbers and $\pi$, especially featuring the numbers $\zeta(n)B_n/n$ unveiled by Bettin and Conrey in 2013. Their main power series identity allows for a short proof of an auxiliary result: the computation of the $k$-th derivatives at $1$ of the "exponential auto-correlation" function studied in a recent paper by the authors. We also provide an elementary and self-contained proof of this secondary result. The starting point of our work is a remarkable identity proven by Ramanujan in 1915. The sequence of moments studied here, not to be confused with the moments of the Riemann zeta function, entirely characterizes $|\zeta|$ on the critical line. They arise in some generalizations of the Nyman-Beurling criterion, but might be of independent interest for the numerous connections concerning the above mentioned numbers

Noncommutative Catalan numbers

The goal of this paper is to introduce and study noncommutative Catalan numbers $C_n$ which belong to the free Laurent polynomial algebra in $n$ generators. Our noncommutative numbers admit interesting (commutative and noncommutative) specializations, one of them related to Garsia-Haiman $(q,t)$-versions, another -- to solving noncommutative quadratic equations. We also establish total positivity of the corresponding (noncommutative) Hankel matrices $H_m$ and introduce accompanying noncommutative binomial coefficients.Comment: 12 pages AM LaTex, a picture and proof of Lemma 3.6 are added, misprints correcte

Toric Varieties with NC Toric Actions: NC Type IIA Geometry

Extending the usual $\mathbf{C}^{\ast r}$ actions of toric manifolds by allowing asymmetries between the various $\mathbf{C}^{\ast}$ factors, we build a class of non commutative (NC) toric varieties $\mathcal{V}$. We construct NC complex $d$ dimension Calabi-Yau manifolds embedded in $\mathcal{V}_{d+1}^{(nc)}$ by using the algebraic geometry method. Realizations of NC $\mathbf{C}^{\ast r}$ toric group are given in presence and absence of quantum symmetries and for both cases of discrete or continuous spectrums. We also derive the constraint eqs for NC Calabi-Yau backgrounds $\mathcal{M}_{d}^{nc}$ embedded in $\mathcal{V}_{d+1}^{nc}$ and work out their solutions. The latters depend on the Calabi-Yau condition $\sum_{i}q_{i}^{a}=0$, $q_{i}^{a}$ being the charges of $\mathbf{C}^{\ast r}$% ; but also on the toric data ${q_{i}^{a},\nu_{i}^{A};p_{I}^{\alpha},\nu _{iA}^{\ast}}$ of the polygons associated to $\mathcal{V}$. Moreover, we study fractional $D$ branes at singularities and show that, due to the complete reducibility property of $\mathbf{C}^{\ast r}$ group representations, there is an infinite number of fractional $D$ branes. We also give the generalized Berenstein and Leigh quiver diagrams for discrete and continuous $\mathbf{C}^{\ast r}$ representation spectrums. An illustrating example is presented.Comment: 25 pages, no figure