Binomial Identities Research Articles

On $$q$$ q -analogues of two-one formulas for multiple harmonic sums and multiple zeta star values

Recently, the present authors jointly with Tauraso found a family of binomial identities for multiple harmonic sums (MHS) on strings \((\{2\}^a,c,\{2\}^b)\) that appeared to be useful for proving new congruences for MHS as well as new relations for multiple zeta values. Very recently, Zhao generalized this set of MHS identities to strings with repetitions of the above patterns and, as an application, proved the two-one formula for multiple zeta star values conjectured by Ohno and Zudilin. In this paper, we extend our approach to \(q\)-binomial identities and prove \(q\)-analogues of two-one formulas for multiple zeta star values.

Some Binomial Identities Arising from a Partition of an n -Dimensional Cube

Some Binomial Identities Arising from a Partition of an <i>n</i> -Dimensional Cube

A short proof of a symmetry identity for the $q$-Hahn distribution

We give a short and elementary proof of a $(q, \mu, \nu)$-deformed Binomial distribution identity arising in the study of the $(q, \mu, \nu)$-Boson process and the $(q, \mu, \nu)$-TASEP. This identity found by Corwin in [4] was a key technical step to prove an intertwining relation between the Markov transition matrices of these two classes of discrete-time Markov chains. This was used in turn to derive exact formulas for a large class of observables of both these processes.

The uniformization of certain algebraic hypergeometric functions

The hypergeometric functions Fn−1n are higher transcendental functions, but for certain parameter values they become algebraic, because the monodromy of the defining hypergeometric differential equation becomes finite. It is shown that many algebraic Fn−1nʼs, for which the finite monodromy is irreducible but imprimitive, can be represented as combinations of certain explicitly algebraic functions of a single variable; namely, the roots of trinomials. This generalizes a result of Birkeland, and is derived as a corollary of a family of binomial coefficient identities that is of independent interest. Any tuple of roots of a trinomial traces out a projective algebraic curve, and it is also determined when this so-called Schwarz curve is of genus zero and can be rationally parametrized. Any such parametrization yields a hypergeometric identity that explicitly uniformizes a family of algebraic Fn−1nʼs. Many examples of such uniformizations are worked out explicitly. Even when the governing Schwarz curve is of positive genus, it is shown how it is sometimes possible to construct explicit single-valued or multivalued parametrizations of individual algebraic Fn−1nʼs, by parametrizing a quotiented Schwarz curve. The parametrization requires computations in rings of symmetric polynomials.

New properties of multiple harmonic sums modulo 𝑝 and 𝑝-analogues of Leshchiner’s series

In this paper we present some new binomial identities for multiple harmonic sums whose indices are the sequences ( { 1 } a , c , { 1 } b ) , (\{1\}^a,c,\{1\}^b), ( { 2 } a , c , { 2 } b ) (\{2\}^a,c,\{2\}^b) and prove a number of congruences for these sums modulo a prime p . p. The congruences obtained allow us to find nice p p -analogues of Leshchiner’s series for zeta values and to refine a result due to M. Hoffman and J. Zhao about the set of generators of the multiple harmonic sums of weight 7 7 and 9 9 modulo p p . As a further application we provide a new proof of Zagier’s formula for ζ ∗ ( { 2 } a , 3 , { 2 } b ) \zeta ^{*}(\{2\}^a,3,\{2\}^b) based on a finite identity for partial sums of the zeta-star series.

Monogenic pseudo-complex power functions and their applications

The use of a non-commutative algebra in hypercomplex function theory requires a large variety of different representations of polynomials suitably adapted to the solution of different concrete problems. Naturally arises the question of their relationships and the advantages or disadvantages of different types of polynomials. In this sense, the present paper investigates the intrinsic relationship between two different types of monogenic Appell polynomials. Several authors payed attention to the construction of complete sets of monogenic Appell polynomials, orthogonal with respect to a certain inner product, and used them advantageously for the study of problems in 3D-elasticity and other problems. Our goal is to show that, as consequence of the binomial nature of those generalized Appell polynomials, their inner structure is determined by interesting combinatorial relations in which the central binomial coefficients play a special role. As a byproduct of own interest, a Riordan–Sofo type binomial identity is also proved. Copyright © 2013 John Wiley & Sons, Ltd.

Combinatorial proofs and algebraic proofs — I

In Part I of this article we considered some binomial identities and also some identities involving the Fibonacci numbers, and proved them using methods which we described as ‘largely’ combinatorial. Now we shift our focus to number theory and to prime numbers in particular, and showcase some proofs having a strong combinatorial element. Throughout this article, p denotes an odd prime.

A Probabilistic Proof of a Binomial Identity

We give an elementary probabilistic proof of a binomial identity. The proof is obtained by computing the probability of a certain event in two different ways, yielding two different expressions for the same quantity.

A Binomial Identity via Differential Equations

In the following we discuss a well-known binomial identity. Many proofs by different methods are known for this identity. Here we present another proof, which uses linear ordinary differential equations of the first order.

Generalized falling factorial and associated binomial coefficient identities of Pascal's type

Generalized falling factorial and associated binomial coefficient identities of Pascal's type

New families of completely regular codes and their corresponding distance regular coset graphs

In this paper three new infinite families of linear binary completely regular codes are constructed. They have covering radius ? = 3 and 4, and are halves of binary Hamming and binary extended Hamming codes of length n = 2 m ?1 and 2 m , where m is even. There are also shown some combinatorial (binomial) identities which are new, to our knowledge.These completely regular codes induce, in the usual way, i.e., as coset graphs, three infinite families of distance-regular graphs of diameter three and four. This description of such graphs is new.

Series transformation formulas of Euler type, Hadamard product of series, and harmonic number identities

The Hadamard multiplication theorem for series is used to establish several Euler-type series transformation formulas. As applications we obtain a number of binomial identities involving harmonic numbers and an identity for the Laguerre polynomials. We also evaluate in a closed form certain power series with harmonic numbers.

An algebraic identity on [formula omitted]-Apéry numbers

By means of partial fraction decomposition, we establish a q -extension of an algebraic identity on rational function due to Chu [W. Chu, A binomial coefficient identity associated with Beukers’ conjecture on Apéry numbers, The Electronic Journal of Combinatorics 11 (2004) #N15]. Its limiting case as q → 1 leads to a harmonic number identity closely related to Beukers’ well-known conjecture on Apéry numbers.

Combinatorial and Automated Proofs of Certain Identities

This paper focuses on two binomial identities. The proofs illustrate the power and elegance in enumerative/algebraic combinatorial arguments, modern machine-assisted techniques of Wilf-Zeilberger and the classical tools of generatingfunctionology.

A family of identities related to zero-sum and team games

We list and prove a family of binomial identities by calculating in two ways the probabilities of approximate saddlepoints occurring in random m × n matrices. The identities are easily seen to be equivalent to the evaluation of a family of Gauss 2 F 1 polynomials according to a formula of Vandermonde. We also consider some implications concerning the number of approximate pure strategy Nash equilibria we can expect in large matrix zero-sum and team games.

Binomial Coefficients Modulo a Prime — A Visualization Approach to Undergraduate Research

In this article we present, as a case study, results of undergraduate research involving binomial coefficients modulo a prime p. We will discuss how undergraduates were involved in the project, even with a minimal mathematical background beforehand. There are two main avenues of exploration described to discover these binomial identities. The first uses a p-adic approach. The second, and more modern, utilizes the self-similarity, i.e., fractal nature, of Pascal's triangle modulo a prime. We will also discuss strategies for involving mathematics students in undergraduate research and provide some ideas which can be incorporated in several of the standard courses in the undergraduate mathematics curriculum. In particular, the identities and techniques discussed can serve as a bridge between classroom activities and future undergraduate research projects.

Hybrid Proofs of the $q$-Binomial Theorem and Other Identities

We give "hybrid" proofs of the $q$-binomial theorem and other identities. The proofs are "hybrid" in the sense that we use partition arguments to prove a restricted version of the theorem, and then use analytic methods (in the form of the Identity Theorem) to prove the full version. We prove three somewhat unusual summation formulae, and use these to give hybrid proofs of a number of identities due to Ramanujan. Finally, we use these new summation formulae to give new partition interpretations of the Rogers-Ramanujan identities and the Rogers-Selberg identities.

Some Binomial Identities Associated with the Generalized Natural Number Sequence

Some Binomial Identities Associated with the Generalized Natural Number Sequence

Constructing Correlation Immune Symmetric Boolean Functions

A Boolean function is said to be correlation immune if its output leaks no information about its input values. Such functions have many applications in computer security practices including the construction of key stream generators from a set of shift registers. Finding methods for easy construction of correlation immune Boolean functions has been an active research area since the introduction of the notion by Siegenthaler. In this paper, we present several constructions of nonpalindromic correlation immune symmetric Boolean functions. Our methods involve finding binomial coefficient identities and obtaining new correlation immune functions from known correlation immune functions. We also consider the construction of higher order correlation immunity symmetric functions and propose a class of third order correlation immune symmetric functions on n variables, where n+1(≥9) is a perfect square.

Binomial Coefficient – Harmonic Sum Identities Associated to Supercongruences

AbstractWe establish two binomial coefficient-generalized harmonic sum identities using the partial fraction decomposition method. These identities are a key ingredient in the proofs of numerous supercongruences. In particular, in other works of the author, they are used to establish modulo