Uniform proofs of q-Series-product identities

In this paper we give a simple proof of the Jacobi triple product identity by using basic properties of cube roots of unity. Then we give a new proof of the quintuple product identity, the septuple product identity and Winquist’s identity by using the Jacobi triple product identity and basic properties of cube and fifth roots of unity. Furthermore, we derive some new product identities by this uniform method. Later, we give some generalizations of those identities. Lastly, we derive some modular equations.     

Bijective proofs of Gould's and Rothe's identities

We first give a bijective proof of Gould's identity in the model of binary words. Then we deduce Rothe's identity from Gould's identity again by a bijection, which also leads to a double-sum extension of the q-Chu-Vandermonde formula.     

Combinatorial proofs of some determinantal identities

In this paper, we give combinatorial proofs of some determinantal identities. In fact, we give a combinatorial proof of a theorem of R. P. Stanley regarding the enumeration of paths in acyclic digraphs along with some interesting applications. We also give an almost visual proof of a recent result of Oliver Knill, namely ‘The generalized Cauchy–Binet Theorem.’     

Bijective proofs of some classical partition identities

A bijective proof of a general partition theorem is given which has as direct corollaries many classical partition theorems due to Euler, Glaisher, Schur, Andrews, Subbarao, and others. It is shown that the bijective proof specializes to give bijective proofs of these classical results and moreover the bijections which result often coincide with bijections which have occurred in the literature. Also given are some sufficient conditions for when two classes of words omitting certain sequences of words are in bijection.     

Bijective proofs of shifted tableau and alternating sign matrix identities

We give a bijective proof of an identity relating primed shifted gl(n)-standard tableaux to the product of a gl(n) character in the form of a Schur function and . This result generalises a number of well-known results due to Robbins and Rumsey, Chapman, Tokuyama, Okada and Macdonald. An analogous result is then obtained in the case of primed shifted sp(2n)-standard tableaux which are bijectively related to the product of a t-deformed sp(2n) character and . All results are also interpreted in terms of alternating sign matrix (ASM) identities, including a result regarding subsets of ASMs specified by conditions on certain restricted column sums.     

Beck-type identities: new combinatorial proofs and a modular refinement

The Ramanujan Journal - Let $${\mathcal {O}}_r(n)$$ be the set of r-regular partitions of n, $${\mathcal {D}}_r(n)$$ the set of partitions of n with parts repeated at most $$r-1$$ times,...     

Bijective proofs of partition identities arising from modular equations

We establish generalizations of certain partition theorems originating with modular equations and give bijective proofs for them. As a special case, we give a bijective proof of the Farkas and Kra partition theorem modulo 7.     

Semigroup identities in the monoid of triangular tropical matrices

We show that the submonoid of all n×n triangular tropical matrices satisfies a nontrivial semigroup identity and provide a generic construction for classes of such identities. The utilization of the Fibonacci number formula gives us an upper bound on the length of these 2-variable semigroup identities.     

Semigroup identities in the monoid of two-by-two tropical matrices

We show that the monoid $M_{2}(\mathbb {T})$ of 2×2 tropical matrices is a regular semigroup satisfying the semigroup identity $$A^2B^4A^2A^2B^2A^2B^4A^2=A^2B^4A^2B^2A^2A^2B^4A^2.$$ Studying reduced identities for subsemigroups of $M_{2}(\mathbb {T})$ , and introducing a faithful semigroup representation for the bicyclic monoid by 2×2 tropical matrices, we reprove Adjan’s identity for the bicyclic monoid in a much simpler way.     

Formal proofs of operator identities by a single formal computation

A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two operators can be added or composed. Algebraically, identities can be modelled by noncommutative polynomials and such a formal computation proves that the polynomial corresponding to the new identity lies in the ideal generated by the polynomials corresponding to the known identities. In order to prove an operator identity, however, just proving membership of the polynomial in the ideal is not enough, since the ring of noncommutative polynomials ignores domains and codomains. We show that it suffices to additionally verify compatibility of this polynomial and of the generators of the ideal with the labelled quiver that encodes which polynomials can be realized as linear operators. Then, for every consistent representation of such a quiver in a linear category, there exists a computation in the category that proves the corresponding instance of the identity. Moreover, by assigning the same label to several edges of the quiver, the algebraic framework developed allows to model different versions of an operator by the same indeterminate in the noncommutative polynomials.     

Computer-assisted proofs of special function identities related to Poisson integrals

As a by-product of his work, E. Symeonidis obtained in (Comment. Math. Univ. Carol. 44(3):437–460, 2003) indirect proofs of two interesting special function identities involving Gegenbauer polynomials. In (Muldoon, 2004 ) the question of direct proofs was posed. We answer this question by presenting proofs obtained with the help of computer algebra algorithms based on WZ theory.     

New proofs of identities of Lebesgue and Göllnitz via tilings

In 1840, V.A. Lebesgue proved the following two series-product identities:

     

Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians

The classic Cayley identity states that

det(∂)(detX)^s=s(s+1)?(s+n−1)(detX)^s−1$\det (\partial ){(\det X)}^s=s(s+1)?(s+n-1){(\det X)}^{s-1}$

where X=(_xij)$X=(x_{ij})$ is an n×n$n\times n$ matrix of indeterminates and ∂=(∂/∂_xij)$\partial =(\partial /\partial x_{ij})$ is the corresponding matrix of partial derivatives. In this paper we present straightforward algebraic/combinatorial proofs of a variety of Cayley-type identities, both old and new. The most powerful of these proofs employ Grassmann algebra (= exterior algebra) and Grassmann–Berezin integration. Among the new identities proven here are a pair of “diagonal-parametrized” Cayley identities, a pair of “Laplacian-parametrized” Cayley identities, and the “product-parametrized” and “border-parametrized” rectangular Cayley identities.