Polynomial generalizations of all 130 of the identities in Slater’s list of identities of the Rogers-Ramanujan type are presented. Furthermore, duality relationships among many of the identities are derived. Some of the these polynomial identities were previously known but many are new. The author has implemented much of the finitization process in a Maple package which is available for free do...

In this paper, we present some formulas of symbolic operator summation, which involving Generalization well-know number sequences or polynomial sequences, and mean while we obtain some identities about the sequences by employing M-R‘s substitution rule. Keywords—Generating functions; Operators sequence group; Riordan arrays; R.G Operator group; Combinatorial identities.

We show the equivalence of the Pieri formula for ag manifolds with certain identities among the structure constants for the Schubert basis of the polynomial ring. This gives new proofs of both the Pieri formula and of these identities. A key step is the association of a symmetric function to a nite poset with labeled Hasse diagram satisfying a symmetry condition. This gives a uniied deenition o...

We consider polynomial identities satisfied by nonhomogeneous subalgebras of Lie and special Jordan superalgebras: we ignore the grading and regard the superalgebra as an ordinary algebra. The Lie case has been studied by Volichenko and Baranov: they found identities in degrees 3, 4 and 5 which imply all the identities in degrees ≤ 6. We simplify their identities in degree 5, and show that ther...

In order to find a general framework for different combinatorial identities and identities that involve special functions, Bernshtein introduced in [1] a notion of holonomic functions. Let us recall this definition (in formulating it, we will follow [10]). A discrete holonomic function f(n) of one variable is defined as a solution of a homogeneous, linear difference equation with polynomial coe...

We show the equivalence of the Pieri formula for flag manifolds with certain identities among the structure constants for the Schubert basis of the polynomial ring. This gives new proofs of both the Pieri formula and of these identities. A key step is the association of a symmetric function to a finite poset with labeled Hasse diagram satisfying a symmetry condition. This gives a unified defini...

We introduce degree n Sabinin algebras, which are defined by the polynomial identities up to degree n in a Sabinin algebra. Degree 4 Sabinin algebras can be characterized by the polynomial identities satisfied by the commutator, associator and two quaternators in the free nonassociative algebra. We consider these operations in a free power associative algebra and show that one of the quaternato...

We conjecture polynomial identities which imply Rogers–Ramanujan type identities for branching functions associated with the cosets (G)l−1⊗(G )1/(G )l, with G=An−1 (l ≥ 2), Dn−1 (l ≥ 2), E6,7,8 (l = 2). In support of our conjectures we establish the correct behaviour under level-rank duality for G=An−1 and show that the A-D-E Rogers–Ramanujan identities have the expected q → 1− asymptotics in t...

The Martin polynomials, introduced by Martin in his 1977 thesis, encode information about the families of circuits in Eulerian graphs and digraphs. The circuit partition polynomials, J (G;x) and j ( G;x), are simple transformations of the Martin polynomials. We give new identities for these polynomials, analogous to Tutte’s identity for the chromatic polynomial. Following a useful expansion of ...

We study the complexity of generating identities of matrix rings. We establish an unconditional lower bound on the minimal number of generators needed to generate a matrix identity, where the generators are substitution instances of elements from any given finite basis of the matrix identities. Based on our findings, we propose to consider matrix identities (and their encoding) as hard instance...