Related papers: Bilateral Bailey pairs and Rogers-Ramanujan type i…
We use the classical umbral calculus to describe Riordan arrays. Here, a Riordan array is generated by a pair of umbrae, and this provides efficient proofs of several basic results of the theory such as the multiplication rule, the…
We derive an identity connecting any two second-order linear recurrence sequences having the same recurrence relation but whose initial terms may be different. Binomial and ordinary summation identities arising from the identity are…
We use both Abel's lemma on summation by parts and Zeilberger's algorithm to find recurrence relations for definite summations. The role of Abel's lemma can be extended to the case of linear difference operators with polynomial…
Using an elementary approach involving the Euler Beta function and the binomial theorem, we derive two polynomial identities; one of which is a generalization of a known polynomial identity. Two well-known combinatorial identities, namely…
We define a length function for a perfect crystal. As an application, we derive a variant of the Rogers-Ramanujan identities which involves (a $q$-analog of) the Fibonacci numbers.
The two Rogers-Ramanujan $q$-series \[ \sum_{n=0}^{\infty}\frac{q^{n(n+\sigma)}}{(1-q)\cdots (1-q^n)}, \] where $\sigma=0,1$, play many roles in mathematics and physics. By the Rogers-Ramanujan identities, they are essentially modular…
In this paper, we prove two identities on the partition of a totally positive algebraic integer over a totally real number field which are the generalization of the Sylvester Theorem and that of the Rogers-Ramanujan Identities.…
The q-generalizations of the two fundamental statements of matrix algebra -- the Cayley-Hamilton theorem and the Newton relations -- to the cases of quantum matrix algebras of an "RTT-" and of a "Reflection equation" types have been…
The Rogers-Ramanujan-Gordon identities generalize the classical partition identities discovered independently by L. J. Rogers and S. Ramanujan. In 2021, Afsharijoo provided a commutative algebra proof of the Rogers-Ramanujan-Gordon…
We define bilateral series related to Ramanujan-like series for $1/\pi^2$. Then, we conjecture a property of them and give some applications.
We obtain identities involving symmetric and doubly symmetric polynomials. These identities provide a way of handling expressions appearing in the Atiyah-Bott-Berline-Vergne formula for Grassmannians. As corollaries, we obtain formulas for…
In this paper we give a combinatorial proof and refinement of a Rogers-Ramanujan type partition identity of Siladi\'c arising from the study of Lie algebras. Our proof uses generating functions and $q$-difference equations.
A new type of polynomial analogue of the Rogers-Ramanujan identities is proven. Here the product-side of the Rogers-Ramanujan identities is replaced by a partial theta sum and the sum-side by a weighted sum over Schur polynomials.
We show that a strange group theoretical formula of Hall follows from the Rogers-Ramanujan identities.
Recently, Garvan obtained two-variable Hecke-Rogers identities for three universal mock theta functions $g_2(z;q),\,g_3(z;q),\,K(z;q)$ by using basic hypergeometric functions, and he proposed a problem of finding direct proofs of these…
Strict partitions are enumerated with respect to the weight, the number of parts, and the number of sequences of odd length. We write this trivariate generating function as a double sum $q$-series. Equipped with such a combinatorial set-up,…
Refinements of the classical Rogers-Ramanujan identities are given in which some parts are weighted. Combinatorial interpretations refining MacMahon's results are corollaries.
We provide the missing member of a family of four $q$-series identities related to the modulus 36, the other members having been found by Ramanujan and Slater. We examine combinatorial implications of the identities in this family, and of…
In this work, we construct a new Bailey pairs for the integral pentagon identity in terms of q-hypergeometric functions. The pentagon identity considered here represents equality of the partition functions of a certain three-dimensional…
Recently Cherednik and Feigin obtained several Rogers-Ramanujan type identities via the nilpotent double affine Hecke algebras (Nil-DAHA). These identities further led to a series of dilogarithm identities, some of which are known, while…