Related papers: Identities Involving Zeros of Ramanujan and Shanks…
A polynomial x^3+px^2+qx+r with the condition pr^(1/3)+ 3r^(2/3)+q=0 we call a Ramanujan cubic polynomial (RCP). We study different interest properties of RCP, in particular, an important role of a parameter pq/r. We prove some new…
Presented are polynomial identities which imply generalizations of Euler and Rogers--Ramanujan identities. Both sides of the identities can be interpreted as generating functions of certain restricted partitions. We prove the identities by…
Ramanujan listed several q-series identities in his lost notebook. The most well known q-series identities are the Rogers-Ramanujan type identities which are first discovered by Rogers and then rediscovered by Ramanujan. In this paper, we…
In 1914, Ramanujan gave a list of 17 identities expressing $1/\pi$ as linear combinations of values of hypergeometric functions at certain rational numbers. Since then, identities of similar nature have been discovered by many authors.…
By using some hypergeometric series identities, we prove two supercongruences on truncated hypergeometric series, one of which is related to a modular Calabi--Yau threefold, and the other is regarded as $p$-adic analogue of an identity due…
We prove two partition identities which are dual to the Rogers-Ramanujan identities. These identities are inspired by (and proved using) a correspondence between three kinds of objects: a new type of partitions (neighborly partitions),…
We give a linear relation between a cubic Gaussian period and a root of Shanks' cubic polynomial in wildly ramified cases.
Starting with Ramanujan's famous taxicab problem, we can study the solvability of the equations $p^n+q^n=r^n+s^n$ and, more generally, $p_1^{k_1}+\dots+p_m^{k_m}=0$ among polynomials.
We find new hypergeometric identities which, in a certain aspect, are stron-ger than others of the same style found by the author in a previous paper. The identities in Section \ref{section-pi} are related to some Ramanujan-type series for…
We give several expansion and identities involving the Ramanujan function $A_q$ and the Stieltjes--Wigert polynomials. Special values of our idenitities give $m$-versions of some of the items on the Slater list of Rogers-Ramanujan type…
We study the generating functions for cylindric partitions with profile $(c_1,c_2,c_3)$ for all $c_1,c_2,c_3$ such that $c_1+c_2+c_3=5$. This allows us to discover and prove seven new $A_2$ Rogers-Ramanujan identities modulo $8$ with…
A cubic Galois polynomial is a cubic polynomial with rational coefficients that defines a cubic Galois field. Its discriminant is a full square and its roots $x_1,x_2,x_3$ (enumerated in some order) are real. There exists (and only one)…
We deduce new q-series identities by applying inverse relations to certain identities for basic hypergeometric series. The identities obtained themselves do not belong to the hierarchy of basic hypergeometric series. We extend two of our…
We prove four new Rogers-Ramanujan-type identities for double series. They follow from the classical Rogers-Ramanujan identities using the constant term method and properties of Rogers-Szeg\H{o} polynomials.
We show that every monic polynomial of degree three with complex coefficients and no repeated roots is either a (vertical and horizontal) translation of $y=x^3$ or can be composed with a linear function to obtain a Ramanujan cubic. As a…
Many classical $q$-series identities, such as the Rogers--Ramanujan identities, yield combinatorial interpretations in terms of integer partitions. Here we consider algebraically manipulating some of the classical $q$-series to yield…
Ramanujan's notebooks contain many elegant identities and one of the celebrated identities is a formula for $\zeta(2k+1)$. In 1972, Grosswald gave an extension of the Ramanujan's formula for $\zeta(2k+1)$, which contains a polynomial of…
We derive the polynomial representations for minimal relations of generating set of numerical semigroups R_n^k=<(n-1)^k,n^k,(n+1)^k>, k=2,3,4, n>2. We find also the polynomial representations for degrees of syzygies in the Hilbert series…
We investigate the polynomials $\sum_{k=0}^{n-1} c_n(k)x^k$ and $\sum_{k=0}^{n-1} |c_n(k)| x^k$, where $c_n(k)$ denote the Ramanujan sums. We point out connections and analogies to the cyclotomic polynomials.
Let $p(n)$ be the number of partition of a positive integer $n$. We derive a new identity for complete Bell polynomials based on a generating function of $p(7n+5)$ given by Ramanujan.