Related papers: On Ramanujan's cubic composition formula
The Ramanujan sum $c_n(k)$ is defined as the sum of $k$-th powers of the primitive $n$-th roots of unity. We investigate arithmetic functions of $r$ variables defined as certain sums of the products $c_{m_1}(g_1(k))...c_{m_r}(g_r(k))$,…
In this note, we will give a short proof of an identity for cubic partitions.
We present several supercongruences that may be viewed as $p$-adic analogues of Ramanujan-type series for $1/\pi$ and $1/\pi^2$, and prove three of these examples.
We describe in details the nxnxn Rubik's Cube, namely a Rubik's Cube with n rotating slices in each face. Then we state and prove the "first law of Cubology", i.e. the solvability criterion, for it
This essay recounts my personal journey towards a deeper understanding of the mathematical foundations of algorithmic music composition. I do not spend much time on specific mathematical algorithms used by composers; rather, I focus on…
In this short note, we prove several new congruences for the overcubic partition triples function, using both elementary techniques and the theory of modular forms. These extend the recent list of such congruences given by Nayaka,…
In this paper we obtain some new transformation formula for Ramanujan summation formula and also establish some eta-function identities. we also deduce a q-Gamma function identity, n q-integral and some interesting series representation.
In this paper we establish several results concerning the generalized Ramanujan primes. For $n\in\mathbb{N}$ and $k \in \mathbb{R}_{> 1}$ we give estimates for the $n$th $k$-Ramanujan prime which lead both to generalizations and to…
Ramanujan wrote the following identity \begin{align*} \sqrt{2 \left(1 - \frac{1}{3^2}\right) \left(1 - \frac{1}{7^2}\right) \left(1 - \frac{1}{11^2}\right) \left(1 - \frac{1}{19^2}\right)} \ = \ \left(1 + \frac{1}{7}\right) \left(1 +…
Srinivasa Ramanujan posed a problem on infinite nested radical of the square root in the Journal of Indian Mathematical Society in 1911. He had generated the problem years before in the form of an example illustrating a more general…
We compute two parametric determinants in which rows and columns are indexed by compositions, where in one determinant the entries are products of binomial coefficients, while in the other the entries are products of powers. These results…
We have found several summation formulas that extend Ramanujan's psi sum. First contains a parameter $\alpha=1/N$, $N$ is a positive integer, and transforms to $q$-beta integral in the limit $N\to\infty$. The other is a $q$-analogue of…
Compositional data are commonly known as multivariate observations carrying relative information. Even though the case of vector or even two-factorial compositional data (compositional tables) is already well described in the literature,…
We define bilateral series related to Ramanujan-like series for $1/\pi^2$. Then, we conjecture a property of them and give some applications.
Roman logarithmic binomial formula analogue has been found . It is presented here also for the case of fibonomial coefficients which recently have been given a combinatorial interpretation by the present author.
We give a one-sentence elementary proof of the combinatorial Fa\`a di Bruno's formula.
An outline of the basic Riemannian structures underlying the separation of variables in the Hamilton-Jacobi equation of natural Hamiltonian systems.
We explain the use and set grounds about applicability of algebraic transformations of arithmetic hypergeometric series for proving Ramanujan's formulae for $1/\pi$ and their generalisations.
We address three questions posed by Bibak \cite{KB20}, and generalize some results of Bibak, Lehmer and K G Ramanathan on solutions of linear congruences $\sum_{i=1}^k a_i x_i \equiv b \Mod{n}$. In particular, we obtain explicit expressions…
We obtain a series of lower bounds for the product set of combinatorial cubes, as well as some non--trivial upper estimates for the multiplicative energy of such sets.