Related papers: On Ramanujan's cubic composition formula
We give a generalization of a Ramanujan's exercise for high school students. Our results can be regarded as a variation of the factorization formula of $x^{n} - 1$.
Ramanujan Master Theorem is a technique developed by the indian mathematician S. Ramanujan to evaluate a class of definite integrals. This technique is used here to calculate the values of integrals associated with specific Feynman…
We give an intrinsic characterization of all subsets of a doubling metric space that can arise as a member of some system of dyadic cubes on the underlying space, as constructed by M. Christ.
In this article we establish some properties regarding the solutions of a linear congruence, bases of solutions of a linear congruence, and the finding of other solutions starting from these bases.
Let R be a ring. A construction method for flexible quadratic algebras with scalar involution over R is presented which unifies various classical constructions in the literature, in particular those to construct composition algebras.
We prove two new series of Ramanujan type for $1/\pi^2$.
We present an infinite family of identities that represent Ramanujan's tau function in terms of convolution sums of twisted divisor functions. Our method involves explicitly constructing non-vanishing level $1$ cusp forms from modular forms…
In this paper we first prove an isomorphism between certain spaces of Jacobi forms. Using this isomorphism, we study the mod $p$ theory of Hermitian Jacobi forms over $\mathbb{Q}(i)$. We then apply the mod $p$ theory of Hermitian Jacobi…
Ramanujan's trigonometric sum $c_q(n)$ can be interpreted as a set of trigonometric moments of a finite measure concentrated at primitive $q$-th roots of unity with equal masses. This gives rise to sets of corresponding polynomials…
We revisit several entries from Ramanujan's notebooks which follow from more elementary arguments than a first glance may suggest. Our goal is to demystify these results through more accessible proofs, while also shining some light on the…
We determine conditions for the existence and non-existence of Ramanujan-type congruences for Jacobi forms. We extend these results to Siegel modular forms of degree 2 and as an application, we establish Ramanujan-type congruences for…
Quadratic forms over Z that represent all positive integers are called universal. Starting with Ramanujan, 54 universal quaternary quadratic forms without cross product terms were discovered. The form that is the sum of four squares was…
We study a number of possible extensions of the Ramanujan master theorem, which is formulated here by using methods of Umbral nature. We discuss the implications of the procedure for the theory of special functions, like the derivation of…
We describe two distinct simple, short and self contained proofs of the composition lemma.
We provide direct elementary proofs of several explicit expressions for Bernoulli numbers and Bernoulli polynomials. As a byproduct of our method of proof, we provide natural definitions for generalized Bernoulli numbers and polynomials of…
Spinors are used in physics quite extensively. The goal of this study is also the spinor structure lying in the basis of the quaternion algebra. In this paper, first, we have introduced spinors mathematically. Then, we have defined…
We present an explicit formula for Witten-Kontsevich tau-function.
Algebraic hyperstructures represent a natural extension of classical algebraic structures. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two…
We prove a closed formula for the derivative, of any order, of a implicit function, in terms of some binomial building blocks, and explain the combinatorics behind the coefficients appearing in the formula.
We obtain explicit upper bounds for the number of irreducible factors for a class of compositions of polynomials in several variables over a given field. In particular, some irreducibility criteria are given for this class of compositions…