General relations between sums of squares and sums of triangular numbers

Let = ( 1, · · · , m) be a partition of k. Let r (n) denote the number of solutions in integers of 1x21 + · · · + mx2 m = n, and let t (n) denote the number of solutions in non negative integers of 1x1(x1 +1)/2+· · ·+ mxm(xm +1)/2 = n. We prove that if 1 k 7, then there is a constant c , depending only on , such that r (8n + k) = c t (n), for all integers n

A Note on 1-Edge Balance Index Set

A graph labeling is an assignment of integers to the vertices or edges or both, subject to certain conditions. Varieties of graph labeling have been investigated by many authors [2], [3] [5] and they serve as useful models for broad range of applications

Quintuple product identity as a special case of Ramanujan's 1ψ1 summation formula.

In this note we observe an interesting fact that the well-known quintuple product identity can be regarded as a special case of the celebrated 1ψ1 summation formula of Ramanujan which is known to unify the Jacobi triple product identity and the q-binomial theorem

On the constant term of the minimal polynomial of cos (2 pi/n) over Q

The algebraic numbers cos (2 pi/n) and 2 cos (pi/n) play an important role in the theory of discrete groups and has many applications because of their relation with Chebycheff polynomials. There are some partial results in literature for the minimal polynomial of the latter number over rationals until 2012 when a complete solution was given in [5]. In this paper we determine the constant term of the minimal polynomial of cos(2 pi/n) over Q by a new method

Modular relations for the Rogers-Ramanujan-Slater type functions of order fifteen and its applications to partitions

In a manuscript of Ramanujan, published with his Lost Notebook [20] there are forty identities involving the Rogers-Ramanujan functions. In this paper, we establish several modular relations involving the Rogers-Ramanujan functions and the Rogers-Ramanujan-Slater type functions of order fifteen which are analogues to Ramanujan’s well known forty identities. Furthermore, we give partition theoretic interpretations of two modular relations

Some new modular relations for the cubic functions

In this paper, we establish certain relations for the cubic functions \begin{linenomath*} \begin{align*} S(q):=& \sum_{n=0}^{\infty}\frac{(-q;q^2)_nq^{n^2+2n}}{(q^4;q^4)_n},\\ T(q):=& \sum_{n=0}^{\infty}\frac{q^{n^2}}{(q^2;q^2)_n} \end{align*}\end{linenomath*} which are analogous to Ramanujan's forty identities for the Rogers-Ramanujan functions. From our relations, we deduce some interesting color partition identities