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

Arithmetic Properties of Partition Quadruples With Odd Parts Distinct

Let $\mathrm{pod}_{-4}(n)$ denote the number of partition quadruples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of $\mathrm{pod}_{-4}(n)$ involving the following infinite family of congruences: for any integers $\alpha \ge 1$ and $n \ge 0$, $$\mathrm{pod}_{-4}\Big({{3}^{\alpha +1}}n+\frac{5\cdot {{3}^{\alpha }}+1}{2}\Big)\equiv 0 \pmod{9}.$$ We also establish some internal congruences and some congruences modulo 2, 5 and 8 satisfied by $\mathrm{pod}_{-4}(n)$.Comment: 10 page

Arithmetic properties of �-regular overpartition pairs

In this paper, we investigate the arithmetic properties of � -regular overpartition pairs. Let B�(n) denote the number of � -regular overpartition pairs of n. We will prove the number of Ramanujan-like congruences and infinite families of congruences modulo 3, 8, 16, 36, 48, 96 for B3(n) and modulo 3, 16, 64, 96 for B4(n) . For example, we find that for all nonnegative integers α and n, B3(3α(3n + 2)) � 0 (mod 3), B3(3α(6n + 4)) � 0 (mod 3), and B4(8n + 7) � 0 (mod 64). © T�BI�TAK

Infinite product representations of some q-series

For integers $a$ and $b$ (not both $0$) we define the integers $c(a,b;n)\ \ (n=0,1,2,\ldots)$ by \sum_{n=0}^{infty} c(a,b,;n)q^n = \prod_{n=1}^\infty \left(1-q^n\right)^a (1-q^{2n})^b \quad (|q|<1). These integers include the numbers $t_k(n) = c(-k,2k;n)$, which count the number of representations of $n$ as a sum of $k$ triangular numbers, and the numbers $(-1)^n r_k(n) = c(2k,-k;n)$, where $r_k(n)$ counts the number of representations of $n$ as a sum of $k$ squares. A computer search was carried out for integers $a$ and $b$, satisfying $-24\leq a,b\leq 24$, such that at least one of the sums \begin{align} \sum_{n=0}^{infty} c(a,b;3n+j)q^n, \quad j=0,1,2, \end{align} (0.1) is either zero or can be expressed as a nonzero constant multiple of the product of a power of $q$ and a single infinite product of factors involving powers of $1-q^{rn}$ with $r\in\{1,2,3,4,6,8,12,24\}$ for all powers of $q$ up to $q^{1000}$. A total of 84 such candidate identities involving 56 pairs of integers $(a,b)$ all satisfying a\equiv b\pmd3 were found and proved in a uniform manner. The proof of these identities is extended to establish general formulas for the sums (0.1). These formulas are used to determine formulas for the sums \[\sum_{n=0}^{infty} t_k(3n+j)q^n, \quad \sum_{n=0}^{infty} r_k(3n+j)q^n, \quad j=0,1,2. \

RELATIONS BETWEEN SQUARES AND TRIANGLES

With rk(n) denoting the number of representations of n as the sum of k squares and tk(n) the number of representations of n as the sum of k triangular numbers, we prove the relation rk(8n + k) = 2 k−1 Classification: 11E2