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Formulas evaluating differences of integer partitions according to the parity of the parts are referred to as Legendre theorems. In this paper we give some formulas of Legendre type for overpartitions.

For positive integers $n$ and $k$, let $r_{k}(n)$ denote the number of representations of $n$ as a sum of $k$ squares, where representations with different orders and different signs are counted as distinct. For a given positive integer $m$, by means of some properties of binomial coefficients, we derive some infinite families of congruences for $r_{k}(n)$ modulo $2^{m}$. Furthermore, in view of these arithmetic properties of $r_{k}(n)$, we establish many infinite families of congruences for the overpartition function and the overpartition pair function.

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THEOREMS OF LEGENDRE TYPE FOR OVERPARTITIONS

SUMS OF SQUARES AND PARTITION CONGRUENCES

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2 results

THEOREMS OF LEGENDRE TYPE FOR OVERPARTITIONS

SUMS OF SQUARES AND PARTITION CONGRUENCES