  In this paper, a family of low-density parity-check (LDPC) degree
distributions, whose decoding threshold on the binary erasure channel (BEC)
admits a simple closed form, is presented. These degree distributions are a
subset of the check regular distributions (i.e. all the check nodes have the
same degree), and are referred to as $p$-positive distributions. It is given
proof that the threshold for a $p$-positive distribution is simply expressed by
$[\lambda'(0)\rho'(1)]^{-1}$. Besides this closed form threshold expression,
the $p$-positive distributions exhibit three additional properties. First, for
given code rate, check degree and maximum variable degree, they are in some
cases characterized by a threshold which is extremely close to that of the best
known check regular distributions, under the same set of constraints. Second,
the threshold optimization problem within the $p$-positive class can be solved
in some cases with analytic methods, without using any numerical optimization
tool. Third, these distributions can achieve the BEC capacity. The last
property is shown by proving that the well-known binomial degree distributions
belong to the $p$-positive family.
