Asymptotics for the reciprocal and shifted quotient of the partition function

Let p ( n ) denote the partition function. In this paper our main goal is to derive an asymptotic expansion up to order N (for any fixed positive integer N ) along with estimates for error bounds for the shifted quotient of the partition function, namely $$p(n+k)/p(n)$$ p ( n + k ) / p ( n ) with k∈ N k ∈ N , which generalizes a result of Gomez, Males, and Rolen. In order to do so, we derive asymptotic expansions with error bounds for the shifted version $$p(n+k)$$ p ( n + k ) and the multiplicative inverse 1/ p ( n ), which is of independent interest.