Partition identities II. The results of Bateman and Erdős

This paper shows that the natural setting for the Bateman and Erdős study of monotonicity of the k  th difference of partition functions a(n)a(n) is the class of partition identitiesA(x)≔∑n=0∞a(n)xn=∏n=1∞(1-xn)-p(n)with polynomially bounded p(n)p(n). The results include a proof of their conjecture generalized to polynomially bounded p(n)p(n)—their conjecture was for p(n)∈{0,1}p(n)∈{0,1}.