From recursions to asymptotics: On Szekeres' formula for the number of partitions (Q1378533)

Summary: We give a new proof of Szekeres' formula for \(P(n,k)\), the number of partitions of the integer \(n\) having \(k\) or fewer positive parts. Our proof is based on the recursion satisfied by \(P(n,k)\) and Taylor's formula. We make no use of the Cauchy integral formula or any complex variables. The derivation is presented as a step-by-step procedure, to facilitate its application in other situations. As corollaries we obtain the main term of the Hardy-Ramanujan formulas for \(p(n)=\) the number of unrestricted partitions of \(n\), and for \(q(n)=\) the number of partitions of \(n\) into distinct parts.