Some new partition identities (Q2713649)

Eleven partition identities involving functions \(p(n)\) (the number of all partitions of \(n\)), \(q(n)\) (distinct parts), \(q_0(n)\) (distinct odd parts), \(g(n)\) (distinct parts and if \(2m\) is a part then \(m\) is odd and is not a part) and pentagonal numbers \(\omega (n)=n(3n-1)/2\) are proved, for example: NEWLINE\[NEWLINEp(n)+2\sum _{j\geq 1}(-1)^jp(n-j^2)= (-1)^nq_0(n),NEWLINE\]NEWLINE NEWLINE\[NEWLINEp(n/3)+\sum _{j\geq 1}(p((n-\omega (j))/3) +p((n-\omega (-j))/3))=g(n),NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\sum _{k=0}^n q((n-k)/3)g(k)=q(n).NEWLINE\]NEWLINE (If \(n/3\) is not an integer, \(p(n/3)=0\) etc.) The proofs are based on the Jacobi triple product identity, quintuple product identity, and power series manipulations.