Some identities involving theta functions

Let φ(q)=∑n=−∞∞qn2(|q|<1)(|q|<1). For k∈Nk∈N it is shown that there exist k   rational numbers A(k,0),…,A(k,k−1)A(k,0),…,A(k,k−1) such that1+4E2k∑n=1∞(∑d∈Nd|n(−4d)d2k)qn=∑j=0k−1A(k,j)φ4j+2(q)φ4k−4j(−q), where E2kE2k is an Euler number. Similarly it is shown that there exist k+1k+1 rational numbers B(k,0),…,B(k,k)B(k,0),…,B(k,k) such that∑n=1∞(∑d∈Nd|n(−4n/d)d2k)qn=∑j=0kB(k,j)φ4j+2(q)φ4k−4j(−q). Recurrence relations are given for the A(k,j)A(k,j) and B(k,j)B(k,j).