New proofs of identities of Lebesgue and Göllnitz via tilings

In 1840, V.A. Lebesgue proved the following two series-product identities:∑n⩾0(−1;q)n(q)nq(n+12)=∏n⩾11+q2n−11−q2n−1,∑n⩾0(−q;q)n(q)nq(n+12)=∏n⩾11−q4n1−qn. These can be viewed as specializations of the following more general result:∑n⩾0(−z;q)n(q)nq(n+12)=∏n⩾1(1+qn)(1+zq2n−1). There are numerous combinatorial proofs of this identity, all of which describe a bijection between different types of integer partitions. Our goal is to provide a new, novel combinatorial proof that demonstrates how both sides of the above identity enumerate the same collection of “weighted Pell tilings.” In the process, we also provide a new proof of the Göllnitz identities.