New proofs of identities of Lebesgue and Gِllnitz via tilings

In 1840, V.A. Lebesgue proved the following two series-product identities: ∑ n ⩾ 0 ( − 1 ; q ) n ( q ) n q ( n + 1 2 ) = ∏ n ⩾ 1 1 + q 2 n − 1 1 − q 2 n − 1 , ∑ n ⩾ 0 ( − q ; q ) n ( q ) n q ( n + 1 2 ) = ∏ n ⩾ 1 1 − q 4 n 1 − q n . These can be viewed as specializations of the following more general result: ∑ n ⩾ 0 ( − z ; q ) n ( q ) n q ( n + 1 2 ) = ∏ n ⩾ 1 ( 1 + q n ) ( 1 + z q 2 n − 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.