New proofs of identities of Lebesgue and Göllnitz via tilings

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