q-Pell sequences and two identities of V.A. Lebesgue

Arithmetic Properties of Overpartitions into Odd Parts

“…Interestingly enough, the generating function for p o (n), which we will denote by P o (q), has appeared quite recently in the works of Ardonne, Kedem, and Stone [2], Bessenrodt [3], Santos and Sills [13]. However, in none of these cases do the authors connect their work to overpartitions into odd parts.…”

Combinatorial Proofs of Various q-Pell Identities via Tilings

“…In a recent work by Santos and Sills [3], the q-Pell numbers P n (q) were defined by P n+1 (q) = 1 + q n+1 P n (q) + q n P n−1 (q), P 0 (q) = 1, P 1 (q) = 1 + q.…”

New proofs of identities of Lebesgue and Göllnitz via tilings

“…Thus At this point, an example should make our construction clear. Suppose that we take the term q 2 · q 4 · q 5 · q 9 · q 12 · q 13 · q 15 · bq 2−1 · bq 6−1 · bq 8−1 = b 3 q 73 (8) from the expansion of the product n 1 1 + q n 1 + bq 2n−1 .…”

“…Thus the first position of T (1) is covered by a black square, followed by the first eight tiles of T (2) , followed by a white square. In other words, the second domino accounts for the factor of bq 5 in (8).…”

“…In 2002, Santos and Sills [8] considered finite analogues for which (1) and (2) are limiting cases. In the process they were able, to a degree, to demonstrate connections between Lebesgue's identities and q-Pell sequences.…”