New proofs of identities of Lebesgue and Göllnitz via tilings
The authors propose a new method for proving the q-identity \[ \sum_{n\geq 0} \frac{(-z,q)_n}{(q)_n} q^{\binom{n+1}{2}} = \prod_{n\geq 1} (1+q^n)(1+z q^{2n-1}). \tag{\(*\)} \] This identity includes the classical identities of V.A. Lebesgue (1840), corresponding to the cases of \(z=1\) and \(z=q\). Moreover, from identities obtained in this paper two known Göllnitz identities are derived. The method proposed in this paper is in fact original, according to the authors:``that demonstrates how both sides of (\(*\)) enumerate the same collection of weighted Pell tilings''. The whole charm of the Little-Sellers method consists in introducting the generating function that counts all Pell tilings by a special weight. It allows, afterwards, to generate many useful identities for this function, from which the \(q\)-identities discussed here easily follow. Let us mention that \textit{J. P. O. Santos} and \textit{A. V. Sills} in their paper [``q-Pell sequences and two identities of V. A. Lebesgue'', Discrete Math. 257, No.\,1, 125--142 (2002; Zbl 1007.05017)] considered finite analogues of Lebesgue's \(q\)-identities. It was in that paper that the connections between Lebesgue's identities and \(q\)-Pell sequences were presented for the first time.
scientific article; zbMATH DE number 5523128