An Iterated Map for the Lebesgue Identity

On a weighted spin of the Lebesgue Identity

Alladi studied partition theoretic implications of a two variable generalization of the Lebesgue identity. In this short note, we focus on a slight variation of the basic hypergeometric sum that Alladi studied. We present two new partition identities involving weights.

q-Hypergeometric proofs of polynomial analogues of the triple product identity, Lebesgue's identity and Euler's pentagonal number theorem

We present alternative, q-hypergeometric proofs of some polynomial analogues of classical q-series identities recently discovered by Alladi and Berkovich, and Berkovich and Garvan.

与 Lebesgue、Schur 和 Capparelli 的分拆定理相关的一些 $q$-超几何恒等式

Some $q$-hypergeometric identities associated with partition theorems of Lebesgue, Schur and Capparelli

本文构造含参数 $L,M$ 的三变量多项式恒等式，统一推广并在极限下导出 Schur 与 Lebesgue 关键恒等式，还给出 Alladi–Schur 精细化及连通 Euler、Lebesgue、Capparelli 定理的新无限层级。

Combinatorial interpretation and proof of Glaisher-Crofton identity

We give a purely combinatorial proof of the Glaisher-Crofton identity which derives from the analysis of discrete structures generated by iterated second derivative. The argument illustrates utility of symbolic and generating function methodology of modern enumerative combinatorics and their applications to computational problems.

A quadratic identity in the shuffle algebra and an alternative proof for de Bruijn's formula

Motivated by a polynomial identity of certain iterated integrals, first observed in [CGM20] in the setting of lattice paths, we prove an intriguing combinatorial identity in the shuffle algebra. It has a close connection to de Bruijn's formula when interpreted in the framework of signatures of paths.