Multiple extensions of a finite Eulerʹs pentagonal number theorem and the Lucas formulas Original Research Article

Motivated by the resemblance of a multivariate series identity and a finite analogue of Eulerʹs pentagonal number theorem, we study multiple extensions of the latter formula. In a different direction we derive a common extension of this multivariate series identity and two formulas of Lucas. Finally we give a combinatorial proof of Lucas’ formulas.