An Extension of Franklin’s Bijection

Abstract

We are dealing here with the power series expansion of the product ITn>m (l_qn). This expansion may be readily obtained from an identity of Sylvester and the latter, in turn, may be given a relatively simple combinatorial proof. Nev­ ertheless, the problem remains to give a combinatorial explanation for the massive cancellations which produce the final result. The case m = 0 is clearly explained by Franklin's proof of the Euler Pentagonal Number Theorem. Efforts to extend the same mechanism of proof to the general case m > 0 have led to the discovery of an extension of the Franklin involution which explains all the components of the final expansion.