A Combinatorial Identity and Its Related Conjecture

Abstract

In recent researches on a discriminant for polynomials, I faced a recursive (combinatorial) sequence ${\lambda }_{n,m}$ whose first four terms and identities are ${\lambda }_{0,m}:=(\genfrac{}{}{0ex}{}{m}{0})$, ${\lambda }_{1,m}:=(\genfrac{}{}{0ex}{}{m}{1})=(\genfrac{}{}{0ex}{}{m}{m-1})$, ${\lambda }_{2,m}:={(\genfrac{}{}{0ex}{}{m}{2})}^2–(\genfrac{}{}{0ex}{}{m}{2})=(\genfrac{}{}{0ex}{}{m+1}{m-1})$, and ${\lambda }_{3,m}={(\genfrac{}{}{0ex}{}{m}{1})}^3–2(\genfrac{}{}{0ex}{}{m}{1})(\genfrac{}{}{0ex}{}{m}{2})+(\genfrac{}{}{0ex}{}{m}{3})=(\genfrac{}{}{0ex}{}{m+2}{m-1})$. In this paper, I introduce this sequence, prove an identity concerning it, and leave a problem and a conjecture regarding its properties.