An application of the representation theory of Lie algebra \(sl(2,\mathbb{C})\) (Q1275865)

The paper deals with Lie algebra \(L=sl(2,\mathbb{C})\). Consider an irreducible \((m+1)\)-dimensional \(L\)-module \(V\) and the element \(\tau=\exp y \cdot \exp(-x)\cdot\exp y\), where \(x,y\) are from the standard basis \(\{x,y,h\}\) of \(L\), i.e. \([x,y]=h\), \([h,x]=2x\), \([h,y]=-2y\). Using the matrix realization of \(x,y\) and \(\tau\) in the weight vector basis of \(V\), the author derives the following combinatorial identity \[ \sum^i_{s=0} C^s_i\sum^{m-s}_{r=0}(-1)^r C^r_{m-s}C^j_{r+s}= \begin{cases} 0, &i+j\neq m\\ (-1)^j, &i+j=m \end{cases}. \] Special cases of this formula are considered, too.