Some identities for binomial coefficients (Q1918675)

The characteristic identities proved for binomial coefficients are the following: Put \(nc_k\equiv c^k_n\equiv{}_nc_k\equiv c_{n,k}\equiv{n\choose k}= n!/(k!(n-k)!)\). Then \[ \sum^{n-1}_{k=0} {(-1)^k\over k+\alpha+1}{n-1\choose k}= B(n,\alpha+1),\quad n\geq 1,\quad\alpha>-1.\tag{1} \] \[ \sum^{n-1-i-\nu}_{k=0}(-1)^k{n\choose k+i+\nu+1}{k+i\choose i}={n-1-i\choose\nu},\;n\geq 1,\;0\leq i+\nu<n-1,\tag{2} \] \[ \sum^\nu_{j=0}{\nu\choose j}{n+m-\nu\choose m-j}={n+m\choose m},\tag{3} \] \[ n\geq 1,\quad m\geq 0,\quad 0\leq\nu\leq\min(m,n). \] \[ \sum^{m-\nu}_{k=0}(-1)^k{n\choose n-m+k+\nu} B(\alpha+1,n-m+k)\cdot{\Gamma(n+1+\alpha+k)\over \Gamma(k+1)}=\tag{4} \] \[ =\Gamma(n+1+\alpha){m\choose\nu} B(n-m,m-\nu+\alpha+1), \] \(n\geq 1\), \(0\leq\nu\leq m\leq n-1\), \(\alpha>-1\); where \(B\) and \(\Gamma\) denote the well-known Euler functions. It may be remarked that \(\sum^\nu_{j=0}\) in (3) should be replaced by \(\sum^m_{j=0}\). Also (3) is special case of the well-known identity \(\sum^n_{j=0}{m_2\choose j}{m_1\choose n-j}={m_1+m_2\choose n}\), cf. ``Summation of series'' collected by \textit{L. B. W. Jolley} [2nd revised ed., New York: Dover Publications (1961; Zbl 0101.28602), p.36].