On an identity of Ky Fan

This nice identity was applied in the same paper. Rather surprisingly, this result did not seem to have caught much attention. The object of this short note is to indicate some other consequences of identity (1), which have not been quoted in [5] or [6]. Before going further, and since no proof of this identity is given in [6], we thought worth to give one. The proof goes as follows. Put for any positive integer n, Sn = X1+ . . .+Xn, and if m is another positive integer let Tn,m = Sn+m−Sn, so that Sn+m = Sn+Tn,m. Then