A new identity. II. (Q5933537)

Let \(f(0)\), \(f(1),f(2),\dots\) be any sequence of complex numbers. Put \(H_n=\sum^n_{m=0} f(m)\) where \(m(\geq 1)\) is any integer. Starting with the identity \(bx^2-(a+b) (x+a)^2+ a(x+a+b)^2 =ab (a+b)\) the author obtains the new identity NEWLINE\[NEWLINE\begin{aligned} f(n+2)H^2_n-\bigl( f(n+1)& +f(n+2)\bigr) H^2_{n+1}+ f(n+1) H^2_{n+2}\\ & =f(n+1) f(n+2) \bigl(f(n+1)+ f(n+2)\bigr).\end{aligned}NEWLINE\]NEWLINE From this he deduces as corollaries: an identity involving the Liouville's function \(\lambda (n)\), an identity involving the van-Mangoldt's function \(\lambda(n)\) and an identity involving the generating function of Waring's problem. Some other identities are also mentioned.NEWLINENEWLINE Part I is not cited.