Summation formulae on reciprocal sequences
Given a sequence \(\{a_n\}_{n\geq0}\) of complex numbers, the reviewer [Eur. J. Comb. 24, 709--718 (2003; Zbl 1024.05010)] introduced its dual sequence \(\{a_n^*\}_{n\geq0}\) with \(a_n^*=\sum_{k=0}^n\binom {n}{k}(-1)^ka_k\), and set \[ A_n(x)=\sum_{k=0}^n\binom{n}{k}(-1)^ka_kx^{n-k}\quad \text{and}\quad A_n^*(x)=\sum_{k=0}^n\binom{n}{k}(-1)^ka_k^*x^{n-k}. \] Note that \(A_n^*(1-x)=(-1)^nA_n(x)\) and \(A'_{n+1}(x)=(n+1)A_n(x)\). Thus we can restate two main identities ((1.5) and (1.4)) in the paper of the reviewer as follows: \[ \sum_{k=0}^m\binom{m}k(x-y)^{m-k}A_{n+k}(y)= \sum_{k=0}^n\binom{n}k(y-x)^{n-k}A_{m+k}(x) \tag \(*\) \] and \[ \begin{multlined} \sum_{k=0}^m\binom mk(x-y)^{m-k}\frac{A_{n+k+1}(y)}{n+k+1} -\sum_{k=0}^n\binom{n}k(y-x)^{n-k}\frac{A_{m+k+1}(x)}{m+k+1} \\=(-1)^m\frac{m!n!}{(m+n+1)!}a_0(y-x)^{m+n+1}. \end{multlined}\tag \(\star\) \] There are several different extensions of the fundamental identities \((*)\) and \((\star)\), see, e.g., \textit{R. Chapman} [An infinite family of dual sequence identities. Integers 5, No. 1, Paper A31 (2005; Zbl 1134.11308)], \textit{K.-W. Chen} [J. Number Theory 124, 142--150 (2007; Zbl 1120.05007)], and \textit{S. J. X. Hou} and \textit{J. Zeng} [Eur. J. Comb. 28, 214--227 (2007; Zbl 1107.05011)]. In the paper under review, the authors define \[ \Omega_{m,n,l}(x,y)=\sum_{k=0}^m\binom mk\frac{A_{n+k+l}(x)}{(n+k+1)_l}(y-x)^{m-k}, \] where \[ (c)_l=\begin{cases} c(c+1)\cdots(c+l-1)&\text{if }\;l>0,\\ 1&\text{if }\;l=0,\\ 1/((c-1)\cdots(c+l))&\text{if }\;l<0.\end{cases} \] Then, they generalize the reviewer's identities \((*)\) and \((\star\)) in the following way: If \(l\leq0\) then \(\Omega_{m,n,l}(x,y)=\Omega_{n,m,l}(y,x)\); if \(l>0\) then \(\Omega_{m,n,l}(x,y)-\Omega_{n,m,l}(y,x)\) equals \[ \frac{(-1)^m m!n!}{(m+n+l)!} \sum_{k=1}^l\binom{m+n+l}{l-k} \binom{m+k-1}m A_{l-k}(y) (x-y)^{m+n+k}. \] In the case \(l\leq0\) this result is equivalent to the authors' Corollary 2 which has the same flavor with Lemma 3.1 of the reviewer's paper. Actually Corollary 2 follows immediately from \((*)\) if we write \(y=x+t\) and take partial derivatives (with respect to \(x\)) of order \(l\). Another result of this type is Theorem 1.2 of a paper by the reviewer and \textit{H. Pan} [Acta Arith. 125, 21--39 (2006; Zbl 1153.11012)].
scientific article; zbMATH DE number 5137597