Quelques identités de l'analyse combinatoire
This paper contains a number of combinatorial identities, the proofs of which depend heavily on the knowledge of certain partial fraction decompositions. Let \[ A(X_ 1,...,X_ n)= \sum(n!/K_ 1!K_ 2!...K_ n!) \prod^{n}_{i=1}(X_ i/i)^{k_ i}, \] where the summation is over all non-negative integral \(k_ i\) for which \(\sum^{n}_{i=1}ik_ i=n\). Typical of the results is the following: let k and r be positive integers and let f(x) be an arbitrary polynomial of degree \(\leq k+r-1\). Then \[ \sum^{k}_{j=1}(-1)^{j+1}\binom{k}{j} f(j)j^{-r}= \frac{1}{r!}(f^{(r)}(0)+ \sum^{r}_{j=1}\binom{r}{j} f^{(r-j)}_{(0)}\quad A(S_ 1,...,S_ j), \] where \(S_ m= \sum^{k}_{i=1}i^{-m}\). Among the corollaries of these results are such identities as \[ \sum^{n}_{k=1}(-1)^{k+1}\binom{n}{k} \{\sum^{k}_{j=1} \frac{1}{j})^ 2+ \sum^{k}_{j=1}\frac{1}{j^ 2}\}=\frac{2}{n^ 2} \] and \[ \begin{multlined} \{\sum^{m}_{k=n} (-1)^ k\binom{m}{k} 2^ kk!/1\cdot 3\cdot...(2k-1)\} \{\sum^{k}_{j=0} \frac{1}{2j-1}\}= \\ =\{(-1)^ n2^ nm!/(m-n)^ 1\cdot(2m-1)^ 2\cdot 1\cdot 3\cdot...\cdot(2n-3)\}\{1+(2m-1) \sum^{n-1}_{j=0} \frac{1}{2j- 1}\}.\end{multlined} \]
scientific article; zbMATH DE number 3847397