How to add a non-integer number of terms, and how to produce unusual infinite summations
In the very interesting paper under review, the authors extend the definition of a sum \(\sum^x_{y=1} f(y)\) to the case when \(x\) is a real or complex number. The fractional sum for polynomials is defined by \(\sum^x_{k=1} p(k):= P(x)\) (where \(p:\mathbb{C}\to\mathbb{C}\) is the given polynomial), \(P\) denoting the unique polynomial that agrees with the classical value of the sum of every positive integer \(x\). For arbitrary \(a,b\in\mathbb{C}\), put \(\sum^b_{k=a} p(k)= P(b)- P(a-1)\). Let now \(U\subset\mathbb{C}\) and \(\sigma\in \mathbb{N}\cup\{-\infty\}\). A function \(f: U\to\mathbb{C}\) will be called right summable of degree \(\sigma\) if: (i) \(U+1\subset U\) (Minkowski summation); (ii) there exists a sequence of polynomials \((p_n)\) of fixed degree \(\sigma\) such that \(|f(n+x)- p_n(n+ x)|\to 0\) as \(n\to\infty\), for all \(x\) in every bounded subset of \(U\); (iii) for \(a,b+1\in U\), the limit \[ \lim_{n\to\infty} \Biggl(\sum^{n+b}_{k= n+a} p_n(k)\Biggr)+ \sum^n_{k=1} (f(k+a-1)- f(k+ b)) \] exists. (Here, unlike the authors, we preserved the same notation for fractional sums, which may be obviously distinguished from a usual sum.) Fractional products are defined by \[ \prod^b_{k=a} f(k)= \exp\Biggl(\sum^b_{k=a} \ln f(k)\Biggr). \] Left fractional sums or products may be defined in a simlar manner. By using some properties of fractional sums and products, the authors deduce in a unitary way certain unusual infinite sums, products, and limits. They state that, while this method has been mentioned already by Euler and Ramanujan, no systematic attempts to work it out were known up to now.
scientific article; zbMATH DE number 2157679