A simple proof of Ramanujan’s $\sb{1}\psi \sb{1}$ sum

A simple proof of Zariski's Lemma

‎Our aim in this very short note is to show that the proof of the‎ ‎following well-known fundamental lemma of Zariski follows from an‎ ‎argument similar to the proof of the fact that the rational field‎ ‎$mathbb{Q}$ is not a finitely generated $mathbb{Z}$-algebra.

On the Edge-Difference and Edge-Sum Chromatic Sum of the Simple Graphs

‎For a coloring $c$ of a graph $G$‎, ‎the edge-difference coloring sum and edge-sum coloring sum with respect to the coloring $c$ are respectively‎ ‎$sum_c D(G)=sum |c(a)-c(b)|$ and $sum_s S(G)=sum (c(a)+c(b))$‎, ‎where the summations are taken over all edges $abin E(G)$‎. ‎The edge-difference chromatic sum‎, ‎denoted by $sum D(G)$‎, ‎and the edge-sum chromatic sum‎, ‎denoted by $sum S(G)$‎, ‎a...

A Simple Combinatorial Proof of

A Simple Proof of The

We give elementary derivations of some classical summation formulae for bilateral (basic) hypergeometric series. In particular, we apply Gauß’ 2F1 summation and elementary series manipulations to give a simple proof of Dougall’s 2H2 summation. Similarly, we apply Rogers’ nonterminating 6φ5 summation and elementary series manipulations to give a simple proof of Bailey’s very-well-poised 6ψ6 summ...

A simple geometric proof that comonotonic risks have the convex-largest sum

In the recent actuarial literature, several proofs have been given for the fact that if a random vector (X1,X2, . . . ,Xn) with given marginals has a comonotonic joint distribution, the sum X1 +X2 + · · · +Xn is the largest possible in convex order. In this note we give a lucid proof of this fact, based on a geometric interpretation of the support of the comonotonic distribution.