An elementary representation of the higher-order Jacobi-type differential equation

We investigate the differential equation for the Jacobi-type polynomials which are orthogonal on the interval $[-1,1]$ with respect to the classical Jacobi measure and an additional point mass at one endpoint. This scale of higher-order equations was introduced by J. and R. Koekoek in 1999 essentially by using special function methods. In this paper, a completely elementary representation of the Jacobi-type differential operator of any even order is given. This enables us to trace the orthogonality relation of the Jacobi-type polynomials back to their differential equation. Moreover, we establish a new factorization of the Jacobi-type operator which gives rise to a recurrence relation with respect to the order of the equation.Comment: 17 page

Net-Loss Reciprocation and the Context Dependency of Economic Choices

This paper proposes a novel explanation for the context dependency of individual choices in two-player games. Context dependency refers to the well-established phenomenon that a player, when choosing from a given opportunity set created by the other player’s strategy, chooses differently in different situations because of different alternatives to the other player’s strategy. The utility model used to explain this kind of context dependency incorporates a preference for net-loss reciprocation. Net-loss reciprocation means that a player’s willingness to impose a net loss (i.e., loss minus gain) on the other player increases in the net loss that he or she derives from the other player’s strategy. I show that net-loss reciprocation together with the method for calculating net losses developed in this paper explains the context dependencies in individual behaviour that have been documented in a number of experimental studies, whereas existing models of intention-based reciprocity fail to explain all the evidence