Eulerian polynomials as moments, via exponential Riordan arrays

Using the theory of exponential Riordan arrays and orthogonal polynomials, we demonstrate that the "descending power" Eulerian polynomials, and their once shifted sequence, are moment sequences for simple families of orthogonal polynomials, which we characterize in terms of their three-term recurrence. We obtain the generating functions of the polynomial sequences in terms of continued fractions, and we also calculate their Hankel transforms.Comment: 12 page

Schizophrenia and the Virtues of Self-Effacement

Michael Stocker’s “The Schizophrenia of Modern Ethical Theories” attacks versions of consequentialism and deontological ethics on the grounds that they are self-effacing. While it is often thought that Stocker’s argument gives us a reason to favour virtue ethics over those other theories, Simon Keller has argued that this is a mistake. He claims that virtue ethics is also self-effacing, and is therefore afflicted with the self-effacement- related problems that Stocker identifies in consequentialism and deontology. This paper defends virtue ethics against this claim. Although there is a kind of self-effacement invol- ved in the exercise of virtue, this is quite different from the so-called schizophrenia that Stocker thinks is induced by modern ethical theory. Importantly, manifesting virtue does not require one to embrace mutually inconsistent moral commitments, as is at times encouraged by consequentialists and deontologists. This paper also considers a reading of the virtue-ethical criterion of right action that is encouraged by Bernard Williams’s distinction between a de re and a de dicto interpretation of the phrase “acting as the virtuous person would.” I argue that such a reading addresses concerns that a virtue-ethi- cal criterion of right action inevitably generates a problematic form of self-effacement