General Convolution Identities for Bernoulli and Euler Polynomials

Using general identities for difference operators, as well as a technique of symbolic computation and tools from probability theory, we derive very general kth order (k \ge 2) convolution identities for Bernoulli and Euler polynomials. This is achieved by use of an elementary result on uniformly distributed random variables. These identities depend on k positive real parameters, and as special cases we obtain numerous known and new identities for these polynomials. In particular we show that the well-known identities of Miki and Matiyasevich for Bernoulli numbers are special cases of the same general formula.Comment: 20 page

Zeros and convergent subsequences of Stern polynomials

We investigate Dilcher and Stolarsky's polynomial analogue of the Stern diatomic sequence. Basic information is obtained concerning the distribution of their zeros in the plane. Also, uncountably many subsequences are found which each converge to a unique analytic function on the open unit disk. We thus generalize a result of Dilcher and Stolarsky from their second paper on the subject.Comment: 10 pages, 1 figur

The juridical roots of Max Weber

O artigo investiga a formação, a carreira e os trabalhos de Max Weber como jurista, assim como delineia as linhas gerais do impacto e das influências de sua formação jurídica na sua obra em geral. Especial atenção merece sua tese de doutorado sobre as sociedades comerciais na Idade Média, que é apresentada em sua problematização geral e em suas principais ideias. Com isso, procura-se indicar como os problemas tratados por Weber em sua tese de doutorado - sua principal obra jurídica - antecipam temas e problemas que serão desenvolvidos na obra posterior de modo ampliado e aprofundado.The article investigates the formation, career and works of Max Weber as a jurist, and outlines the impact and influences of his legal training on his work in general. Special attention is given to his doctoral thesis on trading societies of the Middle Ages, presenting his overall problematization and his main ideas. The objective is to show how the problems explored by Weber in his doctoral thesis - his main juridical work - anticipate themes and problems developed more widely and deeply in his later work

The Ebenezers, 1843-1859

Students of history are actively engaged in processes and meanings of memory, heritage and history. The differences between memory, heritage and history generate constructive conflict and inquiry about the past. Contextualization and corroboration of primary sources accesses the past. Contextualization and corroboration of inquiry primary sources, as with other primary sources, informs understandings about the past. Construction of a webpage using primary sources of memory, heritage, history, and contextualized and corroborated inquiry may provide one model for inquiry within memory, heritage, and history

Shortened recurrence relations for Bernoulli numbers

AbstractStarting with two little-known results of Saalschütz, we derive a number of general recurrence relations for Bernoulli numbers. These relations involve an arbitrarily small number of terms and have Stirling numbers of both kinds as coefficients. As special cases we obtain explicit formulas for Bernoulli numbers, as well as several known identities