General convolution identities for Bernoulli and Euler polynomials

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• 作者：Karl Dilcher^a ; ¹ ; ^{dilcher@mathstat.dal.ca" class="auth_mail" title="E-mail the corresponding author} ; Christophe Vignat^b ; ^c ; ^{cvignat@tulane.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Bernoulli polynomials ; Euler polynomials ; Bernoulli numbers ; Euler numbers ; Convolution identities
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：435
• 期：2
• 页码：1478-1498
• 全文大小：443 K

文摘

Using general identities for difference operators, as well as a technique of symbolic computation and tools from probability theory, we derive very general k  th order (k≥2iner hidden">$k\ge 2$) 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.