Identities for Bernoulli polynomials and Bernoulli numbers |
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Authors: | Horst Alzer Man Kam Kwong |
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Institution: | 1. Morsbacher Str. 10, 51545, Waldbr?l, Germany 2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Hong Kong
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Abstract: | We prove that if m and \({\nu}\) are integers with \({0 \leq \nu \leq m}\) and x is a real number, then - $$\sum_{k=0 \atop k+m \, \, odd}^{m-1} {m \choose k}{k+m \choose \nu} B_{k+m-\nu}(x) = \frac{1}{2} \sum_{j=0}^m (-1)^{j+m} {m \choose j}{j+m-1 \choose \nu} (j+m) x^{j+m-\nu-1},$$ where B n (x) denotes the Bernoulli polynomial of degree n. An application of (1) leads to new identities for Bernoulli numbers B n . Among others, we obtain
- $$\sum_{k=0 \atop k+m \, \, odd}^{m -1} {m \choose k}{k+m \choose \nu} {k+m-\nu \choose j}B_{k+m-\nu-j} =0 \quad{(0 \leq j \leq m-2-\nu)}. $$ This formula extends two results obtained by Kaneko and Chen-Sun, who proved (2) for the special cases j = 1, \({\nu=0}\) and j = 3, \({\nu=0}\) , respectively.
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