• Journal for History of Mathematics
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• v.20 no.4
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• pp.71-84
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• 2007

J. Bernoulli first discovered the method which one can produce those formulae for the sum $S_n(k)=\sum_{{\iota}=1}^n\;{\iota}^k$ for any natural numbers k. After then, there has been increasing interest in Bernoulli and Euler numbers associated with Riemann zeta functions. Recently, Kim have been studied extended q-Bernoulli numbers and q-Euler numbers associated with p-adic q-integral on $\mathbb{Z}_p$, and sums of powers of consecutive q-integers, etc. In this paper, we investigate for the historical background and evolution process of the sums of powers of consecutive q-integers and discuss for Euler zeta functions subjects which are studying related to these areas in the recent.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.10 no.7
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• pp.1188-1195
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• 2006

This paper presents an efficient design method of 2-D state-space digital filter based on an improved branch-and -bound algorithm. The resultant 2-D state-space digital filters whose coefficients are represented as the sum of two power-of-two terms, are attractive for high-speed operation and simple implementation. The feasibility of the proposed method is demonstrated by several experiments. The results show that the approximation error and group delay characteristic of the resultant filters are similar to those of the digital filters which designed in the continuous coefficient space.