数学物理学报(英文版) ›› 2011, Vol. 31 ›› Issue (2): 408-418.doi: 10.1016/S0252-9602(11)60241-5

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EXACT EVALUATIONS OF FINITE |TRIGONOMETRIC SUMS BY SAMPLING THEOREMS

M.H. Annaby|R.M. Asharabi   

  1. 1.Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt;2.Department of Mathematics, Faculty of Education Al-Mahweet, Sana'a University, Yemen
  • 收稿日期:2007-05-22 修回日期:2009-11-01 出版日期:2011-03-20 发布日期:2011-03-20

EXACT EVALUATIONS OF FINITE |TRIGONOMETRIC SUMS BY SAMPLING THEOREMS

M.H. Annaby|R.M. Asharabi   

  1. 1.Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt;2.Department of Mathematics, Faculty of Education Al-Mahweet, Sana'a University, Yemen
  • Received:2007-05-22 Revised:2009-11-01 Online:2011-03-20 Published:2011-03-20

摘要:

We use the sampling representations associated with Sturm-Liouville difference operators to derive generalized integral-valued
trigonometric sums. This extends the known results where zeros of Chebyshev polynomials of the first kind are involved to the use of
the eigenvalues of difference operators, which leads to new identities. In these identities Bernoulli's numbers play a role similar to that of Euler's in the old ones. Our technique differs from that of Byrne-Smith (1997) and Berndt-Yeap (2002).

关键词: Trigonometric sums, difference equations, sampling theorem

Abstract:

We use the sampling representations associated with Sturm-Liouville difference operators to derive generalized integral-valued
trigonometric sums. This extends the known results where zeros of Chebyshev polynomials of the first kind are involved to the use of
the eigenvalues of difference operators, which leads to new identities. In these identities Bernoulli's numbers play a role similar to that of Euler's in the old ones. Our technique differs from that of Byrne-Smith (1997) and Berndt-Yeap (2002).

Key words: Trigonometric sums, difference equations, sampling theorem

中图分类号: 

  • 26A09