Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014, Article ID 289187, 6 pages
http://dx.doi.org/10.1155/2014/289187
Research Article

Euler Polynomials and Combinatoric Convolution Sums of Divisor Functions with Even Indices

1National Institute for Mathematical Sciences, Daejeon 305-811, Republic of Korea
2Département de Mathématiques, Université d'Evry Val d'Essonne, France
3Woosuk University, Samlae, Wanju, Jeonbuk 565-701, Republic of Korea

Received 30 May 2014; Accepted 29 July 2014; Published 27 August 2014

Academic Editor: Elena Litsyn

Copyright © 2014 Daeyeoul Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Erdelyi, Higher Transcendental Functions, vol. 1, McGraw Hill, New York, NY, USA, 1953.
  2. H. M. Srivastava and A. Pinter, “Remarks on some relationships between the Bernoulli and Euler polynomials,” Applied Mathematics Letters, vol. 17, no. 4, pp. 375–380, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. M. Besge, “Extrait d'une lettre de M. Besge à M. Liouville,” Journal de Mathématiques Pures et Appliquées, vol. 7, p. 256, 1862. View at Google Scholar
  4. S. Ramanujan, “On certain arithmetical functions,” Transactions of the Cambridge Philosophical Society, vol. 22, pp. 159–184, 1916. View at Google Scholar
  5. B. Cho, D. Kim, and H. Park, “Evaluation of a certain combinatorial convolution sum in higher level cases,” Journal of Mathematical Analysis and Applications, vol. 406, no. 1, pp. 203–210, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. D. Kim and A. Bayad, “Convolution identities for twisted Eisenstein series and twisted divisor functions,” Fixed Point Theory and Applications, vol. 2013, article 81, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  7. W. Palubicki, K. Horel, S. Longay et al., “Self-organizing tree models for image synthesis,” ACM Transactions on Graphics, vol. 28, no. 3, article 58, 2009. View at Publisher · View at Google Scholar · View at Scopus
  8. P. Tan, T. Fang, J. Xiao, P. Zhao, and L. Quan, “Single image tree modeling,” ACM Transactions on Graphics, vol. 27, no. 5, article 108, 2008. View at Publisher · View at Google Scholar · View at Scopus
  9. J. Kim, D. Kim, and H. Cho, “Procedural modeling of trees based on convolution sums of divisor functions for real-time virtual ecosystems,” Computer Animation and Virtual Worlds, vol. 24, no. 3-4, pp. 237–246, 2013. View at Publisher · View at Google Scholar · View at Scopus
  10. http://open.nims.re.kr/JrDr/.
  11. J. Kim, D. Kim, and H. Cho, “Tree growth model design for realistic game landscape production (Korean),” Journal of Korea Game Society, vol. 13, no. 2, pp. 49–58, 2013. View at Google Scholar
  12. H. Xiao and X. Chen, “Modeling and simulation of curled dry leaves,” Soft Matter, vol. 7, no. 22, pp. 10794–10802, 2011. View at Publisher · View at Google Scholar · View at Scopus
  13. J. Kim and D. Kim, “The procedural modeling and visualization of multiple leaves,” Submitted.
  14. D. Kim and Y. K. Park, “Bernoulli identities and combinatoric convolution sums with odd divisor functions,” Abstract and Applied Analysis, vol. 2014, Article ID 890973, 8 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  15. B. Cho, D. Kim, and H. Park, “Certain combinatorial convolu tion sums for divisor functions and Bernoulli numbers,” accepted to Bulletin of the Korean Mathematical Society.
  16. W. Chu and R. R. Zhou, “Convolution of Bernoulli and Euler polynomials,” Sarajevo Journal of Mathematics, vol. 6, no. 18, pp. 147–163, 2010. View at Google Scholar · View at MathSciNet