Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2009, Article ID 168672, 18 pages
http://dx.doi.org/10.1155/2009/168672
Review Article

Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals

Department of Mathematics, Ohio Northern University, Ada, OH 45810, USA

Received 15 May 2009; Accepted 4 August 2009

Academic Editor: Lance Littlejohn

Copyright © 2009 Khristo N. Boyadzhiev. 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. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Vol. 1. Elementary Functions, Gordon & Breach Science, New York, NY, USA, 1986. View at MathSciNet
  2. H. W. Gould, Topics in Combinatorics, H. W. Gould, 2nd edition, 2000.
  3. G. Dobinski, “Summerung der Reihe nm/n!, für m=1,2,3,4,5,,” Archiv der Mathematik und Physik, vol. 61, pp. 333–336, 1877. View at Google Scholar
  4. W. Ligowski, “Zur summerung der Reihe...,” Archiv der Mathematik und Physik, vol. 62, pp. 334–335, 1878. View at Google Scholar
  5. J. A. Grunert, “Uber die Summerung der Reihen...,” Journal für die reine und angewandte Mathematik, vol. 25, pp. 240–279, 1843. View at Google Scholar
  6. L. H. Harper, “Stirling behavior is asymptotically normal,” Annals of Mathematical Statistics, vol. 38, pp. 410–414, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. B. C. Berndt, Ramanujan's Notebooks. Part I, Springer, New York, NY, USA, 1985. View at Zentralblatt MATH
  8. B. C. Berndt, Ramanujan's Notebooks. Part II, Springer, New York, NY, USA, 1989. View at Zentralblatt MATH
  9. K. N. Boyadzhiev, “Polyexponentials,” http://www.arxiv.org/pdf/0710.1332.
  10. E. T. Bell, “Exponential polynomials,” Annals of Mathematics, vol. 35, no. 2, pp. 258–277, 1934. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. J. Touchard, “Nombres exponentiels et nombres de Bernoulli,” Canadian Journal of Mathematics, vol. 8, pp. 305–320, 1956. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. Touchard, “Proprietes arithmetiques de certains nombres recurrents,” Annales de la Société Scientifique de Bruxelles A, vol. 53, pp. 21–31, 1933. View at Google Scholar
  13. G.-C. Rota, Finite Operator Calculus, Academic Press, New York, NY, USA, 1975. View at MathSciNet
  14. L. Carlitz, “Single variable Bell polynomials,” Collectanea Mathematica, vol. 14, pp. 13–25, 1962. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. H. W. Gould and J. Quaintance, “A linear binomial recurrence and the Bell numbers and polynomials,” Applicable Analysis and Discrete Mathematics, vol. 1, no. 2, pp. 371–385, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  16. H. W. Gould and J. Quaintance, “Implications of Spivey's Bell number formula,” Journal of Integer Sequences, vol. 11, no. 3, article 08.3.7, 2008. View at Google Scholar · View at MathSciNet
  17. Y.-Q. Zhao, “A uniform asymptotic expansion of the single variable Bell polynomials,” Journal of Computational and Applied Mathematics, vol. 150, no. 2, pp. 329–355, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. L. Toscano, “Una classe di polinomi della matematica attuariale,” Rivista di Matematica della Università di Parma, vol. 1, pp. 459–470, 1950. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. R. P. Boas Jr. and R. C. Buck, Polynomial Expansions of Analytic Functions, Academic Press, New York, NY, USA, 1964. View at MathSciNet
  20. K. N. Boyadzhiev, “A series transformation formula and related polynomials,” International Journal of Mathematics and Mathematical Sciences, vol. 2005, no. 23, pp. 3849–3866, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. K. N. Boyadzhiev, “On Taylor's coefficients of the Hurwitz zeta function,” JP Journal of Algebra, Number Theory and Applications, vol. 12, no. 1, pp. 103–112, 2008. View at Google Scholar · View at MathSciNet
  22. A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, New York, NY, USA, 1991.
  23. J. Riordan, Combinatorial Identities, John Wiley & Sons, New York, NY, USA, 1968. View at MathSciNet
  24. J. Riordan, An Introduction to Combinatorial Analysis, John Wiley & Sons, New York, NY, USA, 1967.
  25. J. Sandor and B. Crstici, Handbook of Number Theory, Part II, Springer/Kluwer Academic Publishers, Dordrecht, The Netherlands, 2004.
  26. E. T. Bell, “Exponential numbers,” The American Mathematical Monthly, vol. 41, no. 7, pp. 411–419, 1934. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. N. G. de Bruijn, Asymptotic Methods in Analysis, Dover, New York, NY, USA, 3rd edition, 1981. View at MathSciNet
  28. M. E. Dasef and S. M. Kautz, “Some sums of some significance,” The College Mathematics Journal, vol. 28, pp. 52–55, 1997. View at Google Scholar
  29. L. F. Epstein, “A function related to the series for eex,” Journal of Mathematics and Physics, vol. 18, pp. 153–173, 1939. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. M. Klazar, “Bell numbers, their relatives, and algebraic differential equations,” Journal of Combinatorial Theory. Series A, vol. 102, no. 1, pp. 63–87, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. J. Pitman, “Some probabilistic aspects of set partitions,” The American Mathematical Monthly, vol. 104, no. 3, pp. 201–209, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. M. Z. Spivey, “A generalized recurrence for Bell numbers,” Journal of Integer Sequences, vol. 11, pp. 1–3, 2008. View at Google Scholar
  33. H. W. Gould, Catalan and Bell Numbers: Research Bibliography of Two Special Number Sequences, H. W. Gould, 5th edition, 1979.
  34. P. L. Butzer, M. Hauss, and M. Schmidt, “Factorial functions and Stirling numbers of fractional orders,” Results in Mathematics, vol. 16, no. 1-2, pp. 16–48, 1989. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. P. L. Butzer and M. Hauss, “On Stirling functions of the second kind,” Studies in Applied Mathematics, vol. 84, no. 1, pp. 71–91, 1991. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, D. Reidel, Dordrecht, The Netherlands, 1974. View at MathSciNet
  37. H. W. Gould, “Euler's formula for nth differences of powers,” The American Mathematical Monthly, vol. 85, no. 6, pp. 450–467, 1978. View at Publisher · View at Google Scholar · View at MathSciNet
  38. R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, Reading, Mass, USA, 2nd edition, 1994. View at MathSciNet
  39. D. E. Knuth, “Two notes on notation,” The American Mathematical Monthly, vol. 99, no. 5, pp. 403–422, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. I. J. Schwatt, An Introduction to the Operations with Series, Chelsea, New York, NY, USA, 1962.
  41. P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, “Fractional calculus in the Mellin setting and Hadamard-type fractional integrals,” Journal of Mathematical Analysis and Applications, vol. 269, no. 1, pp. 1–27, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  42. P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, “Compositions of Hadamard-type fractional integration operators and the semigroup property,” Journal of Mathematical Analysis and Applications, vol. 269, no. 2, pp. 387–400, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, “Mellin transform analysis and integration by parts for Hadamard-type fractional integrals,” Journal of Mathematical Analysis and Applications, vol. 270, no. 1, pp. 1–15, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  44. F. Oberhettinger, Tables of Mellin Transforms, Springer, New York, NY, USA, 1974. View at MathSciNet