162
Views
14
CrossRef citations to date
0
Altmetric
Original Articles

Stirling Functions of the Second Kind in the Setting of Difference and Fractional Calculus

, &
Pages 673-711 | Published online: 07 Feb 2007

References

  • Abramowitz , M. and Stegun , I. A. 1965 . Handbook of Mathematical Functions New York : Dover Publ. .
  • Adamchik , V. S. 1997 . On Stirling numbers and Euler sums . J. Comput. Appl. Math. , 79 ( n. 1 ) : 119 – 130 .
  • Aigner , M. 1975 . Kombinatorik Vol. 1 , Berlin : Springer-Verlag .
  • Butzer , P. L. and Hauss , M. 1991a . On Stirling functions of the second kind . Stud. Appl. Math. , 84 ( n. 1 ) : 71 – 91 .
  • Butzer , P. L. and Hauss , M. Stirling functions of first and second kind: some new applications. Approximation, Interpolation, and Summability . Israel Math. Conf. Proc. 4. Tel Aviv, Israel . pp. pp. 89 – 108 .
  • Butzer , P. L. and Jansche , S. 1997 . A direct approach to the Mellin transform . J. Fourier Anal. Appl. , 3 ( n. 4 ) : 325 – 376 .
  • Butzer , P. L. and Hauss , M. 1999 . “ Application of sampling theory to combinatorical analysis, Stirling numbers, special functions, and the Riemann zeta function ” . In Sampling Theory in Fourier Analysis: Advanced Topics Edited by: Higgins , J. R. and Sterns , R. L. pp. 1 – 37 . Oxford : Oxford Univ. Press .
  • Butzer , P. L. and Westphal , U. 2000 . “ An introduction to fractional calculus ” . In Applications of Fractional Calculus in Physics Edited by: Hifler , H . pp. 1 – 85 . Singapore : Word Scientific Publ. .
  • Butzer , P. L. , Hauss , M and Schmidt , M . 1989 . Factorial functions and Stirling numbers of fractional orders . Results Math. , 16 ( n.1 – 2 ) : 16 – 48 .
  • Butzer , P. L. , Kilbas , A. A. and Trujillo , J. J. 2002a . Fractional calculus in the Mellin setting and Hadamard-type fractional integrals . J. Math. Anal. Appl. , 269 ( n. 1 ) : 1 – 27 .
  • Butzer , P. L. , Kilbas , A. A. and Trujillo , J. J. 2002b . Compositions of Hadamard-type fractional integration operators and the semigroup property . J. Math. Anal. Appl. , 269 ( n. 2 ) : 387 – 400 .
  • Butzer , P. L. , Kilbas , A. A. and Trujillo , J. J. 2002c . Mellin transform analysis and integration by parts for Hadamard-type fractional integrals . J. Math. Anal. Appl. , 270 ( n. 1 ) : 1 – 15 .
  • Butzer , P. L. , Kilbas , A. A. and Trujillo , J. J. 2003 . Generalized Stirling functions of second kind and representations of fractional order differences via derivatives . J. Diff. Equ. Appl. , 9 : 503 – 533 .
  • Chaundy , T. W. 1969 . Elementary Differential Equations Oxford : Clarendon Press .
  • Chelluri , R. , Richmond , L. B. and Temme , N. M. 2000 . Asymptotic estimates for generalized Stirling numbers . Analysis , 20 ( n. 1 ) : 1 – 13 . (CWI Report MAS-R9923)
  • Comtet , L. 1974 . Advanced Combinatorics: The art of finite and infinite expansions Holland : Kluwer Acad. Publ. .
  • Corcino , R. B. and Hsu , L. C. 2001 . Combinatorial and statistical applications of generalized Stirling numbers . J. Math. Res. Exposition , 21 ( n. 3 ) : 337 – 343 .
  • Egorychev , G. P. 1977 . Integral Representations and the Computation of Combinatorial Sums Vol. 59 , Providence, Rode Island : Math. Monographs Amer. Math. Soc. . 1984 (Russian edn. 1977)
  • Erdelyi , A , Magnus , W , Oberhettinger , F and Tricomi , F. G. 1953 . Higher Transcendental Functions Vol. I , New York : McGraw-Hill .
  • Flajolet , P and Prodinger , H . 1999 . On Stirling numbers for complex arguments and Hankel contours . SIAM J. Discrete Math. , 12 ( n. 2 ) : 155 – 159 .
  • Graham , R. L. , Knuth , D. E. and Patashnik , O . 1994 . Concrete Mathematics: A Foundation for Computer Science Boston : Addison-Wesley .
  • Hauss , M . 1975 . Verallgemeinerte Stirling, Bernoulli und Euler Zahlen, deren Anwendungen und schnell konvergente Reihen fuer Zeta Funktionen RWTH Aachen, Shaker Verlag, Aachen . Ph. D. Thesis
  • Hilfer , R . 2000 . Applications of Fractional Calculus in Physics Singapore : Word Scient. publ. .
  • Hsu , L. C. and Shiue , P. J.-S . 1998 . A unified approach to generalized Stirling numbers . Adv. in Appl. Math. , 20 ( n. 3 ) : 366 – 384 .
  • Jordan , Ch . 1950 . Calculus of Finite Differences New York : Chelsea .
  • Kilbas , A. A. , Srivastava , H. M. and Trujillo , J. J. 2002 . “ Fractional differential equations: an emergent field in applied and mathematical sciences ” . In Factorization, Singular Operators, and Relative topics Conf. Proc. Funchal'2002 (accepted for publication)
  • Lang , W. 2000 . On generalizations of the Stirling number triangles . J. Integer Seq. , 3 ( n. 2 Art. 00.2.4 ) : 1 – 19 .
  • Loeb , D. E. 1992 . A generalization of the Stirling numbers . Discrete Math. , 103 ( n. 4 ) : 325 – 376 .
  • Matignon , D and Montseny , G . “ Fractional Differential Systems: Models, Methods, and Applications ” . In European Society for Applied and Industrial Mathematics (ESAIM), electronic book
  • Medicis , A and Leroux , R . 1995 . Generalized Stirling numbers, convolution formulae and p, q-analogues . Canad. J. Math. , 47 ( n. 3 ) : 474 – 499 .
  • Metzler , R and Klafter , J . 2000 . The random walk's guide to anomalous diffusion: a fractional dynamics approach . Phys. Rep. , 339 ( n. 1 ) : 1 – 77 .
  • Platonov , M. L. 1977 . “ The relations between generalized Stirling numbers constructed on different basis ” . In Combinatorial and Asymptotic Analysis 142 – 152 . Krasnojarsk : N. 2, Krasnojarsk. Gos. Univ. .
  • Podlubny , I . 1999 . Fractional Differential Equations San-Diego : Mathematics in Sciences and Engineering, Academic Press .
  • Richmond , R and Merlini , D . 1997 . Stirling numbers for complex arguments . SIAM J. Discrete Math. , 10 ( n. 1 ) : 73 – 82 .
  • Riordan , J . 1979 . Combinatorial Identities New York, Huntington : Krieger Publ. .
  • Rucinski , A and Voigt , B . 1990 . A local limit theorem for generalized Stirling numbers . Rev. Roumaine Math. Pures Appl. , 35 ( n. 2 ) : 161 – 172 .
  • Samko , S. G. , Kilbas , A. A. and Marichev , O. I. 1993 . Fractional Integrals and Derivatives. Theory and Applications Yverdon : Gordon and Breach .
  • Singh Chandel , R. S. 1977 . Generalized Stirling numbers and polynomials . Publ. Inst. Math. Beograd (N.S.) , 22 ( n. 36 ) : 43 – 48 .
  • Sinha , V. P. and Dhawan , G. K. 1969 . On generalized Stirling numbers and polynomials . Riv. Math. Univ. Parma. , 10 ( n. 2 ) : 95 – 100 .
  • Sixdeniers , J. M. , Penson , K. A. and Solomon , A. L. 2001 . Extended Bell and Stirling numbers from hypergeometric exponentiation . J. Integer Seq. , 4 ( n. 1, Art. 01.1.4 ) : 1 – 11 .
  • Sokolov , I. M. , Klafter , J and Blumen , A . 2002 . Fractional Kinetics, Phys 48 – 54 . Today, Nov
  • Toader , S. and Toader , Gh . 1988 . Generalized Stirling numbers . Studia Univ. Babes-Bolyai Math. , 33 ( n. 1 ) : 50 – 53 .
  • Wagner , C. G. 1996 . Generalized Stirling and Lah numbers . Discrete Math. , 160 ( n. 1–3 ) : 199 – 218 .
  • Zhang , C . 1994 . Quelques études en théorie des équations fonctionnelles et en analyse combinatoire Université Louis Pasteur Strasbourg . Ph.D. dissertation

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.