Stirling functions of the second kind in the setting of difference and fractional calculus

Numerical Functional Analysis and Optimization

Volumn 24, Issue 7-8, 2003, Pages 673-711

  Butzer, Paul L ^a   Kilbas, Anatoly A ^b   Trujillo, Juan J ^c  

^a RWTH AACHEN UNIVERSITY   (Germany)

^b BELARUSIAN STATE UNIVERSITY   (Belarus)

^c UNIVERSIDAD DE LA LAGUNA   (Spain)

Author keywords

Fractional order differences; Generalized stirling numbers; Liouville fractional integrals and derivatives; Marchaud and Hadamard type fractional derivatives and integrals; Mellin transforms; Stirling functions of the second kind

Indexed keywords

FUNCTION EVALUATION; MATHEMATICAL OPERATORS; NUMBER THEORY; PROBLEM SOLVING; THEOREM PROVING;

FRACTIONAL CALCULUS; STIRLING FUNCTIONS; STIRLING NUMBERS;

DIFFERENTIATION (CALCULUS);

EID: 0347603937     PISSN: 01630563     EISSN: None     Source Type: Journal    
DOI: 10.1081/NFA-120026366     Document Type: Article

References (44)

1. Abramowitz, M., Stegun, I. A. (1965). Handbook of Mathematical Functions. New York: Dover Publ.
2. Adamchik, V. S. (1997). On Stirling numbers and Euler sums. J. Comput. Appl. Math. 79(n. 1):119-130.
3. Aigner, M. (1975). Kombinatorik. Vol. 1. Berlin: Springer-Verlag.
4. Butzer, P. L., Hauss, M. (1991a). On Stirling functions of the second kind. Stud. Appl. Math. 84(n. 1):71-91.
5. Butzer, P. L., Hauss, M. (1991b). Stirling functions of first and second kind: some new applications. Approximation, Interpolation, and Summability. In: Israel Math. Conf. Proc. 4. Tel Aviv, Israel, pp. 89-108.
6. Butzer, P. L., Jansche, S. (1997). A direct approach to the Mellin transform. J. Fourier Anal. Appl. 3(n. 4):325-376.
7. Butzer, P. L., Hauss, M. (1999). Application of sampling theory to combinatorical analysis, Stirling numbers, special functions, and the Riemann zeta function. In: Higgins, J. R., Sterns, R. L., eds. Sampling Theory in Fourier Analysis: Advanced Topics. Oxford: Oxford Univ. Press, pp. 1-37.
8. Butzer, P. L., Westphal, U. (2000). An introduction to fractional calculus. In Hifler, H., ed. Applications of Fractional Calculus in Physics. Singapore: Word Scientific Publ., pp. 1-85.
9. Butzer, P. L., Hauss, M., Schmidt, M. (1989). Factorial functions and Stirling numbers of fractional orders. Results Math. 16(n. 1-2):16-48.
10. Butzer, P. L., Kilbas, A. A., Trujillo, J. J. (2002a). Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 269(n. 1):1-27.
11. Butzer, P. L., Kilbas, A. A., Trujillo, J. J. (2002b). Compositions of Hadamard-type fractional integration operators and the semigroup property. J. Math. Anal. Appl. 269(n. 2):387-400.
12. Butzer, P. L., Kilbas, A. A., 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.
13. Butzer, P. L., Kilbas, A. A., 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.
14. Chaundy, T. W. (1969). Elementary Differential Equations. Oxford: Clarendon Press.
15. Chelluri, R., Richmond, L. B., Temme, N. M. (2000). Asymptotic estimates for generalized Stirling numbers. Analysis 20(n. 1):1-13, (CWI Report MAS-R9923).
16. Comtet, L. (1974). Advanced Combinatorics: The art of finite and infinite expansions. Holland: Kluwer Acad. Publ.
17. Corcino, R. B., Hsu, L. C. (2001). Combinatorial and statistical applications of generalized Stirling numbers. J. Math. Res. Exposition 21(n. 3):337-343.
18. 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).
19. Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G. (1953). Higher Transcendental Functions. Vol. I. New York: McGraw-Hill.
20. Frajolet, P., Prodinger, H. (1999). On Stirling numbers for complex arguments and Hankel contours. SIAM J. Discrete Math. 12(n. 2):155-159.
21. Graham, R. L., Knuth, D. E., Patashnik, O. (1994). Concrete Mathematics: A Foundation for Computer Science. Boston: Addison-Wesley.
22. Hauss, M. (1975). Verallgemeinerte Stirling, Bernoulli und Euler Zahlen, deren Anwendungen und schnell konvergente Reihen fuer Zeta Funktionen Ph. D. Thesis, RWTH Aachen, Shaker Verlag, Aachen.
23. Hilfer, R. (2000). Applications of Fractional Calculus in Physics. Singapore: Word Scient. publ.
24. Hsu, L. C., Shiue, P. J.-S. (1998). A unified approach to generalized Stirling numbers. Adv. in Appl. Math. 20(n. 3):366-384.
25. Jordan, Ch. (1950). Calculus of Finite Differences. New York: Chelsea.
26. Kilbas, A. A., Srivastava, H. M., 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).
27. Lang, W. (2000). On generalizations of the Stirling number triangles. J. Integer Seq. 3(n. 2 Art. 00.2.4):1-19.
28. Loeb, D. E. (1992). A generalization of the Stirling numbers. Discrete Math. 103(n. 4):325-376.
29. Matignon, D., Montseny, G., (ed.) Fractional Differential Systems: Models, Methods, and Applications, European Society for Applied and Industrial Mathematics (ESAIM), electronic book.
30. Medicis, A., Leroux, R. (1995). Generalized Stirling numbers, convolution formulae and p, q-analogues. Canad. J. Math. 47(n. 3):474-499.
31. Metzler, R., Klafter, J. (2000). The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(n. 1):1-77.
32. Platonov, M. L. (1977). The relations between generalized Stirling numbers constructed on different basis. Combinatorial and Asymptotic Analysis. N. 2, Krasnojarsk. Gos. Univ., Krasnojarsk, 142-152.
33. Podlubny, I. (1999). Fractional Differential Equations. San-Diego: Mathematics in Sciences and Engineering, Academic Press.
34. Richmond, R., Merlini, D. (1997). Stirling numbers for complex arguments. SIAM J. Discrete Math. 10(n. 1):73-82.
35. Riordan, J. (1979). Combinatorial Identities. New York, Huntington: Krieger Publ.
36. Rucinski, A., Voigt, B. (1990) A local limit theorem for generalized Stirling numbers. Rev. Roumaine Math. Pures Appl. 35(n. 2):161-172.
37. Samko, S. G., Kilbas, A. A., Marichev, O. I. (1993). Fractional Integrals and Derivatives. Theory and Applications. Yverdon: Gordon and Breach.
38. Sing Chandel, R. S. (1977). Generalized Stirling numbers and polynomials. Publ. Inst. Math. Beograd (N.S.) 22(n. 36):43-48.
39. Sinha, V. P., Dhawan, G. K. (1969). On generalized Stirling numbers and polynomials. Riv. Math. Univ. Parma. 10(n. 2):95-100.
40. Sixdeniers, J. M., Penson, K. A., Solomon, A. L. (2001). Extended Bell and Stirling numbers from hypergeometric exponentiation. J. Integer Seq. 4(n. 1, Art. 01.1.4):1-11.
41. Sokolov, I. M., Klafter, J., Blumen, A. (2002). Fractional Kinetics, Phys. Today, Nov. 48-54.
42. Toader, S. Toader, Gh. (1988). Generalized Stirling numbers. Studia Univ. Babes-Bolyai Math. 33(n. 1):50-53.
43. Wagner, C. G. (1996). Generalized Stirling and Lah numbers. Discrete Math. 160(n. 1-3):199-218.
44. Zhang, C. (1994). Quelques études en théorie des équations fonctionnelles et en analyse combinatoire. Ph.D. dissertation, Université Louis Pasteur Strasbourg.