aequationes mathematicae

, Volume 46, Issue 1–2, pp 119–142

# Eulerian numbers with fractional order parameters

• P. L. Butzer
• M. Hauss
Research Papers

## Summary

The aim of this paper is to generalize the well-known “Eulerian numbers”, defined by the recursion relationE(n, k) = (k + 1)E(n − 1, k) + (n − k)E(n − 1, k − 1), to the case thatn ∈ ℕ is replaced by α ∈ ℝ. It is shown that these “Eulerian functions”E(α, k), which can also be defined in terms of a generating function, can be represented as a certain sum, as a determinant, or as a fractional Weyl integral. TheE(α, k) satisfy recursion formulae, they are monotone ink and, as functions of α, are arbitrarily often differentiable. Further, connections with the fractional Stirling numbers of second kind, theS(α, k), α > 0, introduced by the authors (1989), are discussed. Finally, a certain counterpart of the famous Worpitzky formula is given; it is essentially an approximation ofx α in terms of a sum involving theE(α, k) and a hypergeometric function.

## AMS (1990) subject classification

Primary 11B83, 11B68, 11B37 Secondary 26A33, 33C05

## References

1. [1]
Abramowitz, M. andStegun, I. A.,Handbook of mathematical functions. Dover Publications, Inc., New York, 1965.Google Scholar
2. [2]
Aigner, M.,Kombinatorik I. Springer-Verlag, Berlin, 1975.Google Scholar
3. [3]
Butzer, P. L. andHauss, M.,On Stirling functions of the second kind. Stud. Appl. Math.84 (1991), 71–79.Google Scholar
4. [4]
Butzer, P. L. andHauss, M.,Stirling functions of first and second kind; some new applications. To appear in “Approximation, Interpolation and Summability”, (Proc. Conf. in honour of Prof. Jakimovski, Research Inst. of Math. Sciences, Bar Ilan University, Tel Aviv, June 4–8, 1990), Israel Mathematical Conference Proceedings, Vol. 4 (1991), Weizmann Press, Israel, pp. 89–108.Google Scholar
5. [5]
Butzer, P. L., Hauss, M. andSchmidt, M.,Factorial functions and Stirling numbers of fractional orders. Resultate Math.16 (1989), 16–48.Google Scholar
6. [6]
Butzer, P. L. andJunggeburth, J.,On Jackson-type inequalities in approximation theory. In: Proc. First Int. Conf. on General Inequalities I. Eds.: J. Aczél, G. Aumann and E. F. Beckenbach. Birkhäuser, Basel, 1978, pp. 85–114.Google Scholar
7. [7]
Carlitz, L.,Eulerian numbers and polynomials. Math. Mag.32 (1959), 247–260.Google Scholar
8. [8]
Carlitz, L.,Extended Bernoulli and Eulerian numbers. Duke Math. J.31 (1964), 667–689.
9. [9]
Carlitz, L.,Some numbers related to the Stirling numbers of the first and second kind. Univ. Beograd. Publ. Elektrotechn. Fak. Ser. Mat. Fiz. No. 544-No. 576 (1976), 49–55.Google Scholar
10. [10]
Carlitz, L, Kurtz, D. C., Scoville, R. andStackelberg, O. P.,Asymptotic properties of Eulerian numbers. Z. Wahrsch. verw. Gebiete23 (1972), 47–54.
11. [11]
Comtet, L.,Advanced combinatorics. D. Reidel Publishing Company, Dordrecht, 1974. (First published 1970 by Presses Universitaires de France, Paris).Google Scholar
12. [12]
Doetsch, G.,Einführung in Theorie und Anwendung der Laplace-Transformation. Birkhäuser, Basel, 1958.Google Scholar
13. [13]
Erdélyi, A., Magnus, W., Oberhettinger, F. andTricomi, F. G.,Tables of integral transforms, Vol. II. McGraw-Hill, New York, 1954.Google Scholar
14. [14]
Gessel, I. andStanley, R. P.,Stirling polynomials. J. Combin. Theory Ser. A24 (1978), 24–33.
15. [15]
16. [16]
Hansen, E. R.,A table of series and products. Prentice-Hall, Englewood Cliffs, N.J. 1975.Google Scholar
17. [17]
Hensley, D.,Eulerian numbers and the unit cube. Fibonacci Quart.20 (1982), 344–348.Google Scholar
18. [18]
Kanold, H.-J.,Über Stirlingsche Zahlen 2. Art. J. Reine Angew. Math.229 (1968), 188–193.Google Scholar
19. [19]
Kimber, A. C.,Eulerian numbers and links with some statistical procedures. Utilitas Math.31 (1987), 57–65.Google Scholar
20. [20]
Riordan, J.,An introduction to combinatorial analysis. John Wiley & Sons, New York, 1958.Google Scholar
21. [21]
Riordan, J.,Combinatorial identities. John Wiley & Sons, New York, 1968.Google Scholar
22. [22]
23. [23]
24. [24]
Schrutka, L. v.,Eine neue Einteilung der Permutationen. Math. Ann.118 (1941), 246–50.
25. [25]
Shanks, E. B.,Iterated sums of powers of the binomial coefficients. Amer. Math. Monthly58 (1951), 404–407.Google Scholar
26. [26]
Slater, L. J.,Generalized hypergeometric functions. Cambridge Univ. Press, Cambridge, London and New York, 1966.Google Scholar
27. [27]
Worpitzky, J.,Studien über die Bernoullischen und Eulerschen Zahlen. J. Reine Angew. Math.94 (1883), 203–232.Google Scholar
28. [28]
Butzer, P. L. andNessel, R. J.,Fourier analysis and approximation. Vol. I:One-Dimensional Theory. Birkhäuser, Basel and Academic Press, New York, 1971.Google Scholar
29. [29]
Westphal, U.,An approach to fractional powers of operators via fractional differences. Proc. London Math. Soc. (3)29 (1974), 557–576.Google Scholar
30. [30]
Butzer, P. L. andWestphal, U.,An access to fractional differentiation via fractional difference quotients, in: Fractional calculus and its applications. [Lecture Notes in Mathematics, No. 457]. Springer-Verlag, Berlin, 1975, pp. 116–145.Google Scholar