In the paper a new combinatorical interpretation of the Jordan numbers is presented. Binomial type formulae connecting both kinds of numbers mentioned in the title are given. The decomposition of the product of polynomial of variable n into the sums of kth powers of consecutive integers from 1 to n is also studied.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmal_1225,
author = {Roman Witu\l a and Konrad Kaczmarek and Piotr Lorenc and Edyta Hetmaniok and Mariusz Pleszczy\'nski},
title = {Jordan numbers, Stirling numbers and sums of powers},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {34},
year = {2014},
pages = {155-166},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmal_1225}
}
Roman Wituła; Konrad Kaczmarek; Piotr Lorenc; Edyta Hetmaniok; Mariusz Pleszczyński. Jordan numbers, Stirling numbers and sums of powers. Discussiones Mathematicae - General Algebra and Applications, Tome 34 (2014) pp. 155-166. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmal_1225/
[000] [1] Z.I. Borevich and I.R. Szafarevich, Number Theory (Nauka, Moscov, 1964, in Russian).
[001] [2] L. Carlitz, Note on the numbers of Jordan and Ward, Duke Math. J. 38 (1971) 783-790. doi: 10.1215/S0012-7094-71-03894-4.
[002] [3] L. Carlitz, Some numbers related to the Stirling numbers of the first and second kind, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 544-576 (1976) 49-55.
[003] [4] L. Carlitz, Some remarks on the Stirling numbers, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 678-715 (1980) 10-14.
[004] [5] K. Dilcher, Bernoulli and Euler Polynomials, 587-600 (in F.W.I. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark, NIST Handbook of Mathematical Functions, Cambridge Univ. Press, 2010).
[005] [6] R.L. Graham, D.E. Knuth and O. Patashnik, Concrete Mathematics (Addison-Wesley, Reading, 1994).
[006] [7] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory (Springer, 1990). doi: 10.1007/978-1-4757-1779-2. | Zbl 0712.11001
[007] [8] C. Jordan, Calculus of Finite Differences (Chelsea, New York, 1960). doi: 10.2307/2333783.
[008] [9] D.E. Knuth, Johann Faulhaber and sums of powers, Math. Comp. 203 (1993) 277-294. doi: 10.2307/2152953. | Zbl 0797.11026
[009] [10] N. Nielsen, Traité élémentaire des nombers de Bernoulli (Gauthier - Villars, Paris, 1923).
[010] [11] S. Rabsztyn, D. Słota and R. Wituła, Gamma and Beta Functions, Part I (Silesian Technical University Press, Gliwice, 2011, in Polish).
[011] [12] J. Riordan, An Introduction to Combinatorial Analysis (John Wiley, 1958). doi: 10.1063/1.3060724. | Zbl 0078.00805
[012] [13] J. Riordan, Combinatorial Identities (Wiley, New York, 1968).
[013] [14] N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences (http://oeis.org/). | Zbl 1274.11001
[014] [15] M. Živković, On a representation of Stirling's numbers of first kind, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 498-541 (1975) 217-221.