Jordan numbers, Stirling numbers and sums of powers

Roman Wituła; Konrad Kaczmarek; Piotr Lorenc; Edyta Hetmaniok; Mariusz Pleszczyński

Discussiones Mathematicae - General Algebra and Applications (2014)

  • Volume: 34, Issue: 2, page 155-166
  • ISSN: 1509-9415

Abstract

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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.

How to cite

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Roman Wituła, et al. "Jordan numbers, Stirling numbers and sums of powers." Discussiones Mathematicae - General Algebra and Applications 34.2 (2014): 155-166. <http://eudml.org/doc/270509>.

@article{RomanWituła2014,
abstract = {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.},
author = {Roman Wituła, Konrad Kaczmarek, Piotr Lorenc, Edyta Hetmaniok, Mariusz Pleszczyński},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {Bernoulli numbers; binomial coefficients; Jordan numbers; Stirling numbers; Živković numbers},
language = {eng},
number = {2},
pages = {155-166},
title = {Jordan numbers, Stirling numbers and sums of powers},
url = {http://eudml.org/doc/270509},
volume = {34},
year = {2014},
}

TY - JOUR
AU - Roman Wituła
AU - Konrad Kaczmarek
AU - Piotr Lorenc
AU - Edyta Hetmaniok
AU - Mariusz Pleszczyński
TI - Jordan numbers, Stirling numbers and sums of powers
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2014
VL - 34
IS - 2
SP - 155
EP - 166
AB - 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.
LA - eng
KW - Bernoulli numbers; binomial coefficients; Jordan numbers; Stirling numbers; Živković numbers
UR - http://eudml.org/doc/270509
ER -

References

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  1. [1] Z.I. Borevich and I.R. Szafarevich, Number Theory (Nauka, Moscov, 1964, in Russian). 
  2. [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. 
  3. [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. 
  4. [4] L. Carlitz, Some remarks on the Stirling numbers, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 678-715 (1980) 10-14. 
  5. [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). 
  6. [6] R.L. Graham, D.E. Knuth and O. Patashnik, Concrete Mathematics (Addison-Wesley, Reading, 1994). 
  7. [7] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory (Springer, 1990). doi: 10.1007/978-1-4757-1779-2. Zbl0712.11001
  8. [8] C. Jordan, Calculus of Finite Differences (Chelsea, New York, 1960). doi: 10.2307/2333783. 
  9. [9] D.E. Knuth, Johann Faulhaber and sums of powers, Math. Comp. 203 (1993) 277-294. doi: 10.2307/2152953. Zbl0797.11026
  10. [10] N. Nielsen, Traité élémentaire des nombers de Bernoulli (Gauthier - Villars, Paris, 1923). 
  11. [11] S. Rabsztyn, D. Słota and R. Wituła, Gamma and Beta Functions, Part I (Silesian Technical University Press, Gliwice, 2011, in Polish). 
  12. [12] J. Riordan, An Introduction to Combinatorial Analysis (John Wiley, 1958). doi: 10.1063/1.3060724. Zbl0078.00805
  13. [13] J. Riordan, Combinatorial Identities (Wiley, New York, 1968). 
  14. [14] N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences (http://oeis.org/). Zbl1274.11001
  15. [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. 

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