Generalized Eulerian Numbers
- 1 Institut für Physik, Humboldt-Universität, Berlin, Germany
Abstract
We generalize the Eulerian numbers to sets of numbers E μ ( k,l ), ( μ =0,1,2, · · ·) where the Eulerian numbers appear as the special case μ =1 . This can be used for the evaluation of generalizations E μ ( k,Z ) of the Geometric series G 0 (k;Z)=G 1 (0;Z) by splitting an essential part (1-Z) -(μK+1) where the numbers E μ (k,l) are then the coefficients of the remainder polynomial. This can be extended for non-integer parameter k to the approximative evaluation of generalized Geometric series. The recurrence relations and for the Generalized Eulerian numbers E 1 (k,l) are derived. The Eulerian numbers are related to the Stirling numbers of second kind S(k,l) and we give proofs for the explicit relations of Eulerian to Stirling numbers of second kind in both directions. We discuss some ordering relations for differentiation and multiplication operators which play a role in our derivations and collect this in Appendices.
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