Abstract
We generalize the Eulerian numbers to sets of numbers Eμ(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 Eμ(k,Z) of the Geometric series G0(k;Z)=G1(0;Z) by splitting an essential part (1-Z)-(μK+1) where the numbers Eμ(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 E1(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.
Highlights
The Eulerian numbers are discussed in the remarkable monograph of Riordan [1] from a combinatorial view and are the special topics in the recently published voluminous and versatile monograph of Petersen [2] with huge material and relations to other topics and with a great number of citations
The Eulerian numbers are taken into account in the article of Bressoud [3] in the NIST Handbook of Mathematical Functions [4]
We looked for references where these numbers which we calculated explicitly are present in literature and found them in the monograph of Riordan [1] about combinatorics1
Summary
The Eulerian numbers are discussed in the remarkable monograph of Riordan [1] from a combinatorial view and are the special topics in the recently published voluminous and versatile monograph of Petersen [2] with huge material and relations to other topics and with a great number of citations.
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