Generalized Eulerian Numbers - Advances in Pure Mathematics

APM > Vol.8 No.3, March 2018

Advances in Pure Mathematics

Volume 8, Issue 3 (March 2018)

ISSN Print: 2160-0368 ISSN Online: 2160-0384

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Generalized Eulerian Numbers ()

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DOI: 10.4236/apm.2018.83018 984 Downloads 2,294 Views Citations

Author(s)

Alfred Wünsche

Affiliation(s)

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 G₀(k;Z)=G₁(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 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.

KEYWORDS

Eulerian Numbers, Eulerian Polynomials, Stirling Numbers, Permutations, Binomials, Hypergeometric Functions, Geometric Series, Vandermonde’s Convolution Identity, Recurrence Relations, Operator Orderings

Share and Cite:

Wünsche, A. (2018) Generalized Eulerian Numbers. Advances in Pure Mathematics, 8, 335-361. doi: 10.4236/apm.2018.83018.

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