TITLE:
Generalized Eulerian Numbers
AUTHORS:
Alfred Wünsche
KEYWORDS:
Eulerian Numbers, Eulerian Polynomials, Stirling Numbers, Permutations, Binomials, Hypergeometric Functions, Geometric Series, Vandermonde’s Convolution Identity, Recurrence Relations, Operator Orderings
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.8 No.3,
March
29,
2018
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.