Generalized Eulerian Numbers

摘要

We generalize the Eulerian numbers src="Edit_42d7f417-09a7-4910-9498-aaabff1cffc1.bmp" alt="" />? "="" style="font-family:Verdana;">to sets of numbers "="" style="font-size:10pt;"> style="white-space:nowrap;"> style="font-family:Verdana;font-size:12px;">E _{style="font-family:Verdana;">μ} style="font-family:Verdana;font-size:12px;">( style="font-family:Verdana;font-size:12px;">k,l style="font-family:Verdana;font-size:12px;">), ( _{style="font-family:Verdana;">μ} style="font-family:Verdana;font-size:12px;">=0,1,2, style="font-family:Verdana;white-space:normal;background-color:#FFFFFF;font-size:12px;">· style="font-family:Verdana;white-space:normal;background-color:#FFFFFF;font-size:12px;">· style="font-family:Verdana;white-space:normal;background-color:#FFFFFF;font-size:12px;">·) "="" style="font-size:10pt;"> style="font-family:Verdana;font-size:12px;">where the Eulerian numbers appear as the special case style="white-space:nowrap;font-family:Verdana;font-size:12px;">_μ=1 style="font-family:Verdana;font-size:12px;">. This can be used for the evaluation of generalizations style="white-space:nowrap;">E_μ(k,Z) style="font-family:Verdana;font-size:12px;">of the Geometric series style="white-space:nowrap;font-family:Verdana;font-size:12px;">G₀(k;Z)=G₁(0;Z) style="font-family:Verdana;font-size:12px;"> style="font-family:Verdana;font-size:12px;">by splitting an style="font-family:Verdana;font-size:12px;"> essential part style="white-space:nowrap;font-family:Verdana;font-size:12px;">(1-Z)-(μK+1) style="font-family:Verdana;font-size:12px;"> where the numbers style="white-space:nowrap;font-family:Verdana;font-size:12px;">E_μ(k,l) style="font-family:Verdana;font-size:12px;"> are then the coefficients of the remainder polynomial. This can be extended for non-integer parameter style="font-family:Verdana;font-size:12px;">k style="font-family:Verdana;font-size:12px;"> to the approximative evaluation of generalized Geometric series. The recurrence relations src="Edit_7a6b0c69-df16-4e1d-9084-d4c896b97b68.bmp" alt="" /> style="font-family:Verdana;font-size:12px;"> and src="Edit_3a7688ab-f7e3-4f44-9ae6-7d97aa1cd3a9.bmp" alt="" /> style="font-family:Verdana;font-size:12px;"> for the Generalized Eulerian numbers style="white-space:nowrap;font-family:Verdana;font-size:12px;">E₁(k,l) style="font-family:Verdana;font-size:12px;">are derived. The Eulerian numbers style="font-family:Verdana;font-size:12px;"> are related to the Stirling numbers of second kind style="white-space:nowrap;font-family:Verdana;font-size:12px;">S(k,l) style="font-family:Verdana;font-size:12px;">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.