Wünsche, A. (2015) Operator Methods and SU (1,1) Symmetry in the Theory of Jacobi and of Ultraspherical Polynomials. Applied Mathematics, 7, 213-261.

has been cited by the following article:

• 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.