A Class of Combinatorial Functions for Eulerian Numbers

In this note, we introduce a new class of functions pertaining to binomial coefficients left( begin{array}{c} n m end{array} right) and Eulerian numbers left( begin{array}{c} n m end{array} right) which arise in the recent study of descents in permutations. Given positive integers a and b, let fi(x)= 2^−i left( begin{array}{c} a+b i end{array} right) left( begin{array}{c} i x end{array} right) and f(x) = Σi fi(x).Based on the generating function methods, some identities involving fi and f are provided.