The nth order difference [Δhn(x)m,g]x=a, where Δh is the difference operator with increment h defined by Δhf(x) = f(x+h)-f(x) and (x)m,g = x(x-g)(x-2g)...(x-mg+g) is the generalized factorial of degree m and increment g, is the subject of this paper. More precisely the numbers G(m,n;r,s)= g-m n!Δnh(X)m,gx=a= 1 n!Δn(rx+s)mx=0, r= h g, s= a g are systematically investigated. Combinatorial interpretations are provided and recurrence relation and generating function are obtained. Moreover connection with other numbers, limiting expressions, orthogonality relations and other properties, useful in combinatorics, are derived. Finally some combinatorial and statistical applications are also discussed. © 1983.
Charalambides, C. A., & Koutras, M. (1983). On the differences of the generalized factorials at an arbitrary point and their combinatorial applications. Discrete Mathematics, 47(C), 183–201. https://doi.org/10.1016/0012-365X(83)90089-4