A Generalized Summation Rule Related to Stirling Numbers

In this section and in section 2 and 3 we will generalize the summation rule obtained by L.C. Hsu [7], involving Stirling numbers of the second kind. Given four real numbers a, b, α and β with α 6= 0 and β 6= 0, L.C. Hsu and H.Q. Yu [11], L.C. Hsu and P.J. Shiue [9] defined the symmetric Stirling-type pairs (〈a, b, α, β〉pairs, for short) {s, s} = {s(n, k), s(n, k)} = {s(n, k, a, b;α, β), s(n, k, b, a;β, α)}