On a New Identity for Double Sum Related to Bernoulli Numbers

Let m, n and l be integers with 0 6 l 6 m + n. It is the main purpose of this paper to give
\nan identity for the sum: m∑
\na=0
\nn∑
\nb=0
\na+b>m+n−l
\nBm−aBn−b
\n(m
\na
\n)(n
\nb
\n)
\na + b + 1
\n(
\na + b + 1
\nm + n − l
\n)
\n,
\nwhere Bm (m = 0, 1, 2, . . . ) is the Bernoulli number. As corollary we prove that the above sum equal to
\n1
\n2 when l = 0