On a New Identity for Double Sum Related to Bernoulli Numbers
Author:
Brahim Mittou
Брахим Митту
Date:
2024-10Journal Name:
Журнал сибирского федерального университета. 2024 17(5). Journal of Siberian Federal University. Mathematics & Physics. 2024 17(5)Abstract:
Let m, n and l be integers with 0 6 l 6 m + n. It is the main purpose of this paper to give
an identity for the sum: m∑
a=0
n∑
b=0
a+b>m+n−l
Bm−aBn−b
(m
a
)(n
b
)
a + b + 1
(
a + b + 1
m + n − l
)
,
where Bm (m = 0, 1, 2, . . . ) is the Bernoulli number. As corollary we prove that the above sum equal to
1
2 when l = 0 Пусть m, n и l — целые числа с 0 6 l 6 m+n. Основной целью данной статьи является
дать тождество для суммы:
m∑
a=0
n∑
b=0
a+b>m+n−l
Bm−aBn−
(m
a
)(n
b
)
a + b + 1
(
a + b + 1
m + n − l
)
,
где Bm (m = 0, 1, 2, . . . ) — число Бернулли. В качестве следствия мы доказываем, что указанная
выше сумма равна 1
2 при l = 0
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