DOI: 10.1515/gmj-2026-3025 ISSN: 1072-947X
On a combinatorial identity of Akyuz and Halici
Horst Alzer, Arjun K. Rathie Abstract
Inspired by a combinatorial identity given by Akyuz and Halici, we present three closely related summation formulas. One of our results states that
∑
ν
=
0
[
n
2
]
-
j
(
ν
+
j
m
)
(
ν
+
j
-
m
ν
)
(
n
2
(
ν
+
j
)
)
=
2
n
-
2
j
-
1
n
n
-
j
(
n
-
j
j
)
(
j
m
)
,
see text
\sum_{\nu=0}^{[\frac{n}{2}]-j}{\nu+j\choose m}{\nu+j-m\choose\nu}{n\choose 2(%
\nu+j)}=2^{n-2j-1}\frac{n}{n-j}{n-j\choose j}{j\choose m},
where
m
≥
0
{m\geq 0}
,
j
≥
0
{j\geq 0}
and
n
≥
1
{n\geq 1}
are integers with
m
≤
j
≤
[
n
2
]
{m\leq j\leq[\frac{n}{2}]}
. We use properties of hypergeometric functions and the method “comparison of coefficients” to prove our theorems.