The aim of this paper is to show that the Minakshisundaram-Pleijel zeta function
Z
k
(
s
)
{Z_k}(s)
of k-dimensional sphere
S
k
,
k
≥
2
{\mathbb {S}^k},k \geq 2
(defined in
ℜ
e
(
s
)
>
k
2
\Re e(s) > \frac {k}{2}
by
\[
Z
k
(
s
)
=
∑
n
=
1
∞
P
k
(
n
)
[
n
(
n
+
k
−
1
)
]
s
{Z_k}(s) = \sum \limits _{n = 1}^\infty {\frac {{{P_k}(n)}}{{{{[n(n + k - 1)]}^s}}}}
\]
with
(
k
−
1
)
!
P
k
(
n
)
=
R
(
n
+
1
,
k
−
2
)
(
2
n
+
k
−
1
)
(k - 1)!{P_k}(n) = \mathcal {R}(n + 1,k - 2)(2n + k - 1)
where the "rising factorial"
R
(
x
,
n
)
=
x
(
x
+
1
)
⋯
(
x
+
n
−
1
)
\mathcal {R}(x,n) = x(x + 1) \cdots (x + n - 1)
is defined for real number x and n nonnegative integer) can be put in the form
\[
(
k
−
1
)
!
Z
k
(
s
)
=
∑
l
=
0
∞
(
−
1
)
l
(
k
−
1
2
)
2
l
(
−
s
l
)
∑
j
=
0
k
−
1
B
k
,
j
ζ
(
2
s
+
2
l
−
j
,
k
+
1
2
)
(k - 1)!{Z_k}(s) = {\sum \limits _{l = 0}^\infty {{{( - 1)}^l}\left ( {\frac {{k - 1}}{2}} \right )} ^{2l}}\left ( {\begin {array}{*{20}{c}} { - s} \\ l \\ \end {array} } \right )\sum \limits _{j = 0}^{k - 1} {{B_{k,}}_j\zeta (2s + 2l - j,\frac {{k + 1}}{2})}
\]
where
B
k
,
j
{B_{k,j}}
are explicitly computed. The above formula allows us to find explicitly the residue of
Z
k
(
s
)
{Z_k}(s)
at the pole
s
=
k
2
−
n
,
n
∈
N
s = \frac {k}{2} - n,n \in \mathbb {N}
,
\[
1
(
k
−
1
)
!
∑
h
=
0
k
2
−
1
∑
l
+
h
=
n
l
≥
0
(
−
1
)
l
(
k
−
1
2
)
2
l
(
n
−
k
2
l
)
B
k
,
k
−
2
h
−
1
.
\frac {1}{{(k - 1)!}}\sum \limits _{h = 0}^{\frac {k}{2} - 1} {{{\sum \limits _{\begin {array}{*{20}{c}} {l + h = n} \\ {l \geq 0} \\ \end {array} } {{{( - 1)}^l}\left ( {\frac {{k - 1}}{2}} \right )} }^{2l}}\left ( {\begin {array}{*{20}{c}} {n - \frac {k}{2}} \\ l \\ \end {array} } \right )} {B_{k,k - 2h - 1}}.
\]
In passing, we also obtain apparently new relations among the Stirling numbers.