On Minakshisundaram-Pleijel zeta functions of spheres
Authors: Carletti E. , Monti Bragadin G. 
Published: July 2010

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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z Subscript k Baseline left-parenthesis s right-parenthesis equals sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts StartFraction upper P Subscript k Baseline left-parenthesis n right-parenthesis Over left-bracket n left-parenthesis n plus k minus 1 right-parenthesis right-bracket Superscript s Baseline EndFraction">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis k minus 1 right-parenthesis factorial upper P Subscript k Baseline left-parenthesis n right-parenthesis equals script upper R left-parenthesis n plus 1 comma k minus 2 right-parenthesis left-parenthesis 2 n plus k minus 1 right-parenthesis">

(
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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis k minus 1 right-parenthesis factorial upper Z Subscript k Baseline left-parenthesis s right-parenthesis equals sigma-summation Underscript l equals 0 Overscript normal infinity Endscripts left-parenthesis negative 1 right-parenthesis Superscript l Baseline left-parenthesis StartFraction k minus 1 Over 2 EndFraction right-parenthesis Superscript 2 l Baseline StartBinomialOrMatrix negative s Choose l EndBinomialOrMatrix sigma-summation Underscript j equals 0 Overscript k minus 1 Endscripts upper B Subscript k comma Baseline Subscript j Baseline zeta left-parenthesis 2 s plus 2 l minus j comma StartFraction k plus 1 Over 2 EndFraction right-parenthesis">

(
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}

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction 1 Over left-parenthesis k minus 1 right-parenthesis factorial EndFraction sigma-summation Underscript h equals 0 Overscript StartFraction k Over 2 EndFraction minus 1 Endscripts sigma-summation Underscript StartLayout 1st Row l plus h equals n 2nd Row l greater-than-or-equal-to 0 EndLayout Endscripts left-parenthesis negative 1 right-parenthesis Superscript l Baseline left-parenthesis StartFraction k minus 1 Over 2 EndFraction right-parenthesis Superscript 2 l Baseline StartBinomialOrMatrix n minus StartFraction k Over 2 EndFraction Choose l EndBinomialOrMatrix upper B Subscript k comma k minus 2 h minus 1 Baseline period">

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.