Affiliation:
1. CMUC, Department of Mathematics, University of Coimbra , 3000-143 Coimbra , Portugal
Abstract
Abstract
Denote by
σ
n
{\sigma }_{n}
the n-th Stirling polynomial in the sense of the well-known book Concrete Mathematics by Graham, Knuth and Patashnik. We show that there exist developments
x
σ
n
(
x
)
=
∑
j
=
0
n
(
2
j
j
!
)
−
1
q
n
−
j
(
j
)
x
j
x{\sigma }_{n}\left(x)={\sum }_{j=0}^{n}{\left({2}^{j}j\!)}^{-1}{q}_{n-j}\left(j){x}^{j}
with polynomials
q
j
{q}_{j}
of degree
j
.
j.
We deduce from this the polynomial identities
∑
a
+
b
+
c
+
d
=
n
(
−
1
)
d
(
x
−
2
a
−
2
b
)
3
n
−
s
−
a
−
c
a
!
b
!
c
!
d
!
(
3
n
−
s
−
a
−
c
)
!
=
0
,
for
s
∈
Z
≥
1
,
\sum _{a+b+c+d=n}{\left(-1)}^{d}\frac{{\left(x-2a-2b)}^{3n-s-a-c}}{a\!b\!c\!d\!\left(3n-s-a-c)\!}=0,\hspace{1.0em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}s\in {{\mathbb{Z}}}_{\ge 1},
found in an attempt to find a simpler formula for the density function in a five-dimensional random flight problem. We point out a probable connection to Riordan arrays.
Subject
Geometry and Topology,Algebra and Number Theory
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