DOI: 10.1515/gmj-2026-3029 ISSN: 1072-947X
Explicit formulas for some product series in the white noise theory and Lévy noise analysis
Hari M. Srivastava, Aleksandar Petojević Abstract
The main object of this article is to provide closed-form expressions for
each of the following sums:
𝒦
q
:=
∑
α
∈
𝒥
(
2
ℕ
)
-
q
α
and
𝒦
¯
q
:=
∑
α
∈
𝒥
(
2
ℕ
+
1
)
-
q
α
,
see text
\mathcal{K}_{q}:=\sum_{\alpha\in\mathcal{J}}(2\mathbb{N})^{-q\alpha}\quad\text%
{and}\quad\overline{\mathcal{K}}_{q}:=\sum_{\alpha\in\mathcal{J}}(2\mathbb{N}+%
1)^{-q\alpha},
which appear in the analysis of stochastic partial differential equations.
Our main results express these sums as finite products of Gamma functions:
𝒦
q
=
∏
j
=
0
q
-
1
Γ
(
1
-
ω
j
2
)
and
𝒦
¯
q
=
2
q
π
q
2
∏
j
=
1
q
-
1
Γ
(
3
2
-
ω
j
2
)
,
see text
\mathcal{K}_{q}=\prod_{j=0}^{q-1}\Gamma\biggl{(}1-\frac{\omega^{j}}{2}\biggr{)%
}\quad\text{and}\quad\overline{\mathcal{K}}_{q}=\frac{2^{q}}{\pi^{\frac{q}{2}}%
}\prod_{j=1}^{q-1}\Gamma\biggl{(}\frac{3}{2}-\frac{\omega^{j}}{2}\biggr{)},