Explicit formulas for some product series in the white noise theory and Lévy noise analysis

The main object of this article is to provide closed-form expressions for each of the following sums: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi mathvariant="script">𝒦</m:mi> <m:mi>q</m:mi> </m:msub> <m:mo>:=</m:mo> <m:mrow> <m:mrow> <m:munder> <m:mo largeop="true" movablelimits="false" symmetric="true">∑</m:mo> <m:mrow> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="script">𝒥</m:mi> </m:mrow> </m:munder> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>ℕ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi>q</m:mi> <m:mo>⁢</m:mo> <m:mi>α</m:mi> </m:mrow> </m:mrow> </m:msup> </m:mrow> <m:mo mathvariant="italic" separator="true"> </m:mo> <m:mtext>and</m:mtext> </m:mrow> </m:mrow> <m:mo mathvariant="italic" separator="true"> </m:mo> <m:mrow> <m:msub> <m:mover accent="true"> <m:mi mathvariant="script">𝒦</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mi>q</m:mi> </m:msub> <m:mo>:=</m:mo> <m:mrow> <m:munder> <m:mo largeop="true" movablelimits="false" symmetric="true">∑</m:mo> <m:mrow> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="script">𝒥</m:mi> </m:mrow> </m:munder> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>ℕ</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi>q</m:mi> <m:mo>⁢</m:mo> <m:mi>α</m:mi> </m:mrow> </m:mrow> </m:msup> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> see text Kq:=∑α(2N)-qα% {and}̄q:=∑α(2N+% 1)-qα, which appear in the analysis of stochastic partial differential equations. Our main results express these sums as finite products of Gamma functions: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi mathvariant="script">𝒦</m:mi> <m:mi>q</m:mi> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:munderover> <m:mo largeop="true" movablelimits="false" symmetric="true">∏</m:mo> <m:mrow> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mi>q</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:munderover> <m:mrow> <m:mi mathvariant="normal">Γ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo maxsize="210%" minsize="210%">(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>-</m:mo> <m:mfrac> <m:msup> <m:mi>ω</m:mi> <m:mi>j</m:mi> </m:msup> <m:mn>2</m:mn> </m:mfrac> </m:mrow> <m:mo maxsize="210%" minsize="210%">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo mathvariant="italic" separator="true"> </m:mo> <m:mtext>and</m:mtext> </m:mrow> </m:mrow> <m:mo mathvariant="italic" separator="true"> </m:mo> <m:mrow> <m:msub> <m:mover accent="true"> <m:mi mathvariant="script">𝒦</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mi>q</m:mi> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mfrac> <m:msup> <m:mn>2</m:mn> <m:mi>q</m:mi> </m:msup> <m:msup> <m:mi>π</m:mi> <m:mfrac> <m:mi>q</m:mi> <m:mn>2</m:mn> </m:mfrac> </m:msup> </m:mfrac> <m:mo>⁢</m:mo> <m:mrow> <m:munderover> <m:mo largeop="true" movableli