乗法的不定和分
∏ x a x = C a x 2 ( x − 1 ) , {\displaystyle \prod _{x}a^{x}=Ca^{{\frac {x}{2}}(x-1)},}
∏ x a 1 x = C a Γ ′ ( x ) Γ ( x ) , {\displaystyle \prod _{x}a^{\frac {1}{x}}=Ca^{\frac {\Gamma '(x)}{\Gamma (x)}},}
∏ x x x = C e ζ ′ ( − 1 , x ) − ζ ′ ( − 1 ) = C e ψ ( − 2 ) ( z ) + z 2 − z 2 − z 2 ln  ( 2 π ) = C K  ( x ) . {\displaystyle \prod _{x}x^{x}=Ce^{\zeta ^{\prime }(-1,x)-\zeta ^{\prime }(-1)}=Ce^{\psi ^{(-2)}(z)+{\frac {z^{2}-z}{2}}-{\frac {z}{2}}\ln(2\pi )}=C\operatorname {\mathit {K}} (x).} (ただし、K(x) はK函数である)
∏ x x 2 + 1 = C Γ ( x − i ) Γ ( x + i ) , {\displaystyle \prod _{x}x^{2}+1=C\Gamma (x-i)\Gamma (x+i),}
∏ x a x 2 + b x = C a x Γ ( x ) Γ ( x + b a ) , {\displaystyle \prod _{x}ax^{2}+bx=Ca^{x}\Gamma (x)\Gamma \left(x+{\frac {b}{a}}\right),}
∏ x x + 1 x = C Γ ( x − i ) Γ ( x + i ) Γ ( x ) , {\displaystyle \prod _{x}x+{\frac {1}{x}}={\frac {C\Gamma (x-i)\Gamma (x+i)}{\Gamma (x)}},}
∏ x x a = C Γ ( x ) a , {\displaystyle \prod _{x}x^{a}=C\Gamma (x)^{a},}