等差数列
初項 a0 で、公差 d の等差数列に対して、初項から 第 n 項までの総乗
P n := a 0 ⋅ a 1 ⋅ ⋯ ⋅ a n = a 0 ⋅ ( a 0 + d ) ⋅ ⋯ ⋅ ( a 0 + n d ) = d a 0 d ⋅ d ( a 0 d + 1 ) ⋅ ⋯ ⋅ d ( a 0 d + n ) = d n + 1 ( a 0 d ) n + 1 ̄ {\displaystyle {\begin{aligned}P_{n}&:=a_{0}\cdot a_{1}\cdot \dotsb \cdot a_{n}\\&=a_{0}\cdot (a_{0}+d)\cdot \dotsb \cdot (a_{0}+nd)\\&=d{\frac {a_{0}}{d}}\cdot d\left({\frac {a_{0}}{d}}+1\right)\cdot \dotsb \cdot d\left({\frac {a_{0}}{d}}+n\right)\\&=d^{n+1}{\left({\frac {a_{0}}{d}}\right)}^{\overline {n+1}}\end{aligned}}} ( x n ̄ {\displaystyle x^{\overline {n}}} は上昇階乗冪)はガンマ関数 Γ を用いて