ハイパー演算子
ただし、他の定義を使うこともある。
n > 4 の場合は次のように定める。これは n > 1 の場合全てに成り立つが、n = 1 では成り立たない。
hyper  n ( a , b ) = hyper  ( a , n , b ) = a ( n ) b = a ( n − 1 ) a ( n − 1 ) ⋯ ( n − 1 ) a ⏟ b copies of a {\displaystyle \operatorname {hyper} n\left(a,b\right)=\operatorname {hyper} \left(a,n,b\right)=a^{\left(n\right)}b=\underbrace {a^{\left(n-1\right)}a^{\left(n-1\right)}\cdots ^{\left(n-1\right)}a} _{b{\text{ copies of }}a}}
hyper0  ( a , b ) = hyper  ( a , 0 , b ) = a ( 0 ) b = b + 1 hyper1  ( a , b ) = hyper  ( a , 1 , b ) = a ( 1 ) b = a + b hyper2  ( a , b ) = hyper  ( a , 2 , b ) = a ( 2 ) b = a b = a + a + ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ + a + a ⏟ 長さ b = ∫ 0 b a d b hyper3  ( a , b ) = hyper  ( a , 3 , b ) = a ( 3 ) b = a b = a ↑ b = a × a × ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ × a × a ⏟ 長さ b hyper4  ( a , b ) = hyper  ( a , 4 , b ) = a ( 4 ) b = b a = a ↑↑ b = a a ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ a a ⏟ 高さ b {\displaystyle {\begin{aligned}\operatorname {hyper0} \left(a,b\right)=\operatorname {hyper} \left(a,0,b\right)=a^{\left(0\right)}b=&b+1\\\operatorname {hyper1} \left(a,b\right)=\operatorname {hyper} \left(a,1,b\right)=a^{\left(1\right)}b=&a+b\\\operatorname {hyper2} \left(a,b\right)=\operatorname {hyper} \left(a,2,b\right)=a^{\left(2\right)}b=&ab=\underbrace {a+{a+{\cdots \cdots \cdots \cdots \cdots \cdots +{a+{a}}}}} _{{\text{長さ}}b}=\int _{0}^{b}a\,db\\\operatorname {hyper3} \left(a,b\right)=\operatorname {hyper} \left(a,3,b\right)=a^{\left(3\right)}b=&a^{b}=a\uparrow b=\underbrace {a\times {a\times {\cdots \cdots \cdots \cdots \cdots \cdots \times {a\times {a}}}}} _{{\text{長さ}}b}\\\operatorname {hyper4} \left(a,b\right)=\operatorname {hyper} \left(a,4,b\right)=a^{\left(4\right)}b=&\,^{b}a=a\uparrow \uparrow b=\underbrace {a^{a^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{a^{a}}}}}}}}}}}}}}}}}}}}}} _{{\text{高さ}}b}\end{aligned}}}