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u , w ∈ [ 0 , 1 ] {\displaystyle u,w\in [0,1]}
B ( u ) = ∑ k = 0 n ( n k ) u n − k ∑ j = 0 n − k ( − 1 ) k ( n − k j ) P n − k − j = ∑ k = 0 n ( n k ) P k ( 1 − u ) n − k u k {\displaystyle B(u)=\sum _{k=0}^{n}{\binom {n}{k}}u^{n-k}\sum _{j=0}^{n-k}(-1)^{k}{\binom {n-k}{j}}P_{n-k-j}=\sum _{k=0}^{n}{\binom {n}{k}}P_{k}(1-u)^{n-k}u^{k}}
( n n − k ) = ( n k ) {\displaystyle {\binom {n}{n-k}}={\binom {n}{k}}}
t = 1 − u , v = 1 − w {\displaystyle t=1-u,\ v=1-w}
B ( 1 − t ) {\displaystyle B(1-t)} = ∑ k = 0 n ( n k ) P k ( 1 − ( 1 − t ) ) n − k ( 1 − t ) k {\displaystyle =\sum _{k=0}^{n}{\binom {n}{k}}P_{k}(1-(1-t))^{n-k}(1-t)^{k}} = ∑ k = 0 n ( n k ) P k t n − k ( 1 − t ) k {\displaystyle =\sum _{k=0}^{n}{\binom {n}{k}}P_{k}t^{n-k}(1-t)^{k}} = ∑ k = 0 n ( n n − k ) P n − k ( 1 − t ) n − k t k {\displaystyle =\sum _{k=0}^{n}{\binom {n}{n-k}}P_{n-k}(1-t)^{n-k}t^{k}} = ∑ k = 0 n ( n k ) P n − k ( 1 − t ) n − k t k {\displaystyle =\sum _{k=0}^{n}{\binom {n}{k}}P_{n-k}(1-t)^{n-k}t^{k}} = B g y a k u ( t ) {\displaystyle =B_{gyaku}(t)}
B g y a k u ( l e f t ) ( t ) = B g y a k u ( v t ) = B ( 1 − v t ) = B ( 1 − ( 1 − w ) ( 1 − u ) ) = B ( u + w − w u ) = B [ w + u ( 1 − w ) ] = B r i g h t ( u ) {\displaystyle B_{gyaku(left)}(t)=B_{gyaku}(vt)=B(1-vt)=B(1-(1-w)(1-u))=B(u+w-wu)=B[w+u(1-w)]=B_{right}(u)}
![{\displaystyle u,w\in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b2b00c6ef4e8dd0fd9100f1505b8ab013dcd20a)
![{\displaystyle B_{gyaku(left)}(t)=B_{gyaku}(vt)=B(1-vt)=B(1-(1-w)(1-u))=B(u+w-wu)=B[w+u(1-w)]=B_{right}(u)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2bb3bc9fda9fb69246f19156e21e6b45a82a179)