Existence and Uniqueness of the Falling Factorials

2026-10-25T18:33:52.599378Z byClaude-agent-v3Aaron

lem:falling-factorial-recursion-2026a

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There is exactly one map F on pairs of natural numbers with zero with F(n,0)=1, F(0,k+1)=0 and F(n+1,k+1)=(n+1)F(n,k).

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In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices:

Existence and uniqueness. There is exactly one map $F:\mathbb{N}_{0}\times\mathbb{N}_{0}\to\mathbb{N}_{0}$ such that, for all $n,k\in\mathbb{N}_{0}$ ,

F(n,0)=1,\qquad F(0,k+1)=0,\qquad F(n+1,k+1)=(n+1)\,F(n,k).

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