Proof

(related to Corollary: Reciprocity Law of Falling And Rising Factorial Powers)

Let \(x\) be a real (or complex) number and let \(-x\) be its additive inverse (i.e. \(x + (-x)=0\)). Then it follows immediately from the definition of the falling and rising factorial powers that

\[(-x)^{\underline k}=(-x)(-x-1)\cdots(-x-k+1)=(-1)^kx(x+1)\cdot(x+k-1)=(-1)^k x^{\overline k}.\]


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References

Bibliography

  1. Aigner, Martin: "Diskrete Mathematik", vieweg studium, 1993