Теорема о разностях

Теорема о разностях — теорема, связывающая понятия производной и прямой конечной разности высших порядков для степенной функции натурального показателя степени.

Теорема

Теорема о разностях утверждает, что для любой степенной функции с натуральным показателем степени справедливо равенство производной и конечной разности порядка, равного показателю степени и равняется показателю степени под знаком факториала.

Доказательство

Рассмотрим степенную функцию вида $f(x_{i})=x_{i}^{n},\ x_{i}=i\cdot \Delta x,\ \Delta x=x_{i+1}-x_{i}$ , где $i,\ n$ — натуральные числа. Прямая конечная разность порядка $n$ ^[1] для такой функции равняется^[2]:

${\frac {\Delta ^{n}(x_{i}^{n})}{(\Delta x)^{n}}}=\sum _{k=0}^{n}{n \choose k}(-1)^{k}\cdot {\big (}i+n-k{\big )}^{n}=n!$

По определению производной, для функции вида $f(x)=x^{n}$ имеем производную порядка $n$ : $f^{(n)}(x)=n!$ . Таким образом, соблюдается равенство ${\frac {d^{n}(x^{n})}{dx^{n}}}={\frac {\Delta ^{n}(x_{i}^{n})}{(\Delta x)^{n}}}=n!$

Примечания

↑ Бахвалов Н.С. Численные методы. — 1- изд. — «Наука», 1975. — С. 66.
↑ Kolosov Petro. Series Representation of Power Function // arXiv:1603.02468 [math]. — 2016-03-08. Архивировано 17 августа 2016 года.

Литература

Richardson, C. H. (1954): An Introduction to the Calculus of Finite Differences (New York: Van Nostrand, 1954) online copy
Jordan, Charles (1939/1965). «Calculus of Finite Differences», Chelsea Publishing. On-line: 1

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