Retkes identities

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematics, the Retkes Identities, named after Zoltán Retkes, are one of the most efficient applications of the Retkes inequality, when f(u)=u^{\alpha}, 0\leq u <\infty , and 0\leq\alpha. In this special setting, one can have for the iterated integrals

F^{(n-1)}(s)=\frac{s^{\alpha+n-1}}{(\alpha+1)(\alpha+2) \cdots (\alpha+n-1)}.

The notation is explained at Hermite–Hadamard inequality.

Particular cases[edit]

Since f is strictly convex if \alpha >1, strictly concave if 0<\alpha<1, linear if \alpha=0,1, the following inequalities and identities hold:

  • 1<\alpha\quad\quad\quad\quad\frac{1}{(\alpha+1)(\alpha+2)\cdots(\alpha+n-1)}\sum_{i=1}^n\frac{x_i^{\alpha+n-1}}{\Pi_k(x_1,\ldots,x_n)}<\frac{1}{n!}\sum_{i=1}^n x_i^{\alpha}
  • \alpha=1\quad\quad\quad\quad\sum_{i=1}^n\frac{x_i^n}{\Pi_i(x_1,\ldots,x_n)}=\sum_{i=1}^n x_i
  • 0<\alpha<1\quad\quad\frac{1}{(\alpha+1)(\alpha+2) \cdots (\alpha+n-1)} \sum_{i=1}^n\frac{x_i^{\alpha+n-1}}{\Pi_k(x_1,\ldots,x_n)}>\frac{1}{n!}\sum_{i=1}^n x_i^{\alpha}
  • \alpha=0\quad\quad\quad\quad\sum_{i=1}^n\frac{x_i^{n-1}}{\Pi_i(x_1,\ldots,x_n)}=1.

Consequences[edit]

One of the consequences of the case \quad\alpha=1 is the Retkes convergence criterion because of the right side of the equality is precisely the nth partial sum of \quad\sum_{i=1}^{\infty}x_i.

Assume henceforth that x_k\neq 0\quad k=1,\ldots,n. Under this condition substituting \quad\frac{1}{x_k} instead of \quad x_k in the second and fourth identities one can have two universal algebraic identities. These four identities are the so-called Retkes identities:

  • \quad\sum_{i=1}^n\frac{x_i^n}{\Pi_i(x_1,\ldots,x_n)}=\sum_{i=1}^n x_i
  • \quad\sum_{i=1}^n\frac{x_i^{n-1}}{\Pi_i(x_1,\ldots,x_n)}=1
  • \quad \sum_{i=1}^n\frac{1}{x_i} = (-1)^{n-1} \prod_{i=1}^n x_i \sum_{i=1}^n \frac{1}{{x_i}^2 \Pi_i(x_1,\ldots,x_n)}
  • \quad\prod_{i=1}^n\frac{1}{x_i}=(-1)^{n-1}\sum_{i=1}^n\frac{1}{x_i\Pi_i(x_1,\ldots,x_n)}

References[edit]