Namespaces
Variants
Actions

Difference between revisions of "Hermite identity"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(LaTeX)
Line 1: Line 1:
−
An identity applied by Ch. Hermite (1873) to certain specially constructed polynomials in the proof that the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046990/h0469901.png" /> is transcendent. In a simplified form it is
+
An identity applied by Ch. Hermite (1873) to certain specially constructed polynomials in the proof that the number $e$ is transcendent. In a simplified form it is
−
 
+
$$
−
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046990/h0469902.png" /></td> </tr></table>
+
e^x F(0) - F(x) = e^x \int_0^x e^{-t} f(t) dt
−
 
+
$$
−
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046990/h0469903.png" /> is a polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046990/h0469904.png" /> and
+
where $f(x)$ is a polynomial in $x$ and
−
 
+
$$
−
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046990/h0469905.png" /></td> </tr></table>
+
F(x) = \sum_{k=0}^\infty f^{(k)}(x) \ .
 +
$$
  
 
====Comments====
 
====Comments====
−
For a proof of the transcendence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046990/h0469906.png" /> and use of Hermite's identity see, e.g., [[#References|[a1]]], Thm. 6.4. There a simplified version of Hermite's original proof is given.
+
For a proof of the transcendence of $e$ and use of Hermite's identity see, e.g., [[#References|[a1]]], Thm. 6.4. There a simplified version of Hermite's original proof is given.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Stewart,  "Galois theory" , Chapman &amp; Hall  (1979)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Stewart,  "Galois theory" , Chapman &amp; Hall  (1979)</TD></TR></table>

Revision as of 18:29, 12 October 2014

An identity applied by Ch. Hermite (1873) to certain specially constructed polynomials in the proof that the number $e$ is transcendent. In a simplified form it is $$ e^x F(0) - F(x) = e^x \int_0^x e^{-t} f(t) dt $$ where $f(x)$ is a polynomial in $x$ and $$ F(x) = \sum_{k=0}^\infty f^{(k)}(x) \ . $$

Comments

For a proof of the transcendence of $e$ and use of Hermite's identity see, e.g., [a1], Thm. 6.4. There a simplified version of Hermite's original proof is given.

References

[a1] I. Stewart, "Galois theory" , Chapman & Hall (1979)
How to Cite This Entry:
Hermite identity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermite_identity&oldid=33581
Retrieved from "https://encyclopediaofmath.org/index.php?title=Hermite_identity&oldid=33581"