Christoffel–Darboux formula

In mathematics, the Christoffel–Darboux formula or Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by Elwin Bruno Christoffel (1858) and Jean Gaston Darboux (1878). It states that

\sum _{j=0}^{n}{\frac {f_{j}(x)f_{j}(y)}{h_{j}}}={\frac {k_{n}}{h_{n}k_{n+1}}}{\frac {f_{n}(y)f_{n+1}(x)-f_{n+1}(y)f_{n}(x)}{x-y}}

where f_j(x) is the jth term of a set of orthogonal polynomials of squared norm h_j and leading coefficient k_j.

There is also a "confluent form" of this identity by taking $y\to x$ limit:

\sum _{j=0}^{n}{\frac {f_{j}^{2}(x)}{h_{j}}}={\frac {k_{n}}{h_{n}k_{n+1}}}\left[f_{n+1}'(x)f_{n}(x)-f_{n}'(x)f_{n+1}(x)\right].

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.