Christoffel–Darboux formula

Let ${\displaystyle p_{n}}$ be a sequence of polynomials orthonormal with respect to a probability measure ${\displaystyle \mu }$, and define

$${\displaystyle a_{n}=\langle xp_{n},p_{n+1}\rangle ,\qquad b_{n}=\langle xp_{n},p_{n}\rangle ,\qquad n\geq 0}$$

(they are called the "Jacobi parameters"), then we have the three-term recurrence[1]

$${\displaystyle {\begin{array}{l l}{p_{0}(x)=1,\qquad p_{1}(x)={\frac {x-b_{0}}{a_{0}}},}\\{xp_{n}(x)=a_{n}p_{n+1}(x)+b_{n}p_{n}(x)+a_{n-1}p_{n-1}(x),\qquad n\geq 1}\end{array}}}$$

Proof: By definition, ${\displaystyle \langle xp_{n},p_{k}\rangle =\langle p_{n},xp_{k}\rangle }$, so if ${\displaystyle k\leq n-2}$, then ${\displaystyle xp_{k}}$ is a linear combination of ${\displaystyle p_{0},...,p_{n-1}}$, and thus ${\displaystyle \langle xp_{n},p_{k}\rangle =0}$. So, to construct ${\displaystyle p_{n+1}}$, it suffices to perform Gram-Schmidt process on ${\displaystyle xp_{n}}$ using ${\displaystyle p_{n},p_{n-1}}$, which yields the desired recurrence.

Proof of Christoffel–Darboux formula:

Since both sides are unchanged by multiplying with a constant, we can scale each ${\displaystyle f_{n}}$ to ${\displaystyle p_{n}}$.

Since ${\displaystyle {\frac {k_{n+1}}{k_{n}}}xp_{n}-p_{n+1}}$ is a degree ${\displaystyle n}$ polynomial, it is perpendicular to ${\displaystyle p_{n+1}}$, and so ${\displaystyle \langle {\frac {k_{n+1}}{k_{n}}}xp_{n},p_{n+1}\rangle =\langle p_{n+1},p_{n+1}\rangle =1}$. Now the Christoffel-Darboux formula is proved by induction, using the three-term recurrence.

Hermite polynomials:

$${\displaystyle \sum _{k=0}^{n}{\frac {H_{k}(x)H_{k}(y)}{k!2^{k}}}={\frac {1}{n!2^{n+1}}}\,{\frac {H_{n}(y)H_{n+1}(x)-H_{n}(x)H_{n+1}(y)}{x-y}}.}$$

$${\displaystyle \sum _{k=0}^{n}{\frac {He_{k}(x)He_{k}(y)}{k!}}={\frac {1}{n!}}\,{\frac {He_{n}(y)He_{n+1}(x)-He_{n}(x)He_{n+1}(y)}{x-y}}.}$$

Associated Legendre polynomials:

${\displaystyle {\begin{aligned}(\mu -\mu ')\sum _{l=m}^{L}\,(2l+1){\frac {(l-m)!}{(l+m)!}}\,P_{lm}(\mu )P_{lm}(\mu ')=\qquad \qquad \qquad \qquad \qquad \\{\frac {(L-m+1)!}{(L+m)!}}{\big [}P_{L+1\,m}(\mu )P_{Lm}(\mu ')-P_{Lm}(\mu )P_{L+1\,m}(\mu '){\big ]}.\end{aligned}}}$

• Turán's inequalities
• Sturm Chain

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