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Christoffel–Darboux formula

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 ${\displaystyle y\to x}$ limit:$${\displaystyle \sum _{j=0}^{n}{\frac {f_{j}^{2}($$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].}

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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• Olver, Frank W. J.; Lozier, Daniel W.; Boisvert, Ronald F.; Clark, Charles W. (2010), "NIST Handbook of Mathematical Functions", NIST Digital Library of Mathematical Functions, Cambridge University Press, p. 438, Eqs. 18.2.12 and 18.2.13, ISBN 978-0-521-19225-5 (Hardback, ISBN 978-0-521-14063-8 Paperback)
• Simons, Frederik J.; Dahlen, F. A.; Wieczorek, Mark A. (2006), "Spatiospectral concentration on a sphere", SIAM Review, 48 (1): 504–536, arXiv:math/0408424, Bibcode:2006SIAMR..48..504S, doi:10.1137/S0036144504445765, S2CID 27519592

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