# Polynomial bases

Placeholder.

## Fun things

Terry Tao on Conversions between standard polynomial bases.

### Well known facts

Xiu and Karniadakis (2002) mention the following “Well known facts”:

All orthogonal polynomials $$\left\{Q_{n}(x)\right\}$$ satisfy a three-term recurrence relation $-x Q_{n}(x)=A_{n} Q_{n+1}(x)-\left(A_{n}+C_{n}\right) Q_{n}(x)+C_{n} Q_{n-1}(x), \quad n \geq 1$ where $$A_{n}, C_{n} \neq 0$$ and $$C_{n} / A_{n-1}>0 .$$ Together with $$Q_{-1}(x)=0$$ and $$Q_{0}(x)=1,$$ all $$Q_{n}(x)$$ can be determined by the recurrence relation.

It is well known that continuous orthogonal polynomials satisfy the second-order differential equation $s(x) y^{\prime \prime}+\tau(x) y^{\prime}+\lambda y=0$ where $$s(x)$$ and $$\tau(x)$$ are polynomials of at most second and first degree, respectively, and $\lambda=\lambda_{n}=-n \tau^{\prime}-\frac{1}{2} n(n-1) s^{\prime \prime}$ are the eigenvalues of the differential equation; the orthogonal polynomials $$y(x)=$$ $$y_{n}(x)$$ are the eigenfunctions.

## Zoo

This list is extracted from a few places including Xiu and Karniadakis (2002).

FamilyOrthogonal wrt
Monomialn/a
Bernsteinn/a
Legendre$$\operatorname{Unif}([-1,1])$$
Hermite$$\mathcal{N}(0,1)$$
Laguerre$$x^{\alpha}\exp -x, \, x>0$$
Jacobi$$(1-x)^{\alpha }(1+x)^{\beta }$$ on $$[-1,1]$$
CharlierPoisson distribution
Meixnernegative binomial distribution
Krawtchoukbinomial distribution
Hahnhypergeometric distribution
Gegenbauer$$\left(1-x^2\right)^{\alpha-\frac{1}{2}}$$.cf Funk-Hecke formula
???Unit ballDoes this have a name?

Natural exponential family with quadratic variance function have infinite divisibility in all cases except binomial .

## References

Ismail, Mourad E.H., and Ruiming Zhang. 2017. Journal of the Egyptian Mathematical Society 25 (2): 91–110.
Morris, Carl N. 1982. The Annals of Statistics 10 (1): 65–80.
———. 1983. The Annals of Statistics 11 (2): 515–29.
Morris, Carl N., and Kari F. Lock. 2009. The American Statistician 63 (3): 247–53.
O’Hagan, Anthony. 2013. “Polynomial Chaos: A Tutorial and Critique from a Statistician’s Perspective,” 20.
Smola, Alex J., Zoltán L. Óvári, and Robert C. Williamson. 2000. In Proceedings of the 13th International Conference on Neural Information Processing Systems, 290–96. NIPS’00. Cambridge, MA, USA: MIT Press.
Solin, Arno, and Simo Särkkä. 2020. Statistics and Computing 30 (2): 419–46.
Voelker, Aaron R, Ivana Kajic, and Chris Eliasmith. n.d. “Legendre Memory Units: Continuous-Time Representation in Recurrent Neural Networks,” 10.
Withers, C. S. 2000. Statistics & Probability Letters 47 (2): 165–69.
Xiu, Dongbin, and George Em Karniadakis. 2002. SIAM Journal on Scientific Computing 24 (2): 619–44.
Xu, Yuan. 2001. Journal of Computational and Applied Mathematics, Numerical Analysis 2000. Vol. V: Quadrature and Orthogonal Polynomials, 127 (1): 349–68.
———. 2004. Advances in Computational Mathematics 20 (1): 247–60.

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