Polynomial bases


You know what is a bloody great introduction to polynomial bases? Golub and Meurant (2010). They cram it into the first chapter and then do computational stuff in the subsequent chapters. It is excellent.

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.


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

FamilyOrthogonal wrt
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 (Morris 1982).


Golub, Gene H., and Gérard Meurant. 2010. Matrices, Moments and Quadrature with Applications. USA: Princeton University Press.
Ismail, Mourad E.H., and Ruiming Zhang. 2017. A Review of Multivariate Orthogonal Polynomials.” Journal of the Egyptian Mathematical Society 25 (2): 91–110.
Morris, Carl N. 1982. Natural Exponential Families with Quadratic Variance Functions.” The Annals of Statistics 10 (1): 65–80.
———. 1983. Natural Exponential Families with Quadratic Variance Functions: Statistical Theory.” The Annals of Statistics 11 (2): 515–29.
Morris, Carl N., and Kari F. Lock. 2009. Unifying the Named Natural Exponential Families and Their Relatives.” 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. Regularization with Dot-Product Kernels.” 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. Hilbert Space Methods for Reduced-Rank Gaussian Process Regression.” 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. A Simple Expression for the Multivariate Hermite Polynomials.” Statistics & Probability Letters 47 (2): 165–69.
Xiu, Dongbin, and George Em Karniadakis. 2002. The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations.” SIAM Journal on Scientific Computing 24 (2): 619–44.
Xu, Yuan. 2001. Orthogonal Polynomials and Cubature Formulae on Balls, Simplices, and Spheres.” Journal of Computational and Applied Mathematics, Numerical Analysis 2000. Vol. V: Quadrature and Orthogonal Polynomials, 127 (1): 349–68.
———. 2004. Polynomial Interpolation on the Unit Sphere and on the Unit Ball.” Advances in Computational Mathematics 20 (1): 247–60.

No comments yet. Why not leave one?

GitHub-flavored Markdown & a sane subset of HTML is supported.