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).

Family | Orthogonal wrt | |
---|---|---|

Monomial | n/a | |

Bernstein | n/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 \([-1,1]\) | |

Charlier | Poisson distribution | |

Meixner | negative binomial distribution | |

Krawtchouk | binomial distribution | |

Hahn | hypergeometric distribution | |

Gegenbauer | \(\left(1-x^2\right)^{\alpha-\frac{1}{2}}\). | cf Funk-Hecke formula |

??? | Unit ball | Does this have a name? |

Natural exponential family with quadratic variance function have infinite divisibility in all cases except binomial (Morris 1982).

## tools

- Orcuslc/OrthNet: TensorFlow, PyTorch and Numpy layers for generating Orthogonal Polynomials
- Julia’s ApproxFun.jl (see a write-up in my julia notebook.
- Chebfun is a classic MATLAB toolkit for this

## References

*Journal of the Egyptian Mathematical Society*25 (2): 91–110.

*The Annals of Statistics*10 (1): 65–80.

*The Annals of Statistics*11 (2): 515–29.

*The American Statistician*63 (3): 247–53.

*Proceedings of the 13th International Conference on Neural Information Processing Systems*, 290–96. NIPS’00. Cambridge, MA, USA: MIT Press.

*Statistics and Computing*30 (2): 419–46.

*Statistics & Probability Letters*47 (2): 165–69.

*SIAM Journal on Scientific Computing*24 (2): 619–44.

*Journal of Computational and Applied Mathematics*, Numerical Analysis 2000. Vol. V: Quadrature and Orthogonal Polynomials, 127 (1): 349–68.

*Advances in Computational Mathematics*20 (1): 247–60.

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