18 Orthogonal PolynomialsClassical Orthogonal Polynomials18.2 General Orthogonal Polynomials18.4 Graphics

Table 18.3.1 provides the definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and normalization (§§18.2(i) and 18.2(iii)). This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre.

Name | ${p}_{n}(x)$ | $(a,b)$ | $w(x)$ | ${h}_{n}$ | ${k}_{n}$ | ${\stackrel{~}{k}}_{n}/{k}_{n}$ | Constraints |
---|---|---|---|---|---|---|---|

Jacobi | ${P}_{n}^{(\alpha ,\beta )}\left(x\right)$ | $(-1,1)$ | ${(1-x)}^{\alpha}{(1+x)}^{\beta}$ | ${\mathcal{A}}_{n}$ | $\frac{{\left(n+\alpha +\beta +1\right)}_{n}}{{2}^{n}n!}$ | $\frac{n(\alpha -\beta )}{2n+\alpha +\beta}$ | $\alpha ,\beta >-1$ |

Ultraspherical (Gegenbauer) | ${C}_{n}^{(\lambda )}\left(x\right)$ | $(-1,1)$ | ${(1-{x}^{2})}^{\lambda -\frac{1}{2}}$ | $\frac{{2}^{1-2\lambda}\pi \mathrm{\Gamma}\left(n+2\lambda \right)}{(n+\lambda ){\left(\mathrm{\Gamma}\left(\lambda \right)\right)}^{2}n!}$ | $\frac{{2}^{n}{\left(\lambda \right)}_{n}}{n!}$ | $0$ | $\lambda >-\frac{1}{2},\lambda \ne 0$ |

Chebyshev of first kind | ${T}_{n}\left(x\right)$ | $(-1,1)$ | ${(1-{x}^{2})}^{-\frac{1}{2}}$ | $\{\begin{array}{cc}\frac{1}{2}\pi ,\hfill & n>0\hfill \\ \pi ,\hfill & n=0\hfill \end{array}$ | $\{\begin{array}{cc}{2}^{n-1},\hfill & n>0\hfill \\ 1,\hfill & n=0\hfill \end{array}$ | $0$ | |

Chebyshev of second kind | ${U}_{n}\left(x\right)$ | $(-1,1)$ | ${(1-{x}^{2})}^{\frac{1}{2}}$ | $\frac{1}{2}\pi $ | ${2}^{n}$ | $0$ | |

Chebyshev of third kind | ${V}_{n}\left(x\right)$ | $(-1,1)$ | ${(1-x)}^{\frac{1}{2}}{(1+x)}^{-\frac{1}{2}}$ | $\pi $ | ${2}^{n}$ | $\frac{1}{2}$ | |

Chebyshev of fourth kind | ${W}_{n}\left(x\right)$ | $(-1,1)$ | ${(1-x)}^{-\frac{1}{2}}{(1+x)}^{\frac{1}{2}}$ | $\pi $ | ${2}^{n}$ | $-\frac{1}{2}$ | |

Shifted Chebyshev of first kind | ${T}_{n}^{*}\left(x\right)$ | $(0,1)$ | ${(x-{x}^{2})}^{-\frac{1}{2}}$ | $\{\begin{array}{cc}\frac{1}{2}\pi ,\hfill & n>0\hfill \\ \pi ,\hfill & n=0\hfill \end{array}$ | $\{\begin{array}{cc}{2}^{2n-1},\hfill & n>0\hfill \\ 1,\hfill & n=0\hfill \end{array}$ | $-\frac{1}{2}n$ | |

Shifted Chebyshev of second kind | ${U}_{n}^{*}\left(x\right)$ | $(0,1)$ | ${(x-{x}^{2})}^{\frac{1}{2}}$ | $\frac{1}{8}\pi $ | ${2}^{2n}$ | $-\frac{1}{2}n$ | |

Legendre | ${P}_{n}\left(x\right)$ | $(-1,1)$ | $1$ | $2/(2n+1)$ | ${2}^{n}{\left(\frac{1}{2}\right)}_{n}/n!$ | $0$ | |

Shifted Legendre | ${P}_{n}^{*}\left(x\right)$ | $(0,1)$ | $1$ | $1/(2n+1)$ | ${2}^{2n}{\left(\frac{1}{2}\right)}_{n}/n!$ | $-\frac{1}{2}n$ | |

Laguerre | ${L}_{n}^{(\alpha )}\left(x\right)$ | $(0,\mathrm{\infty})$ | ${\mathrm{e}}^{-x}{x}^{\alpha}$ | $\mathrm{\Gamma}\left(n+\alpha +1\right)/n!$ | ${(-1)}^{n}/n!$ | $-n(n+\alpha )$ | $\alpha >-1$ |

Hermite | ${H}_{n}\left(x\right)$ | $(-\mathrm{\infty},\mathrm{\infty})$ | ${\mathrm{e}}^{-{x}^{2}}$ | ${\pi}^{\frac{1}{2}}{2}^{n}n!$ | ${2}^{n}$ | $0$ | |

Hermite | ${\mathit{He}}_{n}\left(x\right)$ | $(-\mathrm{\infty},\mathrm{\infty})$ | ${\mathrm{e}}^{-\frac{1}{2}{x}^{2}}$ | ${(2\pi )}^{\frac{1}{2}}n!$ | $1$ | $0$ |

For exact values of the coefficients of the Jacobi polynomials ${P}_{n}^{(\alpha ,\beta )}\left(x\right)$, the ultraspherical polynomials ${C}_{n}^{(\lambda )}\left(x\right)$, the Chebyshev polynomials ${T}_{n}\left(x\right)$ and ${U}_{n}\left(x\right)$, the Legendre polynomials ${P}_{n}\left(x\right)$, the Laguerre polynomials ${L}_{n}\left(x\right)$, and the Hermite polynomials ${H}_{n}\left(x\right)$, see Abramowitz and Stegun (1964, pp. 793–801). The Jacobi polynomials are in powers of $x-1$ for $n=0,1,\mathrm{\dots},6$. The ultraspherical polynomials are in powers of $x$ for $n=0,1,\mathrm{\dots},6$. The other polynomials are in powers of $x$ for $n=0,1,\mathrm{\dots},12$. See also §18.5(iv).

In this chapter, formulas for the Chebyshev polynomials of the second, third, and fourth kinds will not be given as extensively as those of the first kind. However, most of these formulas can be obtained by specialization of formulas for Jacobi polynomials, via (18.7.4)–(18.7.6).

In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials ${T}_{n}\left(x\right)$, $n=0,1,\mathrm{\dots},N$, are orthogonal on the discrete point set comprising the zeros ${x}_{N+1,n},n=1,2,\mathrm{\dots},N+1$, of ${T}_{N+1}\left(x\right)$:

18.3.1 | $$\sum _{n=1}^{N+1}{T}_{j}\left({x}_{N+1,n}\right){T}_{k}\left({x}_{N+1,n}\right)=0,$$ | ||

$0\le j\le N$, $0\le k\le N$, $j\ne k$, | |||

where

18.3.2 | $${x}_{N+1,n}=\mathrm{cos}\left((n-\frac{1}{2})\pi /(N+1)\right).$$ | ||

When $j=k\ne 0$ the sum in (18.3.1) is $\frac{1}{2}(N+1)$. When $j=k=0$ the sum in (18.3.1) is $N+1$.

For proofs of these results and for similar properties of the Chebyshev polynomials of the second, third, and fourth kinds see Mason and Handscomb (2003, §4.6). Note that in this reference the definitions of the Chebyshev polynomials of the third and fourth kinds ${V}_{n}\left(x\right)$ and ${W}_{n}\left(x\right)$ are the converse of the definitions in this chapter.

For another version of the discrete orthogonality property of the polynomials ${T}_{n}\left(x\right)$ see (3.11.9).