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

The classical OP’s comprise the Jacobi, Laguerre and Hermite polynomials. There are many ways of characterizing the classical OP’s within the general OP’s $\{{p}_{n}(x)\}$, see Al-Salam (1990). The three most important characterizations are:

Table 18.3.1 provides the *traditional* definitions of Jacobi, Laguerre, and
Hermite polynomials via orthogonality and
standardization
(§§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 | ${T}_{n}^{\ast}\left(x\right)$ | $(0,1)$ | ${(x-{x}^{2})}^{-\frac{1}{2}}$ | $\{\begin{array}{cc}\frac{1}{2}\pi ,\hfill & n>0\hfill \\ \pi ,\hfill & n=0\hfill \end{array}$ | $\{\begin{array}{cc}{2}^{2n-1},\hfill & n>0\hfill \\ 1,\hfill & n=0\hfill \end{array}$ | $-\frac{1}{2}n$ | |

Shifted Chebyshev of second kind | ${U}_{n}^{\ast}\left(x\right)$ | $(0,1)$ | ${(x-{x}^{2})}^{\frac{1}{2}}$ | $\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 | ${P}_{n}^{\ast}\left(x\right)$ | $(0,1)$ | $1$ | $1/(2n+1)$ | ${2}^{2n}{\left(\frac{1}{2}\right)}_{n}/n!$ | $-\frac{1}{2}n$ | |

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

For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). For representations of the polynomials in Table 18.3.1 by Rodrigues formulas, see §18.5(ii). For finite power series of the Jacobi, ultraspherical, Laguerre, and Hermite polynomials, see §18.5(iii) (in powers of $x-1$ for Jacobi polynomials, in powers of $x$ for the other cases). Explicit power series for Chebyshev, Legendre, Laguerre, and Hermite polynomials for $n=0,1,\mathrm{\dots},6$ are given in §18.5(iv). For explicit power series coefficients up to $n=12$ for these polynomials and for coefficients up to $n=6$ for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801).

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).

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

For $-1-\beta >\alpha >-1$ a finite system of Jacobi polynomials ${P}_{n}^{(\alpha ,\beta )}\left(x\right)$ is orthogonal on $(1,\mathrm{\infty})$ with weight function $w(x)={(x-1)}^{\alpha}{(x+1)}^{\beta}$. For $\nu \in \mathbb{R}$ and $N>-\frac{1}{2}$ a finite system of Jacobi polynomials ${P}_{n}^{(-N-1+\mathrm{i}\nu ,-N-1-\mathrm{i}\nu )}\left(\mathrm{i}x\right)$ (called pseudo Jacobi polynomials or Routh–Romanovski polynomials) is orthogonal on $(-\mathrm{\infty},\mathrm{\infty})$ with $w(x)={\left(1+{x}^{2}\right)}^{-N-1}{\mathrm{e}}^{2\nu \mathrm{arctan}x}$. For further details see Koekoek et al. (2010, (9.8.3) and §9.9).

Bessel polynomials are often included among the classical OP’s. However, in general they are not orthogonal with respect to a positive measure, but a finite system has such an orthogonality. See §18.34.