§18.3 Definitions

Notes:: In Table 18.3.1, for Row 2 see Szegö (1975, §2.4, Item 1, (4.3.3), and (4.21.6)): the entry in the last column follows from (18.5.7); for Row 3 see Szegö (1975, (4.7.1), (4.7.14), and (4.7.9)): the entry in the last column follows from the symmetry in the fourth row of Table 18.6.1; for Rows 4–7 see Andrews et al. (1999, §5.1 and Remark 2.5.3); for Row 10 specialize Row 2 to $\alpha=\beta=0$ ; for Row 12 see Szegö (1975, §2.4, Item 2, (5.1.1), and (5.1.8)): the entry in the last column follows from (18.5.12); for Row 13 see Szegö (1975, §2.4, Item 3, (5.5.1), and (5.5.6)): the entry in the last column follows from the symmetry in the tenth row of Table 18.6.1.
Keywords:: Chebyshev polynomials, Hermite polynomials, Jacobi polynomials, Laguerre polynomials, Legendre polynomials, classical orthogonal polynomials, ultraspherical polynomials
Referenced by:: ¶ ‣ §1.17(iii), ¶ ‣ §1.17(iii), ¶ ‣ §1.17(iii), ¶ ‣ §10.23(ii), §10.54, §10.59, §12.7(i), §13.18(v), §13.6(v), §15.9(i), §15.9(ii), §15.9(iii), §18.2(ii), §18.35(iii), ¶ ‣ §18.38(ii), §28.8(ii), §28.9, §29.15(ii), §29.15(ii), ¶ ‣ §3.5(v), ¶ ‣ §3.5(v), ¶ ‣ §3.5(v), ¶ ‣ §3.5(v), §3.5(iv), §3.5(v), §31.11(iv), §31.16(ii), §32.10(iv), §34.3(vii), §7.10, §7.18(iv), §7.6(ii), §8.7
Permalink:: http://dlmf.nist.gov/18.3

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.

Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. In the second row $\mathcal{A}$ denotes $\ifrac{2^{{\alpha+\beta+1}}\mathop{\Gamma\/}\nolimits\!\left(n+\alpha+1\right)\mathop{\Gamma\/}\nolimits\!\left(n+\beta+1\right)}{((2n+\alpha+\beta+1)\mathop{\Gamma\/}\nolimits\!\left(n+\alpha+\beta+1\right)n!)}$ . For further implications of the parameter constraints see the Note in §18.5(iii).

Name	$p_{n}(x)$	$(a,b)$	$w(x)$	$h_{n}$	$k_{n}$	$\ifrac{\tilde{k}_{n}}{k_{n}}$	Constraints

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

Ultraspherical (Gegenbauer)	$\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$	$(-1,1)$	$(1-x^{2})^{{\lambda-\frac{1}{2}}}$	$\dfrac{2^{{1-2\lambda}}\pi\mathop{\Gamma\/}\nolimits\!\left(n+2\lambda\right)}{(n+\lambda)\left(\mathop{\Gamma\/}\nolimits\!\left(\lambda\right)\right)^{2}n!}$	$\dfrac{2^{n}\left(\lambda\right)_{{n}}}{n!}$	0	$\lambda>-\tfrac{1}{2},\lambda\neq 0$


Chebyshev of first kind	$\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$	$(-1,1)$	$(1-x^{2})^{{-\frac{1}{2}}}$	$\begin{cases}\tfrac{1}{2}\pi,&\text{$n>0$}\\ \pi,&\text{$n=0$}\end{cases}$	$\begin{cases}2^{{n-1}},&\text{$n>0$}\\ 1,&\text{$n=0$}\end{cases}$	0


Chebyshev of second kind	$\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$	$(-1,1)$	$(1-x^{2})^{{\frac{1}{2}}}$	$\tfrac{1}{2}\pi$	$2^{n}$	0


Chebyshev of third kind	$\mathop{V_{{n}}\/}\nolimits\!\left(x\right)$	$(-1,1)$	$(1-x)^{{\frac{1}{2}}}(1+x)^{{-\frac{1}{2}}}$	$\pi$	$2^{n}$	$\tfrac{1}{2}$


Chebyshev of fourth kind	$\mathop{W_{{n}}\/}\nolimits\!\left(x\right)$	$(-1,1)$	$(1-x)^{{-\frac{1}{2}}}(1+x)^{{\frac{1}{2}}}$	$\pi$	$2^{n}$	$-\tfrac{1}{2}$


Shifted Chebyshev of first kind	$\mathop{T^{{*}}_{{n}}\/}\nolimits\!\left(x\right)$	$(0,1)$	$(x-x^{2})^{{-\frac{1}{2}}}$	$\begin{cases}\tfrac{1}{2}\pi,&\text{$n>0$}\\ \pi,&\text{$n=0$}\end{cases}$	$\begin{cases}2^{{2n-1}},&\text{$n>0$}\\ 1,&\text{$n=0$}\end{cases}$	$-\tfrac{1}{2}n$


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

Legendre	$\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$	$(-1,1)$	1	$\ifrac{2}{(2n+1)}$	$\ifrac{2^{n}\left(\frac{1}{2}\right)_{{n}}}{n!}$	0
Shifted Legendre	$\mathop{P^{{*}}_{{n}}\/}\nolimits\!\left(x\right)$	$(0,1)$	1	$\ifrac{1}{(2n+1)}$	$\ifrac{2^{{2n}}\left(\frac{1}{2}\right)_{{n}}}{n!}$	$-\frac{1}{2}n$
Laguerre	$\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$	$(0,\infty)$	$e^{{-x}}x^{{\alpha}}$	$\ifrac{\mathop{\Gamma\/}\nolimits\!\left(n+\alpha+1\right)}{n!}$	$\ifrac{(-1)^{n}}{n!}$	$-n(n+\alpha)$	$\alpha>-1$
Hermite	$\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$	$(-\infty,\infty)$	$e^{{-x^{2}}}$	$\pi^{{\frac{1}{2}}}2^{n}n!$	$2^{n}$	0
Hermite	$\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$	$(-\infty,\infty)$	$e^{{-\frac{1}{2}x^{2}}}$	$(2\pi)^{{\frac{1}{2}}}n!$	1	0

Defines:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{W_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the fourth kind, $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $\mathop{V_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the third kind, $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\mathop{T^{{*}}_{{n}}\/}\nolimits\!\left(x\right)$ : shifted Chebyshev polynomial of the first kind, $\mathop{U^{{*}}_{{n}}\/}\nolimits\!\left(x\right)$ : shifted Chebyshev polynomial of the second kind, $\mathop{P^{{*}}_{{n}}\/}\nolimits\!\left(x\right)$ : shifted Legendre polynomial and $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $e$ : base of exponential function, $!$ : $n!$ : factorial, $(a,b)$ : open interval, $w(x)$ : weight function, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ , $x$ : real variable, $h_{n}$ and $k_{n}$
A&S Ref:: 22.2.1, 22.2.3, 22.2.4, 22.2.5, 22.2.8, 22.2.9, 22.2.10, 22.2.11, 22.2.12, 22.2.13, 22.2.14, 22.2.15, and 22.3.1, 22.3.2, 22.3.4, 22.3.6, 22.3.7, 22.3.8, 22.3.9, 22.3.10, 22.3.11 see column for $k_{n}$
Keywords:: Chebyshev polynomials, Hermite polynomials, Jacobi polynomials, Laguerre polynomials, Legendre polynomials, classical orthogonal polynomials, ultraspherical polynomials, weight functions
Referenced by:: §18.15(vi), ¶ ‣ §18.3, §18.3, §18.3, §18.5(i), §18.5(ii), §18.5(iv), §18.7(i), §3.5(v)
Permalink:: http://dlmf.nist.gov/18.3.T1

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

¶ Chebyshev

Keywords:: Chebyshev polynomials

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 $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ , $n=0,1,\dots,N$ , are orthogonal on the discrete point set comprising the zeros $x_{{N+1,n}},n=1,2,\dots,N+1$ , of $\mathop{T_{{N+1}}\/}\nolimits\!\left(x\right)$ :

18.3.1 ${\sum _{{n=1}}^{{N+1}}\mathop{T_{{j}}\/}\nolimits\!\left(x_{{N+1,n}}\right)\mathop{T_{{k}}\/}\nolimits\!\left(x_{{N+1,n}}\right)=0},$ $0\leq j\leq N$ , $0\leq k\leq N$ , $j\neq k$ ,

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $n$ : nonnegative integer, $N$ : positive integer and $x$ : real variable
Referenced by:: ¶ ‣ §18.3
Permalink:: http://dlmf.nist.gov/18.3.E1
Encodings:: TeX, pMML, png

where

18.3.2 $x_{{N+1,n}}=\mathop{\cos\/}\nolimits\!\left((n-\tfrac{1}{2})\pi/(N+1)\right).$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $n$ : nonnegative integer, $N$ : positive integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.3.E2
Encodings:: TeX, pMML, png

When $j=k\neq 0$ the sum in (18.3.1) is $\tfrac{1}{2}(N+1)$ . 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 $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ see (3.11.9).

¶ Legendre

Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). In consequence, additional properties are included in Chapter 14.