Jacobi polynomials

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1: 18.3 Definitions

§18.3 Definitions

… ►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. …

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Name	$p_n(x)$	$(a,b)$	$w(x)$	$h_n$	$k_n$	${\tilde{k}}_n/k_n$	Constraints
Jacobi	$P_n^{(\alpha ,\beta )}\left(x\right)$	$(-1,1)$	${(1-x)}^{\alpha }{(1+x)}^{\beta }$	${𝒜}_n$	${\displaystyle \frac{{\left(n+\alpha +\beta +1\right)}_n}{2^{n}n!}}$	${\displaystyle \frac{n(\alpha -\beta )}{2n+\alpha +\beta }}$	$\alpha ,\beta >-1$
…

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►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). … ►However, most of these formulas can be obtained by specialization of formulas for Jacobi polynomials, via (18.7.4)–(18.7.6). …

2: 18.14 Inequalities

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Jacobi

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Jacobi

►Let $R_n(x)=P_n^{(\alpha ,\beta )}\left(x\right)/P_n^{(\alpha ,\beta )}\left(1\right)$. … ►

Jacobi

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Szegő–Szász Inequality

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3: 18.7 Interrelations and Limit Relations

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18.7.3 $$T_n\left(x\right)=P_n^{(-\frac{1}{2},-\frac{1}{2})}\left(x\right)/P_n^{(-\frac{1}{2},-\frac{1}{2})}\left(1\right),$$

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.3.15, 22.5.31

Referenced by:

§18.17(viii), §18.18(i), §18.5(iii), §18.7(i), §18.7(ii)

Permalink:

http://dlmf.nist.gov/18.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

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18.7.4 $$U_n\left(x\right)=C_n^{(1)}\left(x\right)=(n+1)P_n^{(\frac{1}{2},\frac{1}{2})}\left(x\right)/P_n^{(\frac{1}{2},\frac{1}{2})}\left(1\right),$$

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Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.3.16, 22.5.32

Referenced by:

§18.12, §18.3, §18.7(i), §18.7(ii)

Permalink:

http://dlmf.nist.gov/18.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

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18.7.13 $$\frac{P_{2n}^{(\alpha ,\alpha )}\left(x\right)}{P_{2n}^{(\alpha ,\alpha )}\left(1\right)}=\frac{P_n^{(\alpha ,-\frac{1}{2})}\left(2x^2-1\right)}{P_n^{(\alpha ,-\frac{1}{2})}\left(1\right)},$$

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.7(ii)

Permalink:

http://dlmf.nist.gov/18.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(ii), §18.7 and Ch.18

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18.7.14 $$\frac{P_{2n+1}^{(\alpha ,\alpha )}\left(x\right)}{P_{2n+1}^{(\alpha ,\alpha )}\left(1\right)}=\frac{x{P}_n^{(\alpha ,\frac{1}{2})}\left(2x^2-1\right)}{P_n^{(\alpha ,\frac{1}{2})}\left(1\right)}.$$

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.7(ii)

Permalink:

http://dlmf.nist.gov/18.7.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(ii), §18.7 and Ch.18

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18.7.23 $$\underset{\alpha \to \mathrm{\infty }}{lim}{\alpha }^{-\frac{1}{2}n}{P}_n^{(\alpha ,\alpha )}\left({\alpha }^{-\frac{1}{2}}x\right)=\frac{H_n\left(x\right)}{2^{n}n!}.$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.10(ii), §18.7(iii)

Permalink:

http://dlmf.nist.gov/18.7.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

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4: 18.6 Symmetry, Special Values, and Limits to Monomials

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Table 18.6.1: Classical OP’s: symmetry and special values.

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$p_n(x)$	$p_n(-x)$	$p_n(1)$	$p_{2n}(0)$	$p_{2n+1}^{\prime }(0)$
$P_n^{(\alpha ,\beta )}\left(x\right)$	${(-1)}^n{P}_n^{(\beta ,\alpha )}\left(x\right)$	${\left(\alpha +1\right)}_n/n!$
$P_n^{(\alpha ,\alpha )}\left(x\right)$	${(-1)}^n{P}_n^{(\alpha ,\alpha )}\left(x\right)$	${\left(\alpha +1\right)}_n/n!$	${(-\frac{1}{4})}^n{\left(n+\alpha +1\right)}_n/n!$	${(-\frac{1}{4})}^n{\left(n+\alpha +1\right)}_{n+1}/n!$
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§18.6(ii) Limits to Monomials

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18.6.2 $$\underset{\alpha \to \mathrm{\infty }}{lim}\frac{P_n^{(\alpha ,\beta )}\left(x\right)}{P_n^{(\alpha ,\beta )}\left(1\right)}={\left(\frac{1+x}{2}\right)}^n,$$

ⓘ

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.6(ii)

Permalink:

http://dlmf.nist.gov/18.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.6(ii), §18.6 and Ch.18

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18.6.3 $$\underset{\beta \to \mathrm{\infty }}{lim}\frac{P_n^{(\alpha ,\beta )}\left(x\right)}{P_n^{(\alpha ,\beta )}\left(-1\right)}={\left(\frac{1-x}{2}\right)}^n,$$

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.6(ii)

Permalink:

http://dlmf.nist.gov/18.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.6(ii), §18.6 and Ch.18

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5: 18.9 Recurrence Relations and Derivatives

… ►For $p_n(x)=P_n^{(\alpha ,\beta )}\left(x\right)$, … ►

18.9.3 $$P_n^{(\alpha ,\beta -1)}\left(x\right)-P_n^{(\alpha -1,\beta )}\left(x\right)=P_{n-1}^{(\alpha ,\beta )}\left(x\right),$$

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.7.20

Referenced by:

§18.9(ii)

Permalink:

http://dlmf.nist.gov/18.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18

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18.9.4 $$(1-x)P_n^{(\alpha +1,\beta )}\left(x\right)+(1+x)P_n^{(\alpha ,\beta +1)}\left(x\right)=2P_n^{(\alpha ,\beta )}\left(x\right).$$

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.7.17

Permalink:

http://dlmf.nist.gov/18.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18

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Jacobi

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18.9.17 $$(2n+\alpha +\beta )(1-x^2)\frac{d}{dx}{P}_n^{(\alpha ,\beta )}\left(x\right)=n\left(\alpha -\beta -(2n+\alpha +\beta )x\right)P_n^{(\alpha ,\beta )}\left(x\right)+2(n+\alpha )(n+\beta )P_{n-1}^{(\alpha ,\beta )}\left(x\right),$$

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\frac{df}{dx}$: derivative, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.11.1

Referenced by:

§18.9(iii)

Permalink:

http://dlmf.nist.gov/18.9.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18

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6: 18.30 Associated OP’s

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Associated Jacobi Polynomials

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18.30.4 $$P_n^{(\alpha ,\beta )}(x;c)=p_n(x;c),$$ $n=0,1,\mathrm{\dots }$,

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Defines:

$P_n^{(\alpha ,\beta )}(x;c)$: associated Jacobi polynomial

Symbols:

$p_n(x)$: polynomial of degree $n$ , $x$: real variable and $n$: continuous nonnegative real

Permalink:

http://dlmf.nist.gov/18.30.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30, §18.30 and Ch.18

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18.30.5 $$\frac{{(-1)}^n{\left(\alpha +\beta +c+1\right)}_{n}n!P_n^{(\alpha ,\beta )}(x;c)}{{\left(\alpha +\beta +2c+1\right)}_n{\left(\beta +c+1\right)}_n}=\sum _{\mathrm{\ell }=0}^n\frac{{\left(-n\right)}_{\mathrm{\ell }}{\left(n+\alpha +\beta +2c+1\right)}_{\mathrm{\ell }}}{{\left(c+1\right)}_{\mathrm{\ell }}{\left(\beta +c+1\right)}_{\mathrm{\ell }}}{\left(\frac{1}{2}x+\frac{1}{2}\right)}^{\mathrm{\ell }}{}_4{}^{}F_3^{}(\genfrac{}{}{0pt}{}{\mathrm{\ell }-n,n+\mathrm{\ell }+\alpha +\beta +2c+1,\beta +c,c}{\beta +\mathrm{\ell }+c+1,\mathrm{\ell }+c+1,\alpha +\beta +2c};1),$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $P_n^{(\alpha ,\beta )}(x;c)$: associated Jacobi polynomial, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $x$: real variable and $n$: continuous nonnegative real

Referenced by:

§18.30

Permalink:

http://dlmf.nist.gov/18.30.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30, §18.30 and Ch.18

… ►For corresponding corecursive associated Jacobi polynomials see Letessier (1995). … ►

18.30.6 $$P_n(x;c)=P_n^{(0,0)}(x;c),$$ $n=0,1,\mathrm{\dots }$.

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Defines:

$P_n(x;c)$: associated Legendre polynomial

Symbols:

$P_n^{(\alpha ,\beta )}(x;c)$: associated Jacobi polynomial, $x$: real variable and $n$: continuous nonnegative real

Permalink:

http://dlmf.nist.gov/18.30.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30, §18.30 and Ch.18

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7: 18.4 Graphics

… ►

►►

Figure 18.4.1: Jacobi polynomials $P_n^{(1.5,-0.5)}\left(x\right)$, $n=1,2,3,4,5$. Magnify

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►►

Figure 18.4.2: Jacobi polynomials $P_n^{(1.25,0.75)}\left(x\right)$, $n=7,8$. This illustrates inequalities for extrema of a Jacobi polynomial; see (18.14.16). … Magnify

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8: 18.8 Differential Equations

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Table 18.8.1: Classical OP’s: differential equations $A(x)f^{\prime \prime }(x)+B(x)f^{\prime }(x)+C(x)f(x)+{\lambda }_{n}f(x)=0$.

► ►►►►►

$f(x)$	$A(x)$	$B(x)$	$C(x)$	${\lambda }_n$
$P_n^{(\alpha ,\beta )}\left(x\right)$	$1-x^2$	$\beta -\alpha -(\alpha +\beta +2)x$	0	$n(n+\alpha +\beta +1)$
$\begin{array}{c}{\left(\sin \frac{1}{2}x\right)}^{\alpha +\frac{1}{2}}{\left(\cos \frac{1}{2}x\right)}^{\beta +\frac{1}{2}}\\ \times P_n^{(\alpha ,\beta )}\left(\cos x\right)\end{array}$	$1$	$0$	${\displaystyle \frac{\frac{1}{4}-{\alpha }^2}{4{\sin }^2\frac{1}{2}x}}+{\displaystyle \frac{\frac{1}{4}-{\beta }^2}{4{\cos }^2\frac{1}{2}x}}$	${\left(n+\frac{1}{2}(\alpha +\beta +1)\right)}^2$
${(\sin x)}^{\alpha +\frac{1}{2}}{P}_n^{(\alpha ,\alpha )}\left(\cos x\right)$	$1$	$0$	$(\frac{1}{4}-{\alpha }^2)/{\sin }^{2}x$	${(n+\alpha +\frac{1}{2})}^2$
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9: 18.18 Sums

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18.18.2 $$f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{P}_n^{(\alpha ,\beta )}\left(z\right),$$

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $z$: complex variable, $n$: nonnegative integer, $f(z)$: analytic function and $a_n$

Referenced by:

§18.18(i), §18.18(i), §18.18(i)

Permalink:

http://dlmf.nist.gov/18.18.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(i), §18.18(i), §18.18 and Ch.18

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18.18.14 $$P_n^{(\gamma ,\beta )}\left(x\right)=\frac{{\left(\beta +1\right)}_n}{{\left(\alpha +\beta +2\right)}_n}\sum _{\mathrm{\ell }=0}^n\frac{\alpha +\beta +2\mathrm{\ell }+1}{\alpha +\beta +1}\frac{{\left(\alpha +\beta +1\right)}_{\mathrm{\ell }}{\left(n+\beta +\gamma +1\right)}_{\mathrm{\ell }}}{{\left(\beta +1\right)}_{\mathrm{\ell }}{\left(n+\alpha +\beta +2\right)}_{\mathrm{\ell }}}\frac{{\left(\gamma -\alpha \right)}_{n-\mathrm{\ell }}}{(n-\mathrm{\ell })!}{P}_{\mathrm{\ell }}^{(\alpha ,\beta )}\left(x\right),$$

ⓘ

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(vi), §18.18(iv)

Permalink:

http://dlmf.nist.gov/18.18.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(iv), §18.18(iv), §18.18 and Ch.18

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Jacobi

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18.18.25 $$\frac{P_n^{(\alpha ,\beta )}\left(x\right)}{P_n^{(\alpha ,\beta )}\left(1\right)}\frac{P_n^{(\alpha ,\beta )}\left(y\right)}{P_n^{(\alpha ,\beta )}\left(1\right)}=\sum _{\mathrm{\ell }=0}^n{b}_{n,\mathrm{\ell }}{(x+y)}^{\mathrm{\ell }}\frac{P_{\mathrm{\ell }}^{(\alpha ,\beta )}\left((1+xy)/(x+y)\right)}{P_{\mathrm{\ell }}^{(\alpha ,\beta )}\left(1\right)},$$

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $y$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.18.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(vi), §18.18(vi), §18.18 and Ch.18

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18.18.26 $$\frac{P_n^{(\alpha ,\beta )}\left(x\right)}{P_n^{(\alpha ,\beta )}\left(1\right)}=\sum _{\mathrm{\ell }=0}^n{b}_{n,\mathrm{\ell }}{(x+1)}^{\mathrm{\ell }}.$$

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.18.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(vi), §18.18(vi), §18.18 and Ch.18

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10: 18.10 Integral Representations

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18.10.1 $$\frac{P_n^{(\alpha ,\alpha )}\left(\cos \theta \right)}{P_n^{(\alpha ,\alpha )}\left(1\right)}=\frac{C_n^{(\alpha +\frac{1}{2})}\left(\cos \theta \right)}{C_n^{(\alpha +\frac{1}{2})}\left(1\right)}=\frac{2^{\alpha +\frac{1}{2}}\mathrm{\Gamma }\left(\alpha +1\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\alpha +\frac{1}{2}\right)}{(\sin \theta )}^{-2\alpha }{\int }_0^{\theta }\frac{\cos \left((n+\alpha +\frac{1}{2})\varphi \right)}{{(\cos \varphi -\cos \theta )}^{-\alpha +\frac{1}{2}}}d\varphi ,$$ , $\alpha >-\frac{1}{2}$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential, $\int$: integral, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial and $n$: nonnegative integer

A&S Ref:

22.10.11

Referenced by:

§18.10(i), §18.10(i)

Permalink:

http://dlmf.nist.gov/18.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(i), §18.10(i), §18.10 and Ch.18

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Jacobi

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18.10.3 $$\frac{P_n^{(\alpha ,\beta )}\left(\cos \theta \right)}{P_n^{(\alpha ,\beta )}\left(1\right)}=\frac{2\mathrm{\Gamma }\left(\alpha +1\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\alpha -\beta \right)\mathrm{\Gamma }\left(\beta +\frac{1}{2}\right)}{\int }_0^1{\int }_0^{\pi }{\left({(\cos \frac{1}{2}\theta )}^2-r^2{(\sin \frac{1}{2}\theta )}^2+\mathrm{i}r\sin \theta \cos \varphi \right)}^n{(1-r^2)}^{\alpha -\beta -1}{r}^{2\beta +1}{(\sin \varphi )}^{2\beta }d\varphi dr,$$ $\alpha >\beta >-\frac{1}{2}$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function and $n$: nonnegative integer

Referenced by:

§18.10(ii)

Permalink:

http://dlmf.nist.gov/18.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

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18.10.4 $$\frac{P_n^{(\alpha ,\alpha )}\left(\cos \theta \right)}{P_n^{(\alpha ,\alpha )}\left(1\right)}=\frac{C_n^{(\alpha +\frac{1}{2})}\left(\cos \theta \right)}{C_n^{(\alpha +\frac{1}{2})}\left(1\right)}=\frac{\mathrm{\Gamma }\left(\alpha +1\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }(\alpha +\frac{1}{2})}{\int }_0^{\pi }{(\cos \theta +\mathrm{i}\sin \theta \cos \varphi )}^n{(\sin \varphi )}^{2\alpha }d\varphi ,$$ $\alpha >-\frac{1}{2}$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial and $n$: nonnegative integer

A&S Ref:

22.10.10

Referenced by:

§18.10(ii)

Permalink:

http://dlmf.nist.gov/18.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

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Table 18.10.1: Classical OP’s: contour integral representations (18.10.8).

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$p_n(x)$	$g_0(x)$	$g_1(z,x)$	$g_2(z,x)$	$c$	Conditions
$P_n^{(\alpha ,\beta )}\left(x\right)$	${(1-x)}^{-\alpha }{(1+x)}^{-\beta }$	${\displaystyle \frac{z^2-1}{2(z-x)}}$	${(1-z)}^{\alpha }{(1+z)}^{\beta }$	$x$	$\pm 1$ outside $C$.
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