Jacobi polynomials

SearchAdvancedHelp

(0.008 seconds)

21—30 of 70 matching pages

21: 18.30 Associated OP’s

… ►

§18.30(i) Associated Jacobi Polynomials

… ►

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

ⓘ

Defines:

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

Symbols:

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

Referenced by:

§18.30

Permalink:

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

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(i), §18.30 and Ch.18

… ►For corresponding corecursive associated Jacobi polynomials, corecursive associated polynomials being discussed in §18.30(vii), see Letessier (1995). For other results for associated Jacobi polynomials, see Wimp (1987) and Ismail and Masson (1991). … ►

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

ⓘ

Defines:

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

Symbols:

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

Referenced by:

§18.30

Permalink:

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

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(ii), §18.30 and Ch.18

…

22: 18.16 Zeros

… ►

§18.16(ii) Jacobi

►Let ${\theta }_{n,m}={\theta }_{n,m}^{(\alpha ,\beta )}$, $m=1,2,\mathrm{\dots },n$, denote the zeros of $P_n^{(\alpha ,\beta )}\left(\cos \theta \right)$ as function of $\theta$ with … ►Then … ►

Asymptotic Behavior

… ►

18.16.19 $$\mathrm{Disc}\left(P_n^{(\alpha ,\beta )}\right)=2^{-n(n-1)}\prod _{j=1}^n{j}^{j-2n+2}{(j+\alpha )}^{j-1}{(j+\beta )}^{j-1}{(n+j+\alpha +\beta )}^{n-j}.$$

ⓘ

Symbols:

$\mathrm{Disc}\left(x\right)$: discriminant function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial and $n$: nonnegative integer

Proved:

Ismail (2009, (3.4.16))(proved)

Referenced by:

Erratum (V1.2.0) Section 18.16

Permalink:

http://dlmf.nist.gov/18.16.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §18.16(vii), §18.16(vii), §18.16 and Ch.18

…

23: 18.38 Mathematical Applications

… ►The Askey–Gasper inequality ►

18.38.3 $$\sum _{m=0}^n{P}_m^{(\alpha ,0)}\left(x\right)=\frac{{\left(\alpha +2\right)}_n}{n!}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-n,n+\alpha +2,\frac{1}{2}(\alpha +1)}{\alpha +1,\frac{1}{2}(\alpha +3)};\frac{1}{2}(1-x))\ge 0,$$ $x\ge -1$, $\alpha \ge -2$, $n=0,1,\mathrm{\dots }$,

ⓘ

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

Proved:

Askey and Gasper (1976, Theorem 3, p.717)(proved)

Referenced by:

§18.14(iv), §18.18(viii), Erratum (V1.2.0) for Equation (18.38.3)

Permalink:

http://dlmf.nist.gov/18.38.E3

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

This equation was updated to include the value of the sum in terms of the ${}_3{}^{}F_2^{}$ function. Also the constraint was previously $\alpha >-1$.

See also:

Annotations for §18.38(ii), §18.38(ii), §18.38 and Ch.18

►also the case $\beta =0$ of (18.14.26), was used in de Branges’ proof of the long-standing Bieberbach conjecture concerning univalent functions on the unit disk in the complex plane. … ►Dunkl type operators and nonsymmetric polynomials have been associated with various other families in the Askey scheme and $q$-Askey scheme, in particular with Wilson polynomials, see Groenevelt (2007), and with Jacobi polynomials, see Koornwinder and Bouzeffour (2011, §7). …

24: 18.26 Wilson Class: Continued

… ►

18.26.7 $$\underset{t\to \mathrm{\infty }}{lim}\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}\alpha +\frac{1}{2},\frac{1}{2}\alpha +\frac{1}{2},\frac{1}{2}\beta +\frac{1}{2}+it,\frac{1}{2}\beta +\frac{1}{2}-it)}{t^{2n}n!}=P_n^{(\alpha ,\beta )}\left(x\right).$$

ⓘ

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $W_n(x;a,b,c,d)$: Wilson polynomial, $!$: factorial (as in $n!$), $t$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.26(ii)

Permalink:

http://dlmf.nist.gov/18.26.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

…

25: 18.28 Askey–Wilson Class

… ►

§18.28(ix) Continuous $q$-Jacobi Polynomials

… ►The continuous $q$-Jacobi polynomial $P_n^{(\alpha ,\beta )}(x|q)$ is defined by … ►

18.28.27 $$\underset{\lambda \to 0}{lim}{r}_n(bqx/(2\lambda );\lambda ,qb{\lambda }^{-1},q,a|q)={(-b)}^n{q}^{n(n+1)/2}\frac{{(qa;q)}_n}{{(qb;q)}_n}{p}_n(x;a,b;q).$$

ⓘ

Symbols:

$p_n(x;a,b;q)$: little $q$-Jacobi polynomial, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real variable, $n$: nonnegative integer and $x$: real variable

Proved:

Koornwinder (1993, (6.4)); The limit formula there is equivalent to the present formula(proved)

Proof sketch:

Proceed in a similar way as in the proof of (18.28.26). After taking the limit one now obtains a ${}_3{}^{}\varphi _2^{}$ which equals (18.27.14_1) by (18.27.13).

Referenced by:

§18.28(x)

Permalink:

http://dlmf.nist.gov/18.28.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

►

18.28.28 $$\underset{\mu \to 0,\lambda /\mu \to 0}{lim}{r}_n(x/(2\lambda \mu );\lambda /\mu ,qa\mu /\lambda ,1/(\lambda \mu ),qb\lambda \mu |q)=p_n(x;a,b;q).$$

ⓘ

Symbols:

$p_n(x;a,b;q)$: little $q$-Jacobi polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Proof sketch:

Proceed in a similar way as in the proof of (18.28.26). After taking the limit one now obtains a ${}_2{}^{}\varphi _1^{}$ which equals (18.27.13).

Referenced by:

§18.28(x)

Permalink:

http://dlmf.nist.gov/18.28.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

… ►

18.28.30 $$\underset{q\to 1}{lim}{P}_n^{(\alpha ,\beta )}(x|q)=P_n^{(\alpha ,\beta )}\left(x\right).$$

ⓘ

Symbols:

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

Source:

Koekoek et al. (2010, (14.10.15))

Referenced by:

§18.28(x)

Permalink:

http://dlmf.nist.gov/18.28.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

…

26: Richard A. Askey

… ►One of his most influential papers Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials (with J. …Another significant contribution was the Askey-Gasper inequality for Jacobi polynomials which was published in Positive Jacobi polynomial sums. II (with G. …

27: 15.9 Relations to Other Functions

… ►

Jacobi

►

15.9.1 $$P_n^{(\alpha ,\beta )}\left(x\right)=\frac{{\left(\alpha +1\right)}_n}{n!}F(\genfrac{}{}{0pt}{}{-n,n+\alpha +\beta +1}{\alpha +1};\frac{1-x}{2}).$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $x$: real variable and $n$: integer

Referenced by:

§15.12(iii)

Permalink:

http://dlmf.nist.gov/15.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

… ►This is a generalization of Jacobi polynomials (§18.3) and has the representation …

28: 3.5 Quadrature

… ►

Gauss–Jacobi Formula

… ►The $p_n(x)$ are the monic Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$ (§18.3). … ►In case of the Jacobi polynomials we have $p_n(x)=P_n^{(\alpha ,\beta )}\left(x\right)/k_n$, $q_n(x)=P_n^{(\alpha ,\beta )}\left(x\right)/\sqrt{h_n}$, and …

29: Bibliography I

… ►

Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

ⓘ

External Links:

ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt

Referenced by:

§18.30(i).

Encodings:

BibTeX

… ►

Asymptotics of the Askey-Wilson and $q$-Jacobi polynomials. SIAM J. Math. Anal. 17 (6), pp. 1475–1482.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§18.29.

Encodings:

BibTeX

…

30: Errata

… ►We have also incorporated material on continuous $q$-Jacobi polynomials, and several new limit transitions. … ►

18.12.2 $${}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\alpha +1};\frac{(x-1)z}{2}){}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\beta +1};\frac{(x+1)z}{2})={\left(\frac{1}{2}(1-x)z\right)}^{-\frac{1}{2}\alpha }{J}_{\alpha }\left(\sqrt{2(1-x)z}\right){\left(\frac{1}{2}(1+x)z\right)}^{-\frac{1}{2}\beta }{I}_{\beta }\left(\sqrt{2(1+x)z}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{P_n^{(\alpha ,\beta )}\left(x\right)}{\mathrm{\Gamma }\left(n+\alpha +1\right)\mathrm{\Gamma }\left(n+\beta +1\right)}{z}^n$$

This equation was updated to include on the left-hand side, its definition in terms of a product of two ${}_0{}^{}𝐅_1^{}$ functions.

… ►

18.38.3 $$\sum _{m=0}^n{P}_m^{(\alpha ,0)}\left(x\right)=\frac{{\left(\alpha +2\right)}_n}{n!}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-n,n+\alpha +2,\frac{1}{2}(\alpha +1)}{\alpha +1,\frac{1}{2}(\alpha +3)};\frac{1}{2}(1-x))\ge 0,$$ $x\ge -1$, $\alpha \ge -2$, $n=0,1,\mathrm{\dots }$

This equation was updated to include the value of the sum in terms of the ${}_3{}^{}F_2^{}$ function. Also the constraint was previously $-1\le x\le 1$, $\alpha >-1$.

… ►

The generalized hypergeometric function of matrix argument ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$, was linked inadvertently as its single variable counterpart ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$. Furthermore, the Jacobi function of matrix argument $P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$, and the Laguerre function of matrix argument $L_{\nu }^{(\gamma )}\left(𝐓\right)$, were also linked inadvertently (and incorrectly) in terms of the single variable counterparts given by $P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$, and $L_{\nu }^{(\gamma )}\left(𝐓\right)$. In order to resolve these inconsistencies, these functions now link correctly to their respective definitions.

… ►

18.27.6 $$P_n^{(\alpha ,\beta )}(x;c,d;q)=\frac{c^n{q}^{-(\alpha +1)n}{(q^{\alpha +1},-q^{\alpha +1}{c}^{-1}d;q)}_n}{{(q,-q;q)}_n}{P}_n(q^{\alpha +1}{c}^{-1}x;q^{\alpha },q^{\beta },-q^{\alpha }{c}^{-1}d;q)$$

Originally the first argument to the big $q$-Jacobi polynomial on the right-hand side was written incorrectly as $q^{\alpha +1}{c}^{-1}dx$.

Reported 2017-09-27 by Tom Koornwinder.

…