limit of Jacobi polynomials

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1: 18.6 Symmetry, Special Values, and Limits to Monomials

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§18.6(ii) Limits to Monomials

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

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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.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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2: 18.11 Relations to Other Functions

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Jacobi

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18.11.5 $$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n^{\alpha }}{P}_n^{(\alpha ,\beta )}\left(1-\frac{z^2}{2n^2}\right)=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n^{\alpha }}{P}_n^{(\alpha ,\beta )}\left(\cos \frac{z}{n}\right)=\frac{2^{\alpha }}{z^{\alpha }}{J}_{\alpha }\left(z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\cos z$: cosine function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

22.15.1

Referenced by:

§18.11(ii), §18.11(ii)

Permalink:

http://dlmf.nist.gov/18.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.11(ii), §18.11(ii), §18.11 and Ch.18

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3: 18.34 Bessel Polynomials

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18.34.8 $$\underset{\alpha \to \mathrm{\infty }}{lim}\frac{P_n^{(\alpha ,a-\alpha -2)}\left(1+\alpha x\right)}{P_n^{(\alpha ,a-\alpha -2)}\left(1\right)}=y_n(x;a).$$

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Symbols:

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

Referenced by:

§18.34(iii)

Permalink:

http://dlmf.nist.gov/18.34.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.34(iii), §18.34 and Ch.18

►In this limit the finite system of Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$ which is orthogonal on $(1,\mathrm{\infty })$ (see §18.3) tends to the finite system of Romanovski–Bessel polynomials which is orthogonal on $(0,\mathrm{\infty })$ (see (18.34.5_5)). …

4: 18.7 Interrelations and Limit Relations

§18.7 Interrelations and Limit Relations

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18.7.21 $$\underset{\beta \to \mathrm{\infty }}{lim}{P}_n^{(\alpha ,\beta )}\left(1-\left(2x/\beta \right)\right)=L_n^{(\alpha )}\left(x\right).$$

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Symbols:

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

A&S Ref:

22.15.5

Referenced by:

§18.10(ii), §18.7(iii)

Permalink:

http://dlmf.nist.gov/18.7.E21

Encodings:

pMML, png, TeX

See also:

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

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18.7.22 $$\underset{\alpha \to \mathrm{\infty }}{lim}{P}_n^{(\alpha ,\beta )}\left((2x/\alpha )-1\right)={(-1)}^n{L}_n^{(\beta )}\left(x\right).$$

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Symbols:

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

Referenced by:

§18.7(iii)

Permalink:

http://dlmf.nist.gov/18.7.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(iii), §18.7(iii), §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

… ► See §18.11(ii) for limit formulas of Mehler–Heine type.

5: 18.27 $q$-Hahn Class

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From Big $q$-Jacobi to Jacobi

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From Big $q$-Jacobi to Little $q$-Jacobi

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From Little $q$-Jacobi to Jacobi

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18.27.14_4 $$\underset{q\to 1}{lim}{p}_n(x;q^{\alpha },q^{\beta };q)=\frac{n!}{{\left(\alpha +1\right)}_n}{P}_n^{(\alpha ,\beta )}\left(1-2x\right).$$

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Symbols:

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

Source:

Koekoek et al. (2010, (14.12.15))

Referenced by:

§18.27(iv)

Permalink:

http://dlmf.nist.gov/18.27.E14_4

Encodings:

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See also:

Annotations for §18.27(iv), §18.27(iv), §18.27 and Ch.18

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18.27.14_6 $$\underset{q\to 1}{lim}{p}_n((1-q)x;q^{\alpha },0;q)=\frac{n!}{{\left(\alpha +1\right)}_n}{L}_n^{(\alpha )}\left(x\right).$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p_n(x;a,b;q)$: little $q$-Jacobi polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (14.12.12), (14.20.12))

Referenced by:

§18.27(iv), §18.27(iv), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E14_6

Encodings:

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Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(iv), §18.27(iv), §18.27 and Ch.18

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6: 18.17 Integrals

… ►The case $x=1$ is a limit case of an integral for Jacobi polynomials; see Askey and Razban (1972). …

7: Errata

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

8: 18.21 Hahn Class: Interrelations

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18.21.5 $$\underset{N\to \mathrm{\infty }}{lim}{Q}_n(Nx;\alpha ,\beta ,N)=\frac{P_n^{(\alpha ,\beta )}\left(1-2x\right)}{P_n^{(\alpha ,\beta )}\left(1\right)}.$$

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Symbols:

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

Referenced by:

§18.21(ii)

Permalink:

http://dlmf.nist.gov/18.21.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

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9: 18.28 Askey–Wilson Class

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18.28.26 $$\underset{\lambda \to 0}{lim}{r}_n(x/(2\lambda );\lambda ,qa{\lambda }^{-1},qc{\lambda }^{-1},b{c}^{-1}\lambda |q)=P_n(x;a,b,c;q).$$

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Symbols:

$P_n(x;a,b,c;q)$: big $q$-Jacobi polynomial, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $z$: complex variable, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (14.1.18))

Proof sketch:

In the terminating $q$-hypergeometric series which defines the ${}_4{}^{}\varphi _3^{}$ in (18.28.1_5), with ${(az{q}^{\mathrm{\ell }},a{z}^{-1}{q}^{\mathrm{\ell }};q)}_{\mathrm{\ell }}$ written as ${\prod }_{j=0}^{\mathrm{\ell }-1}(1-2a{q}^{j}x+a^2{q}^{2j})$, substitute for $x$ and the parameters according to the left-hand side of the present formula, take the limit, and obtain the ${}_3{}^{}\varphi _2^{}$ as in (18.27.5).

Referenced by:

(18.28.27), (18.28.28), §18.28(x)

Permalink:

http://dlmf.nist.gov/18.28.E26

Encodings:

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See also:

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

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

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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

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

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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

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

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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

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10: 18.10 Integral Representations

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Ultraspherical

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Legendre

… ►Generalizations of (18.10.1) for $P_n^{(\alpha ,\beta )}$ are given in Gasper (1975, (6),(8)) and Koornwinder (1975a, (5.7),(5.8)). … ►

Jacobi

… ►for the Jacobi, Laguerre, and Hermite polynomials. …