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: 37.10 Other Orthogonal Polynomials of Two Variables

… ►There is also a limit to Jacobi polynomials on the triangle (see (37.3.3)): ►

37.10.7 $$\underset{N\to \mathrm{\infty }}{lim}{Q}_{k,n}(Nx,Ny;\alpha ,\beta ,\gamma ,N)=\frac{P_{k,n}^{\alpha ,\beta ,\gamma }(x,y)}{P_{k,n}^{\alpha ,\beta ,\gamma }(0,0)}.$$

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Symbols:

: two-variable Jacobi polynomial on the triangle, $n$: nonnegative integer, $k$: nonnegative integer, $N$: nonnegative integer, $\alpha$: real parameter, $\beta$: real parameter, $Q_{k,n}(x,y;\alpha ,\beta ,\gamma ,N)$: Hahn polynomial of two-variables, $\gamma$: real parameter, $x$: real variable and $y$: real variable

Proof sketch:

Use (18.20.5), (18.5.7), the symmetry of Jacobi polynomials given in Table 18.6.1, and (37.3.3).

Permalink:

http://dlmf.nist.gov/37.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §37.10(iv), §37.10 and Ch.37

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

ⓘ

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.

6: 37.5 Quarter Plane with Weight Function $x^{\alpha }{y}^{\beta }{\mathrm{e}}^{-x-y}$

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37.5.11 $$\underset{\gamma \to \mathrm{\infty }}{lim}{\gamma }^k{R}_{k,n}^{\alpha ,\beta ,\gamma }({\gamma }^{-1}x,{\gamma }^{-1}y)={(-1)}^k{Q}_{k,n}^{(\alpha ,\beta )}(x,y),$$

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Symbols:

: two-variable $R$-Jacobi polynomial on the triangle, $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable, $\alpha$: real parameter, $\gamma$: real parameter, $\beta$: real parameter and $Q_{k,n}^{(\alpha ,\beta )}$: two-variable Laguerre–Jacobi polynomials on ${ℝ}_{+}^2$

Permalink:

http://dlmf.nist.gov/37.5.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §37.5(iii), §37.5 and Ch.37

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37.5.12 $$\underset{\gamma \to \mathrm{\infty }}{lim}{P}_{k,n}^{\alpha ,\beta ,\gamma }({\gamma }^{-1}x,{\gamma }^{-1}y)={(-1)}^n{L}_{n-k}^{(\alpha )}\left(x\right)L_k^{(\beta )}\left(y\right),$$

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Symbols:

: two-variable Jacobi polynomial on the triangle, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable, $\alpha$: real parameter, $\gamma$: real parameter and $\beta$: real parameter

Permalink:

http://dlmf.nist.gov/37.5.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §37.5(iii), §37.5 and Ch.37

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37.5.13 $$\underset{\gamma \to \mathrm{\infty }}{lim}{Q}_{k,n}^{\alpha ,\beta ,\gamma }({\gamma }^{-1}x,{\gamma }^{-1}y)={(-1)}^{n-k}{L}_k^{(\alpha )}\left(x\right)L_{n-k}^{(\beta )}\left(y\right),$$

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Symbols:

: two-variable $Q$-Jacobi polynomial on the triangle, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable, $\alpha$: real parameter, $\gamma$: real parameter and $\beta$: real parameter

Permalink:

http://dlmf.nist.gov/37.5.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §37.5(iii), §37.5 and Ch.37

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37.5.14 $$\underset{\gamma \to \mathrm{\infty }}{lim}{\gamma }^n{V}_{k,n}^{\alpha ,\beta ,\gamma }({\gamma }^{-1}x,{\gamma }^{-1}y)={(-1)}^{n}k!(n-k)!L_k^{(\alpha )}\left(x\right)L_{n-k}^{(\beta )}\left(y\right),$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $!$: factorial (as in $n!$), : monic basis element of two-variable Jacobi polynomial on the triangle, $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable, $\alpha$: real parameter, $\gamma$: real parameter and $\beta$: real parameter

Permalink:

http://dlmf.nist.gov/37.5.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §37.5(iii), §37.5 and Ch.37

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37.5.15 $$\underset{\gamma \to \mathrm{\infty }}{lim}{U}_{k,n}^{\alpha ,\beta ,\gamma }({\gamma }^{-1}x,{\gamma }^{-1}y)=k!(n-k)!L_k^{(\alpha )}\left(x\right)L_{n-k}^{(\beta )}\left(y\right).$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, : co-monic basis element of two-variable Jacobi polynomial on the triangle, $!$: factorial (as in $n!$), $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable, $\alpha$: real parameter, $\gamma$: real parameter and $\beta$: real parameter

Permalink:

http://dlmf.nist.gov/37.5.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §37.5(iii), §37.5 and Ch.37

7: 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:

pMML, png, TeX

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) §18.27

Permalink:

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

Encodings:

pMML, png, TeX

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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8: 37.12 Orthogonal Polynomials on Quadratic Surfaces

… ►Then OPs in the basis (37.12.4) are given in terms of restandardized Jacobi polynomials (37.4.10): … ► … ► … ►

Limits

►The polynomials (37.12.14) are limits of the polynomials (37.12.9): …

9: 18.17 Integrals

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

10: Errata

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