18 Orthogonal Polynomials Classical Orthogonal Polynomials 18.6 Symmetry, Special Values, and Limits to Monomials 18.8 Differential Equations

§18.7 Interrelations and Limit Relations

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Keywords:: Chebyshev polynomials, Hermite polynomials, Jacobi polynomials, interrelations with other classical orthogonal polynomials, interrelations with other orthogonal polynomials, ultraspherical polynomials
Permalink:: http://dlmf.nist.gov/18.7
See also:: Annotations for Ch.18

§18.7(i) Linear Transformations

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Keywords:: classical orthogonal polynomials, interrelations, linear
Notes:: For (18.7.1) and (18.7.2) see Szegő (1975, (4.7.1)). For (18.7.3) and (18.7.4) see Szegő (1975, (4.1.7)). For (18.7.5) and (18.7.6) see Szegő (1975, (4.1.8)). (18.7.7)–(18.7.12) follow from the definitions given by Table 18.3.1.
Referenced by:: §18.3
Permalink:: http://dlmf.nist.gov/18.7.i
See also:: Annotations for §18.7 and Ch.18

Ultraspherical and Jacobi

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

18.7.1	$\displaystyle C^{(\lambda)}_{n}\left(x\right)$	$\displaystyle=\frac{{\left(2\lambda\right)_{n}}}{{\left(\lambda+\frac{1}{2}% \right)_{n}}}P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_{n}\left(x\right),$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.5.27 Referenced by: §18.15(ii), §18.7(i), §18.7(ii), §18.7(iii) Permalink: http://dlmf.nist.gov/18.7.E1 Encodings: TeX, pMML, png See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18
18.7.2	$\displaystyle P^{(\alpha,\alpha)}_{n}\left(x\right)$	$\displaystyle=\frac{{\left(\alpha+1\right)_{n}}}{{\left(2\alpha+1\right)_{n}}}% C^{(\alpha+\frac{1}{2})}_{n}\left(x\right).$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.5.20 Referenced by: §18.6(i), §18.7(i) Permalink: http://dlmf.nist.gov/18.7.E2 Encodings: TeX, pMML, png See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

Chebyshev, Ultraspherical, and Jacobi

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Keywords:: Chebyshev polynomials, Jacobi polynomials, relations to other functions
See also:: Annotations for §18.7(i), §18.7 and Ch.18

18.7.3		$T_{n}\left(x\right)=\ifrac{P^{(-\frac{1}{2},-\frac{1}{2})}_{n}\left(x\right)}{% P^{(-\frac{1}{2},-\frac{1}{2})}_{n}\left(1\right)},$
ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : 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(i), §18.7(ii) Permalink: http://dlmf.nist.gov/18.7.E3 Encodings: TeX, pMML, png See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

18.7.4		$U_{n}\left(x\right)=C^{(1)}_{n}\left(x\right)=\ifrac{(n+1)P^{(\frac{1}{2},% \frac{1}{2})}_{n}\left(x\right)}{P^{(\frac{1}{2},\frac{1}{2})}_{n}\left(1% \right)},$
ⓘ Symbols: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : 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: TeX, pMML, png See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

18.7.5		$V_{n}\left(x\right)=\ifrac{P^{(-\frac{1}{2},\frac{1}{2})}_{n}\left(x\right)}{P% ^{(-\frac{1}{2},\frac{1}{2})}_{n}\left(1\right)},$
ⓘ Symbols: $W_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the fourth kind, $V_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the third kind, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(viii), §18.5(i), (18.7.5), (18.7.6), §18.7(i), §18.7(ii), Erratum (V1.0.28) for Table 18.3.1 Permalink: http://dlmf.nist.gov/18.7.E5 Encodings: TeX, pMML, png Correction (effective with 1.0.28): The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ , having been interchanged, the right-hand sides of (18.7.5) and (18.7.6) have been interchanged. For further details see Errata. See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

18.7.6		$W_{n}\left(x\right)=\ifrac{(2n+1)P^{(\frac{1}{2},-\frac{1}{2})}_{n}\left(x% \right)}{P^{(\frac{1}{2},-\frac{1}{2})}_{n}\left(1\right)}.$
ⓘ Symbols: $W_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the fourth kind, $V_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the third kind, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.18(i), §18.3, §18.5(iii), §18.5(i), (18.7.5), (18.7.6), §18.7(i), §18.7(ii), Erratum (V1.0.28) for Table 18.3.1 Permalink: http://dlmf.nist.gov/18.7.E6 Encodings: TeX, pMML, png Correction (effective with 1.0.28): The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ , having been interchanged, the right-hand sides of (18.7.5) and (18.7.6) have been interchanged. For further details see Errata. See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

18.7.7	$\displaystyle T^{*}_{n}\left(x\right)$	$\displaystyle=T_{n}\left(2x-1\right),$
ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $T^{*}_{\NVar{n}}\left(\NVar{x}\right)$ : shifted Chebyshev polynomial of the first kind, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.5.14 Referenced by: §18.7(i) Permalink: http://dlmf.nist.gov/18.7.E7 Encodings: TeX, pMML, png See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18
18.7.8	$\displaystyle U^{*}_{n}\left(x\right)$	$\displaystyle=U_{n}\left(2x-1\right).$
ⓘ Symbols: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $U^{*}_{\NVar{n}}\left(\NVar{x}\right)$ : shifted Chebyshev polynomial of the second kind, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.5.15 Permalink: http://dlmf.nist.gov/18.7.E8 Encodings: TeX, pMML, png See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

See also (18.9.9)–(18.9.12). For (18.7.3) see also (18.7.25).

Legendre, Ultraspherical, and Jacobi

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Keywords:: Legendre polynomials, interrelations with other orthogonal polynomials
See also:: Annotations for §18.7(i), §18.7 and Ch.18

18.7.9		$P_{n}\left(x\right)=C^{(\frac{1}{2})}_{n}\left(x\right)=P^{(0,0)}_{n}\left(x% \right).$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.5.35, 22.5.36 Referenced by: §10.60(iii), §18.18(ii), §18.5(iii) Permalink: http://dlmf.nist.gov/18.7.E9 Encodings: TeX, pMML, png See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

18.7.10		$P^{*}_{n}\left(x\right)=P_{n}\left(2x-1\right).$
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $P^{*}_{\NVar{n}}\left(\NVar{x}\right)$ : shifted Legendre polynomial, $n$ : nonnegative integer and $x$ : real variable Permalink: http://dlmf.nist.gov/18.7.E10 Encodings: TeX, pMML, png See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

Hermite

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Keywords:: classical orthogonal polynomials, interrelations, linear
See also:: Annotations for §18.7(i), §18.7 and Ch.18

18.7.11	$\displaystyle\mathit{He}_{n}\left(x\right)$	$\displaystyle=2^{-\frac{1}{2}n}H_{n}\left(2^{-\frac{1}{2}}x\right),$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.5.18 Referenced by: §18.5(iii) Permalink: http://dlmf.nist.gov/18.7.E11 Encodings: TeX, pMML, png See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18
18.7.12	$\displaystyle H_{n}\left(x\right)$	$\displaystyle=2^{\frac{1}{2}n}\mathit{He}_{n}\left(2^{\frac{1}{2}}x\right).$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.5.19 Referenced by: §18.7(i) Permalink: http://dlmf.nist.gov/18.7.E12 Encodings: TeX, pMML, png See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

§18.7(ii) Quadratic Transformations

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Keywords:: classical orthogonal polynomials, interrelations, quadratic, quadratic transformations
Notes:: For (18.7.13) and (18.7.14) see Szegő (1975, (4.1.5)). (18.7.15) and (18.7.16) follow from (18.7.13) and (18.7.14), combined with (18.7.1). (18.7.17) follows from (18.7.4), (18.7.5), and (18.7.13). (18.7.18) follows from (18.7.3), (18.7.6), and (18.7.14). For (18.7.19) and (18.7.20) see Szegő (1975, (5.6.1)).
Permalink:: http://dlmf.nist.gov/18.7.ii
See also:: Annotations for §18.7 and Ch.18

18.7.13	$\displaystyle\frac{P^{(\alpha,\alpha)}_{2n}\left(x\right)}{P^{(\alpha,\alpha)}% _{2n}\left(1\right)}$	$\displaystyle=\frac{P^{(\alpha,-\frac{1}{2})}_{n}\left(2x^{2}-1\right)}{P^{(% \alpha,-\frac{1}{2})}_{n}\left(1\right)},$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.7(ii), §18.7(ii) Permalink: http://dlmf.nist.gov/18.7.E13 Encodings: TeX, pMML, png See also: Annotations for §18.7(ii), §18.7 and Ch.18
18.7.14	$\displaystyle\frac{P^{(\alpha,\alpha)}_{2n+1}\left(x\right)}{P^{(\alpha,\alpha% )}_{2n+1}\left(1\right)}$	$\displaystyle=\frac{xP^{(\alpha,\frac{1}{2})}_{n}\left(2x^{2}-1\right)}{P^{(% \alpha,\frac{1}{2})}_{n}\left(1\right)}.$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.7(ii) Permalink: http://dlmf.nist.gov/18.7.E14 Encodings: TeX, pMML, png See also: Annotations for §18.7(ii), §18.7 and Ch.18

18.7.15	$\displaystyle C^{(\lambda)}_{2n}\left(x\right)$	$\displaystyle=\frac{{\left(\lambda\right)_{n}}}{{\left(\tfrac{1}{2}\right)_{n}% }}P^{(\lambda-\frac{1}{2},-\frac{1}{2})}_{n}\left(2x^{2}-1\right),$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.5.25 Referenced by: §18.7(ii), §18.9(ii) Permalink: http://dlmf.nist.gov/18.7.E15 Encodings: TeX, pMML, png See also: Annotations for §18.7(ii), §18.7 and Ch.18
18.7.16	$\displaystyle C^{(\lambda)}_{2n+1}\left(x\right)$	$\displaystyle=\frac{{\left(\lambda\right)_{n+1}}}{{\left(\frac{1}{2}\right)_{n% +1}}}xP^{(\lambda-\frac{1}{2},\frac{1}{2})}_{n}\left(2x^{2}-1\right).$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.5.26 Referenced by: §18.7(ii), §18.9(ii) Permalink: http://dlmf.nist.gov/18.7.E16 Encodings: TeX, pMML, png See also: Annotations for §18.7(ii), §18.7 and Ch.18

18.7.17	$\displaystyle U_{2n}\left(x\right)$	$\displaystyle=W_{n}\left(2x^{2}-1\right),$
ⓘ Symbols: $W_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the fourth kind, $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $V_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the third kind, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.5.30 Referenced by: §18.7(ii), Erratum (V1.0.28) for Table 18.3.1 Permalink: http://dlmf.nist.gov/18.7.E17 Encodings: TeX, pMML, png Correction (effective with 1.0.28): The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ , having been interchanged, on the right-hand side we replaced $V_{n}\left(2x^{2}-1\right)$ , with $W_{n}\left(2x^{2}-1\right)$ . For further details see Errata. See also: Annotations for §18.7(ii), §18.7 and Ch.18
18.7.18	$\displaystyle T_{2n+1}\left(x\right)$	$\displaystyle=xV_{n}\left(2x^{2}-1\right).$
ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $W_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the fourth kind, $V_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the third kind, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.5.29 Referenced by: §18.7(ii), Erratum (V1.0.28) for Table 18.3.1 Permalink: http://dlmf.nist.gov/18.7.E18 Encodings: TeX, pMML, png Correction (effective with 1.0.28): The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ , having been interchanged, on the right-hand side we replaced $xW_{n}\left(2x^{2}-1\right)$ , with $xV_{n}\left(2x^{2}-1\right)$ . For further details see Errata. See also: Annotations for §18.7(ii), §18.7 and Ch.18

18.7.19	$\displaystyle H_{2n}\left(x\right)$	$\displaystyle=(-1)^{n}2^{2n}n!L^{(-\frac{1}{2})}_{n}\left(x^{2}\right),$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.5.40 Referenced by: §18.16(v), §18.18(iii), §18.18(iv), §18.18(viii), §18.7(ii) Permalink: http://dlmf.nist.gov/18.7.E19 Encodings: TeX, pMML, png See also: Annotations for §18.7(ii), §18.7 and Ch.18
18.7.20	$\displaystyle H_{2n+1}\left(x\right)$	$\displaystyle=(-1)^{n}2^{2n+1}n!\,xL^{(\frac{1}{2})}_{n}\left(x^{2}\right).$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.5.41 Referenced by: §18.16(v), §18.18(iii), §18.18(iv), §18.7(ii), §18.7(ii) Permalink: http://dlmf.nist.gov/18.7.E20 Encodings: TeX, pMML, png See also: Annotations for §18.7(ii), §18.7 and Ch.18

Equations (18.7.13)–(18.7.20) are special cases of (18.2.22)–(18.2.23).

§18.7(iii) Limit Relations

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Keywords:: classical orthogonal polynomials, interrelations, limiting forms
Notes:: For (18.7.21) see Szegő (1975, (5.3.4)). (18.7.22) follows from (18.7.21) and the symmetry in Row 2 of Table 18.6.1. (18.7.23) follows from (18.7.24) and (18.7.1). For (18.7.24) see Szegő (1975, (5.6.3)). For (18.7.25) see Szegő (1975, (4.7.8)). For (18.7.26) see Calogero (1978).
Referenced by:: §18.19, §18.21(ii)
Permalink:: http://dlmf.nist.gov/18.7.iii
See also:: Annotations for §18.7 and Ch.18

Jacobi $\to$ Laguerre

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Keywords:: Laguerre polynomials, interrelations with other orthogonal polynomials
See also:: Annotations for §18.7(iii), §18.7 and Ch.18

18.7.21		$\lim_{\beta\to\infty}P^{(\alpha,\beta)}_{n}\left(1-\left(\ifrac{2x}{\beta}% \right)\right)=L^{(\alpha)}_{n}\left(x\right).$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : 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: TeX, pMML, png See also: Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

18.7.22		$\lim_{\alpha\to\infty}P^{(\alpha,\beta)}_{n}\left((2x/\alpha)-1\right)=(-1)^{n% }L^{(\beta)}_{n}\left(x\right).$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : 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: TeX, pMML, png See also: Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

Jacobi $\to$ Hermite

ⓘ

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

18.7.23		$\lim_{\alpha\to\infty}\alpha^{-\frac{1}{2}n}P^{(\alpha,\alpha)}_{n}\left(% \alpha^{-\frac{1}{2}}x\right)=\frac{H_{n}\left(x\right)}{2^{n}n!}.$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : 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: TeX, pMML, png See also: Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

Ultraspherical $\to$ Hermite

ⓘ

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

18.7.24		$\lim_{\lambda\to\infty}\lambda^{-\frac{1}{2}n}C^{(\lambda)}_{n}\left(\lambda^{% -\frac{1}{2}}x\right)=\frac{H_{n}\left(x\right)}{n!}.$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $!$ : factorial (as in $n!$ ), $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.15.6 Referenced by: §18.7(iii) Permalink: http://dlmf.nist.gov/18.7.E24 Encodings: TeX, pMML, png See also: Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

Ultraspherical $\to$ Chebyshev

ⓘ

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

18.7.25		$\lim_{\lambda\to 0}\frac{n+\lambda}{\lambda}C^{(\lambda)}_{n}\left(x\right)=% \begin{cases}1,&\text{$n=0$,}\\ 2T_{n}\left(x\right),&\text{$n=1,2,\dots$.}\end{cases}$
ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable Source: Andrews et al. (1999, (6.4.13)) Referenced by: §18.7(i), §18.7(iii), Erratum (V1.2.0) for Equation (18.7.25) Permalink: http://dlmf.nist.gov/18.7.E25 Encodings: TeX, pMML, png Addition (effective with 1.2.0): We included the case $n=0$ . See also: Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

Laguerre $\to$ Hermite

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Keywords:: Chebyshev polynomials, Hermite polynomials, Jacobi polynomials, Laguerre polynomials, interrelations with other classical orthogonal polynomials, interrelations with other orthogonal polynomials, ultraspherical polynomials
See also:: Annotations for §18.7(iii), §18.7 and Ch.18

18.7.26		$\lim_{\alpha\to\infty}\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n}L^{(\alpha)}% _{n}\left((2\alpha)^{\frac{1}{2}}x+\alpha\right)=\frac{(-1)^{n}}{n!}H_{n}\left% (x\right).$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.7(iii) Permalink: http://dlmf.nist.gov/18.7.E26 Encodings: TeX, pMML, png See also: Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

See Figure 18.21.1 for the Askey schematic representation of most of these limits. See §18.11(ii) for limit formulas of Mehler–Heine type.

18.6 Symmetry, Special Values, and Limits to Monomials 18.8 Differential Equations

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§18.7 Interrelations and Limit Relations

Contents

§18.7(i) Linear Transformations

Ultraspherical and Jacobi

Chebyshev, Ultraspherical, and Jacobi

Legendre, Ultraspherical, and Jacobi

Hermite

§18.7(ii) Quadratic Transformations

§18.7(iii) Limit Relations

Jacobi $\to$ Laguerre

Jacobi $\to$ Hermite

Ultraspherical $\to$ Hermite

Ultraspherical $\to$ Chebyshev

Laguerre $\to$ Hermite

pFad - (p)hone/(F)rame/(a)nonymizer/(d)eclutterfier! Saves Data!

§18.7 Interrelations and Limit Relations

Contents

§18.7(i) Linear Transformations

Ultraspherical and Jacobi

Chebyshev, Ultraspherical, and Jacobi

Legendre, Ultraspherical, and Jacobi

Hermite

§18.7(ii) Quadratic Transformations

§18.7(iii) Limit Relations

Jacobi → Laguerre

Jacobi → Hermite

Ultraspherical → Hermite

Ultraspherical → Chebyshev

Laguerre → Hermite

pFad - (p)hone/(F)rame/(a)nonymizer/(d)eclutterfier! Saves Data!

Jacobi $\to$ Laguerre

Jacobi $\to$ Hermite

Ultraspherical $\to$ Hermite

Ultraspherical $\to$ Chebyshev

Laguerre $\to$ Hermite