18 Orthogonal Polynomials Classical Orthogonal Polynomials 18.10 Integral Representations 18.12 Generating Functions

§18.11 Relations to Other Functions

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Permalink:: http://dlmf.nist.gov/18.11
See also:: Annotations for Ch.18

§18.11(i) Explicit Formulas
§18.11(ii) Formulas of Mehler–Heine Type

§18.11(i) Explicit Formulas

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Notes:: For (18.11.1) see Andrews et al. (1999, (9.6.7)). (18.11.2) is a rewriting of (18.5.12). For (18.11.3) see Temme (1990a, (3.1)).
Permalink:: http://dlmf.nist.gov/18.11.i
See also:: Annotations for §18.11 and Ch.18

See §§18.5(i) and 18.5(iii) for relations to trigonometric functions, the hypergeometric function, and generalized hypergeometric functions.

Ultraspherical

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Keywords:: Ferrers functions, interrelations with other orthogonal polynomials, relations to other functions, ultraspherical polynomials
See also:: Annotations for §18.11(i), §18.11 and Ch.18

18.11.1		$\mathsf{P}^{m}_{n}\left(x\right)={\left(\tfrac{1}{2}\right)_{m}}(-2)^{m}(1-x^{% 2})^{\frac{1}{2}m}C^{(m+\frac{1}{2})}_{n-m}\left(x\right)={\left(n+1\right)_{m% }}(-2)^{-m}(1-x^{2})^{\frac{1}{2}m}P^{(m,m)}_{n-m}\left(x\right),$
$0\leq m\leq n$ .
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : 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, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable Referenced by: §14.3(iv), §18.11(i) Permalink: http://dlmf.nist.gov/18.11.E1 Encodings: TeX, pMML, png See also: Annotations for §18.11(i), §18.11(i), §18.11 and Ch.18

For the Ferrers function $\mathsf{P}^{m}_{n}\left(x\right)$ , see §14.3(i).

Compare also (14.3.21) and (14.3.22).

Laguerre

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Keywords:: Laguerre polynomials, relation to confluent hypergeometric functions
See also:: Annotations for §18.11(i), §18.11 and Ch.18

18.11.2		$L^{(\alpha)}_{n}\left(x\right)=\frac{{\left(\alpha+1\right)_{n}}}{n!}M\left(-n% ,\alpha+1,x\right)=\frac{(-1)^{n}}{n!}U\left(-n,\alpha+1,x\right)=\frac{{\left% (\alpha+1\right)_{n}}}{n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}x}M_{n+\frac% {1}{2}(\alpha+1),\frac{1}{2}\alpha}\left(x\right)=\frac{(-1)^{n}}{n!}x^{-\frac% {1}{2}(\alpha+1)}e^{\frac{1}{2}x}W_{n+\frac{1}{2}(\alpha+1),\frac{1}{2}\alpha}% \left(x\right).$
ⓘ Symbols: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $x$ : real variable Referenced by: §13.6(v), §18.11(i) Permalink: http://dlmf.nist.gov/18.11.E2 Encodings: TeX, pMML, png See also: Annotations for §18.11(i), §18.11(i), §18.11 and Ch.18

For the confluent hypergeometric functions $M\left(a,b,x\right)$ and $U\left(a,b,x\right)$ , see §13.2(i), and for the Whittaker functions $M_{\kappa,\mu}\left(x\right)$ and $W_{\kappa,\mu}\left(x\right)$ see §13.14(i).

Hermite

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Keywords:: Hermite polynomials, confluent hypergeometric functions, relations to other functions
See also:: Annotations for §18.11(i), §18.11 and Ch.18

18.11.3	$\displaystyle H_{n}\left(x\right)$	$\displaystyle=2^{n}U\left(-\tfrac{1}{2}n,\tfrac{1}{2},x^{2}\right)=2^{n}xU% \left(-\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{3}{2},x^{2}\right)=2^{\frac{1}{2}n}e^% {\frac{1}{2}x^{2}}U\left(-n-\tfrac{1}{2},2^{\frac{1}{2}}x\right).$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.5.55, 22.5.58 ((factor $x$ missing on RHS of 22.5.55)) Referenced by: §18.11(i), §18.15(v) Permalink: http://dlmf.nist.gov/18.11.E3 Encodings: TeX, pMML, png See also: Annotations for §18.11(i), §18.11(i), §18.11 and Ch.18
18.11.4	$\displaystyle\mathit{He}_{n}\left(x\right)$	$\displaystyle=2^{\frac{1}{2}n}U\left(-\tfrac{1}{2}n,\tfrac{1}{2},\tfrac{1}{2}x% ^{2}\right)=2^{\frac{1}{2}(n-1)}xU\left(-\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{3}{% 2},\tfrac{1}{2}x^{2}\right)=e^{\tfrac{1}{4}x^{2}}U\left(-n-\tfrac{1}{2},x% \right).$
ⓘ Symbols: $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.5.59 Permalink: http://dlmf.nist.gov/18.11.E4 Encodings: TeX, pMML, png See also: Annotations for §18.11(i), §18.11(i), §18.11 and Ch.18

For the parabolic cylinder function $U\left(a,z\right)$ , see §12.2.

§18.11(ii) Formulas of Mehler–Heine Type

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Keywords:: Mehler–Heine type formulas, classical orthogonal polynomials, limiting forms
Notes:: For (18.11.5) see Szegő (1975, Theorem 8.1.1). For (18.11.6) see Szegő (1975, Theorem 8.22.4). For (18.11.7) and (18.11.8) see Szegő (1975, Theorem 8.22.8).
Permalink:: http://dlmf.nist.gov/18.11.ii
See also:: Annotations for §18.11 and Ch.18

Jacobi

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Keywords:: Jacobi polynomials, as Bessel functions, limiting form
See also:: Annotations for §18.11(ii), §18.11 and Ch.18

18.11.5		$\lim_{n\to\infty}\frac{1}{n^{\alpha}}P^{(\alpha,\beta)}_{n}\left(1-\frac{z^{2}% }{2n^{2}}\right)=\lim_{n\to\infty}\frac{1}{n^{\alpha}}P^{(\alpha,\beta)}_{n}% \left(\cos\frac{z}{n}\right)=\frac{2^{\alpha}}{z^{\alpha}}J_{\alpha}\left(z% \right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\cos\NVar{z}$ : 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: TeX, pMML, png See also: Annotations for §18.11(ii), §18.11(ii), §18.11 and Ch.18

Laguerre

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Keywords:: Laguerre polynomials, limiting form as a Bessel function
See also:: Annotations for §18.11(ii), §18.11 and Ch.18

18.11.6		$\lim_{n\to\infty}\frac{1}{n^{\alpha}}L^{(\alpha)}_{n}\left(\frac{z}{n}\right)=% \frac{1}{z^{\frac{1}{2}\alpha}}J_{\alpha}\left(2z^{\frac{1}{2}}\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $z$ : complex variable and $n$ : nonnegative integer A&S Ref: 22.15.2 Referenced by: §18.11(ii) Permalink: http://dlmf.nist.gov/18.11.E6 Encodings: TeX, pMML, png See also: Annotations for §18.11(ii), §18.11(ii), §18.11 and Ch.18

Hermite

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Keywords:: Hermite polynomials, limiting forms as trigonometric functions
See also:: Annotations for §18.11(ii), §18.11 and Ch.18

18.11.7	$\displaystyle\lim_{n\to\infty}\frac{(-1)^{n}n^{\frac{1}{2}}}{2^{2n}n!}H_{2n}% \left(\frac{z}{2n^{\frac{1}{2}}}\right)$	$\displaystyle=\frac{1}{\pi^{\frac{1}{2}}}\cos z,$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ), $z$ : complex variable and $n$ : nonnegative integer A&S Ref: 22.15.3 Referenced by: §18.11(ii) Permalink: http://dlmf.nist.gov/18.11.E7 Encodings: TeX, pMML, png See also: Annotations for §18.11(ii), §18.11(ii), §18.11 and Ch.18
18.11.8	$\displaystyle\lim_{n\to\infty}\frac{(-1)^{n}}{2^{2n}n!}H_{2n+1}\left(\frac{z}{% 2n^{\frac{1}{2}}}\right)$	$\displaystyle=\frac{2}{\pi^{\frac{1}{2}}}\sin z.$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : sine function, $z$ : complex variable and $n$ : nonnegative integer A&S Ref: 22.15.4 Referenced by: §18.11(ii), §18.11(ii) Permalink: http://dlmf.nist.gov/18.11.E8 Encodings: TeX, pMML, png See also: Annotations for §18.11(ii), §18.11(ii), §18.11 and Ch.18

For the Bessel function $J_{\nu}\left(z\right)$ , see §10.2(ii). The limits (18.11.5)–(18.11.8) hold uniformly for $z$ in any bounded subset of $\mathbb{C}$ .

§18.11 Relations to Other Functions

Contents

§18.11(i) Explicit Formulas

Ultraspherical

Laguerre

Hermite

§18.11(ii) Formulas of Mehler–Heine Type

Jacobi

Laguerre

Hermite