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

§18.11 Relations to Other Functions

ⓘ

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

§18.11(i) Explicit Formulas

ⓘ

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

ⓘ

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

ⓘ

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

18.11.2		$L^{(\alpha)}_{n}\left(x\right)=\frac{{\left(\alpha+1\right)_{n}}}{n!}M\left(-n% ,\alpha+1,x\right)=\frac{(-1)^{n}}{n!}U\left(-n,\alpha+1,x\right)=\frac{{\left% (\alpha+1\right)_{n}}}{n!}x^{-\frac{1}{2}(\alpha+1)}{\mathrm{e}}^{\frac{1}{2}x% }M_{n+\frac{1}{2}(\alpha+1),\frac{1}{2}\alpha}\left(x\right)=\frac{(-1)^{n}}{n% !}x^{-\frac{1}{2}(\alpha+1)}{\mathrm{e}}^{\frac{1}{2}x}W_{n+\frac{1}{2}(\alpha% +1),\frac{1}{2}\alpha}\left(x\right).$
ⓘ 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

ⓘ

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	$\displaystyle H_{n}\left(x\right)$	$\displaystyle=2^{n}U\left(-\tfrac{1}{2}n,\tfrac{1}{2},x^{2}\right)=2^{n}xU% \left(-\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{3}{2},x^{2}\right)=2^{\frac{1}{2}n}{% \mathrm{e}}^{\frac{1}{2}x^{2}}U\left(-n-\tfrac{1}{2},2^{\frac{1}{2}}x\right),$
ⓘ 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	$\displaystyle\mathit{He}_{n}\left(x\right)$	$\displaystyle=2^{\frac{1}{2}n}U\left(-\tfrac{1}{2}n,\tfrac{1}{2},\tfrac{1}{2}x% ^{2}\right)=2^{\frac{1}{2}(n-1)}xU\left(-\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{3}{% 2},\tfrac{1}{2}x^{2}\right)={\mathrm{e}}^{\tfrac{1}{4}x^{2}}U\left(-n-\tfrac{1% }{2},x\right).$
ⓘ 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

ⓘ

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).
Referenced by:: §18.7(iii)
Permalink:: http://dlmf.nist.gov/18.11.ii
See also:: Annotations for §18.11 and Ch.18

Jacobi

ⓘ

Keywords:: Bessel functions, Jacobi polynomials, as Bessel functions, limit of Jacobi polynomials, 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

ⓘ

Keywords:: Bessel functions, Laguerre polynomials, limit of 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

ⓘ

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 $\mathbb{C}$ .

18.10 Integral Representations 18.12 Generating Functions

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§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