18 Orthogonal Polynomials Classical Orthogonal Polynomials 18.9 Recurrence Relations and Derivatives 18.11 Relations to Other Functions

§18.10 Integral Representations

Keywords:: classical orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.10

§18.10(i) Dirichlet-Mehler-Type Integral Representations
§18.10(ii) Laplace-Type Integral Representations
§18.10(iii) Contour Integral Representations
§18.10(iv) Other Integral Representations

§18.10(i) Dirichlet-Mehler-Type Integral Representations

Notes:: For (18.10.1) combine (14.12.1) and (14.3.21). For (18.10.2) see Szegö (1975, (4.8.6)).
Permalink:: http://dlmf.nist.gov/18.10.i

¶ Ultraspherical

Keywords:: ultraspherical polynomials

18.10.1 $\frac{\mathop{P^{{(\alpha,\alpha)}}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\theta\right)}{\mathop{P^{{(\alpha,\alpha)}}_{{n}}\/}\nolimits\!\left% (1\right)}=\frac{\mathop{C^{{(\alpha+\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(% \mathop{\cos\/}\nolimits\theta\right)}{\mathop{C^{{(\alpha+\frac{1}{2})}}_{{n}% }\/}\nolimits\!\left(1\right)}=\frac{2^{{\alpha+\frac{1}{2}}}\mathop{\Gamma\/}% \nolimits\!\left(\alpha+1\right)}{\pi^{{\frac{1}{2}}}\mathop{\Gamma\/}% \nolimits\!\left(\alpha+\frac{1}{2}\right)}(\mathop{\sin\/}\nolimits\theta)^{{% -2\alpha}}\int_{0}^{\theta}\frac{\mathop{\cos\/}\nolimits\!\left((n+\alpha+% \tfrac{1}{2})\phi\right)}{(\mathop{\cos\/}\nolimits\phi-\mathop{\cos\/}% \nolimits\theta)^{{-\alpha+\frac{1}{2}}}}d\phi,$ $0<\theta<\pi$ , $\alpha>-\tfrac{1}{2}$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial and : nonnegative integer
A&S Ref:: 22.10.11
Referenced by:: ¶ ‣ §18.10(i), §18.10(i)
Permalink:: http://dlmf.nist.gov/18.10.E1
Encodings:: TeX, pMML, png

¶ Legendre

Keywords:: Legendre polynomials

18.10.2 $\mathop{P_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)=\frac% {2^{{\frac{1}{2}}}}{\pi}\int_{0}^{\theta}\frac{\mathop{\cos\/}\nolimits\!\left% ((n+\tfrac{1}{2})\phi\right)}{(\mathop{\cos\/}\nolimits\phi-\mathop{\cos\/}% \nolimits\theta)^{{\frac{1}{2}}}}d\phi,$ $0<\theta<\pi$ .

Symbols:: $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral and : nonnegative integer
A&S Ref:: 22.10.13
Referenced by:: §18.10(i)
Permalink:: http://dlmf.nist.gov/18.10.E2
Encodings:: TeX, pMML, png

Generalizations of (18.10.1) are given in Gasper (1975, (6),(8)) and Koornwinder (1975a, (5.7),(5.8)).

§18.10(ii) Laplace-Type Integral Representations

Notes:: For (18.10.3) see Askey (1975, (4.20)). For (18.10.4) see Andrews et al. (1999, Theorem 6.7.4). For (18.10.5) see Szegö (1975, (4.8.10)). (18.10.6) can be obtained as a limit case of (18.10.3) in view of (18.7.21). (18.10.7) can be obtained as a limit case of (18.10.4) in view of (18.7.23).
Permalink:: http://dlmf.nist.gov/18.10.ii

¶ Jacobi

Keywords:: Jacobi polynomials

18.10.3 $\frac{\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\theta\right)}{\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(% 1\right)}=\frac{2\mathop{\Gamma\/}\nolimits\!\left(\alpha+1\right)}{\pi^{{% \frac{1}{2}}}\mathop{\Gamma\/}\nolimits\!\left(\alpha-\beta\right)\mathop{% \Gamma\/}\nolimits\!\left(\beta+\tfrac{1}{2}\right)}\*\int_{0}^{1}\int_{0}^{% \pi}\left((\mathop{\cos\/}\nolimits\tfrac{1}{2}\theta)^{2}-r^{2}(\mathop{\sin% \/}\nolimits\tfrac{1}{2}\theta)^{2}+ir\mathop{\sin\/}\nolimits\theta\mathop{% \cos\/}\nolimits\phi\right)^{n}(1-r^{2})^{{\alpha-\beta-1}}r^{{2\beta+1}}(% \mathop{\sin\/}\nolimits\phi)^{{2\beta}}d\phi dr,$ $\alpha>\beta>-\tfrac{1}{2}$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and : nonnegative integer
Referenced by:: §18.10(ii)
Permalink:: http://dlmf.nist.gov/18.10.E3
Encodings:: TeX, pMML, png

¶ Ultraspherical

Keywords:: ultraspherical polynomials

18.10.4 ${\frac{\mathop{P^{{(\alpha,\alpha)}}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\theta\right)}{\mathop{P^{{(\alpha,\alpha)}}_{{n}}\/}\nolimits\!\left% (1\right)}=\frac{\mathop{C^{{(\alpha+\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(% \mathop{\cos\/}\nolimits\theta\right)}{\mathop{C^{{(\alpha+\frac{1}{2})}}_{{n}% }\/}\nolimits\!\left(1\right)}}=\frac{\mathop{\Gamma\/}\nolimits\!\left(\alpha% +1\right)}{\pi^{{\frac{1}{2}}}\mathop{\Gamma\/}\nolimits{(\alpha+\tfrac{1}{2})% }}\*{\int_{0}^{\pi}(\mathop{\cos\/}\nolimits\theta+i\mathop{\sin\/}\nolimits% \theta\mathop{\cos\/}\nolimits\phi)^{n}\*(\mathop{\sin\/}\nolimits\phi)^{{2% \alpha}}d\phi},$ $\alpha>-\frac{1}{2}$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial and : nonnegative integer
A&S Ref:: 22.10.10
Referenced by:: §18.10(ii)
Permalink:: http://dlmf.nist.gov/18.10.E4
Encodings:: TeX, pMML, png

¶ Legendre

18.10.5 $\mathop{P_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)=\frac% {1}{\pi}\int_{0}^{\pi}(\mathop{\cos\/}\nolimits\theta+i\mathop{\sin\/}% \nolimits\theta\mathop{\cos\/}\nolimits\phi)^{n}d\phi.$

Symbols:: $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and : nonnegative integer
A&S Ref:: 22.10.12
Referenced by:: §18.10(ii)
Permalink:: http://dlmf.nist.gov/18.10.E5
Encodings:: TeX, pMML, png

¶ Laguerre

Keywords:: Laguerre polynomials

18.10.6 $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x^{2}\right)=\frac{2(-1)^{n}}{% \pi^{{\frac{1}{2}}}\mathop{\Gamma\/}\nolimits\!\left(\alpha+\tfrac{1}{2}\right% )n!}\*\int_{0}^{\infty}\int_{0}^{\pi}{(x^{2}-r^{2}+2ixr\mathop{\cos\/}% \nolimits\phi)^{n}}\*e^{{-r^{2}}}r^{{2\alpha+1}}(\mathop{\sin\/}\nolimits\phi)% ^{{2\alpha}}d\phi dr,$ $\alpha>-\frac{1}{2}$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , : base of exponential function, : : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : nonnegative integer and : real variable
Referenced by:: §18.10(ii)
Permalink:: http://dlmf.nist.gov/18.10.E6
Encodings:: TeX, pMML, png

¶ Hermite

Keywords:: Hermite polynomials

18.10.7 $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)=\frac{2^{n}}{\pi^{{\frac{1}{2}}}}% \int_{{-\infty}}^{\infty}(x+it)^{n}e^{{-t^{2}}}dt.$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, : differential of , : base of exponential function, $\int$ : integral, : nonnegative integer and : real variable
Referenced by:: §18.10(ii)
Permalink:: http://dlmf.nist.gov/18.10.E7
Encodings:: TeX, pMML, png

§18.10(iii) Contour Integral Representations

Notes:: These representations follow from corresponding Rodrigues formulas (§18.5(ii)) or generating functions (§18.12); see Szegö (1975, (4.4.6), (4.82.1), (4.8.16), (4.8.1), (5.4.8)).
Permalink:: http://dlmf.nist.gov/18.10.iii

Table 18.10.1 gives contour integral representations of the form

18.10.8 $p_{n}(x)=\frac{g_{0}(x)}{2\pi i}\int_{C}\left(g_{1}(z,x)\right)^{n}g_{2}(z,x)(% z-c)^{{-1}}dz$

Symbols:: : differential of , $\int$ : integral, : complex variable, : nonnegative integer, $p_{n}(x)$ : polynomial of degree , $g_{0}(x)$ : factors, $g_{1}(z,x)$ : factors, $g_{2}(z,x)$ : factor, : center, : closed contour and : real variable
Referenced by:: Table 18.10.1
Permalink:: http://dlmf.nist.gov/18.10.E8
Encodings:: TeX, pMML, png

for the Jacobi, Laguerre, and Hermite polynomials. Here is a simple closed contour encircling once in the positive sense.

Table 18.10.1: Classical OP’s: contour integral representations (18.10.8).

$p_{n}(x)$	$g_{0}(x)$	$g_{1}(z,x)$	$g_{2}(z,x)$		Conditions

$\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$	$(1-x)^{{-\alpha}}(1+x)^{{-\beta}}$	$\dfrac{z^{2}-1}{2(z-x)}$	$(1-z)^{{\alpha}}(1+z)^{{\beta}}$		$\pm 1$ outside .
$\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$	1	$z^{{-1}}$	$(1-2xz+z^{2})^{{-\lambda}}$	0	$e^{{\pm i\theta}}$ outside (where $x=\mathop{\cos\/}\nolimits\theta$ ).
$\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$	1	$z^{{-1}}$	$\dfrac{1-xz}{1-2xz+z^{2}}$	0
$\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$	1	$z^{{-1}}$	$(1-2xz+z^{2})^{{-1}}$	0
$\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$	1	$z^{{-1}}$	$(1-2xz+z^{2})^{{-\frac{1}{2}}}$	0

$\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$	1	$\dfrac{z^{2}-1}{2(z-x)}$	1
$\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$	$e^{x}x^{{-\alpha}}$	$z(z-x)^{{-1}}$	$z^{{\alpha}}e^{{-z}}$		0 outside .
$\mathop{H_{{n}}\/}\nolimits\!\left(x\right)/n!$	1	$z^{{-1}}$	$e^{{2xz-z^{2}}}$	0
$\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)/n!$	1	$z^{{-1}}$	$e^{{xz-\frac{1}{2}z^{2}}}$	0

§18.10(iv) Other Integral Representations

Notes:: For (18.10.9) see Andrews et al. (1999, (6.2.15)). For (18.10.10) see Andrews et al. (1999, (6.1.4)).
Permalink:: http://dlmf.nist.gov/18.10.iv

¶ Laguerre

Keywords:: Bessel functions, Laguerre polynomials

18.10.9 $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)=\frac{e^{x}x^{{-\frac% {1}{2}\alpha}}}{n!}\int_{0}^{\infty}e^{{-t}}t^{{n+\frac{1}{2}\alpha}}\mathop{J% _{{\alpha}}\/}\nolimits\!\left(2\sqrt{xt}\right)dt,$ $\alpha>-1$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : differential of , : base of exponential function, : : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\int$ : integral, : nonnegative integer and : real variable
A&S Ref:: 22.10.14
Referenced by:: §18.10(iv)
Permalink:: http://dlmf.nist.gov/18.10.E9
Encodings:: TeX, pMML, png

For the Bessel function $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ see §10.2(ii).

¶ Hermite

Keywords:: Hermite polynomials, classical orthogonal polynomials

18.10.10 $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)=\frac{(-2i)^{n}e^{{x^{2}}}}{\pi^{{% \frac{1}{2}}}}\int_{{-\infty}}^{\infty}e^{{-t^{2}}}t^{n}e^{{2ixt}}dt=\frac{2^{% {n+1}}}{\pi^{{\frac{1}{2}}}}e^{{x^{2}}}\int_{0}^{\infty}e^{{-t^{2}}}t^{n}% \mathop{\cos\/}\nolimits\!\left(2xt-\tfrac{1}{2}n\pi\right)dt.$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , : base of exponential function, $\int$ : integral, : nonnegative integer and : real variable
A&S Ref:: 22.10.15
Referenced by:: §18.10(iv)
Permalink:: http://dlmf.nist.gov/18.10.E10
Encodings:: TeX, pMML, png

§18.10 Integral Representations

Contents

§18.10(i) Dirichlet-Mehler-Type Integral Representations

¶ Ultraspherical

¶ Legendre

§18.10(ii) Laplace-Type Integral Representations

¶ Jacobi

¶ Ultraspherical

¶ Legendre

¶ Laguerre

¶ Hermite

§18.10(iii) Contour Integral Representations

§18.10(iv) Other Integral Representations

¶ Laguerre

¶ Hermite