18 Orthogonal Polynomials Classical Orthogonal Polynomials 18.16 Zeros 18.18 Sums

§18.17 Integrals

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Keywords:: classical orthogonal polynomials, integrals
Referenced by:: §18.10(iv), Erratum (V1.2.0) §18.17
Permalink:: http://dlmf.nist.gov/18.17
See also:: Annotations for Ch.18

§18.17(i) Indefinite Integrals
§18.17(ii) Integral Representations for Products
§18.17(iii) Nicholson-Type Integrals
§18.17(iv) Fractional Integrals
§18.17(v) Fourier Transforms
§18.17(vi) Laplace Transforms
§18.17(vii) Mellin Transforms
§18.17(viii) Other Integrals
§18.17(ix) Compendia

§18.17(i) Indefinite Integrals

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Notes:: For (18.17.1) see Erdélyi et al. (1953b, §10.8(38)). For the first equation in (18.17.2) apply the convolution property of the Laplace transform (§1.14(iii)) to (18.17.34) with $\alpha=0$ . For the second equation combine (18.9.23), (18.9.13), and (18.6.1). For (18.17.3) and (18.17.4) use (18.9.25) and (18.9.26).
Permalink:: http://dlmf.nist.gov/18.17.i
See also:: Annotations for §18.17 and Ch.18

Jacobi

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Keywords:: Jacobi polynomials, indefinite, integrals
See also:: Annotations for §18.17(i), §18.17 and Ch.18

18.17.1		$2n\int_{0}^{x}(1-y)^{\alpha}(1+y)^{\beta}P^{(\alpha,\beta)}_{n}\left(y\right)% \,\mathrm{d}y=P^{(\alpha+1,\beta+1)}_{n-1}\left(0\right)-(1-x)^{\alpha+1}(1+x)% ^{\beta+1}P^{(\alpha+1,\beta+1)}_{n-1}\left(x\right).$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(i), §18.17(i) Permalink: http://dlmf.nist.gov/18.17.E1 Encodings: TeX, pMML, png See also: Annotations for §18.17(i), §18.17(i), §18.17 and Ch.18

Laguerre

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Keywords:: Laguerre polynomials, indefinite, integrals
See also:: Annotations for §18.17(i), §18.17 and Ch.18

18.17.2		$\int_{0}^{x}L_{m}\left(y\right)L_{n}\left(x-y\right)\,\mathrm{d}y=\int_{0}^{x}% L_{m+n}\left(y\right)\,\mathrm{d}y=L_{m+n}\left(x\right)-L_{m+n+1}\left(x% \right).$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $L_{\NVar{n}}\left(\NVar{x}\right)=L^{(0)}_{n}\left(x\right)$ : Laguerre polynomial, $y$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(i) Permalink: http://dlmf.nist.gov/18.17.E2 Encodings: TeX, pMML, png See also: Annotations for §18.17(i), §18.17(i), §18.17 and Ch.18

Hermite

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Keywords:: Hermite polynomials, indefinite, integrals
See also:: Annotations for §18.17(i), §18.17 and Ch.18

18.17.3		$\int_{0}^{x}H_{n}\left(y\right)\,\mathrm{d}y=\frac{1}{2(n+1)}(H_{n+1}\left(x% \right)-H_{n+1}\left(0\right)),$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(i), §18.17(i) Permalink: http://dlmf.nist.gov/18.17.E3 Encodings: TeX, pMML, png See also: Annotations for §18.17(i), §18.17(i), §18.17 and Ch.18

18.17.4		$\int_{0}^{x}{\mathrm{e}}^{-y^{2}}H_{n}\left(y\right)\,\mathrm{d}y=H_{n-1}\left% (0\right)-{\mathrm{e}}^{-x^{2}}H_{n-1}\left(x\right).$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(i), §18.17(i) Permalink: http://dlmf.nist.gov/18.17.E4 Encodings: TeX, pMML, png See also: Annotations for §18.17(i), §18.17(i), §18.17 and Ch.18

Just as the indefinite integrals (18.17.1), (18.17.3) and (18.17.4), many similar formulas can be obtained by applying (1.4.26) to the differentiation formulas (18.9.15), (18.9.16) and (18.9.19)–(18.9.28).

§18.17(ii) Integral Representations for Products

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Keywords:: classical orthogonal polynomials, for products, integral representations
Notes:: For (18.17.5) see Ismail (2009, (9.6.2)). (18.17.6) is the case $\alpha=0$ of (18.17.5).
Permalink:: http://dlmf.nist.gov/18.17.ii
See also:: Annotations for §18.17 and Ch.18

Ultraspherical

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Keywords:: for products, integral representations, ultraspherical polynomials
See also:: Annotations for §18.17(ii), §18.17 and Ch.18

18.17.5		$\frac{C^{(\lambda)}_{n}\left(\cos\theta_{1}\right)}{C^{(\lambda)}_{n}\left(1% \right)}\frac{C^{(\lambda)}_{n}\left(\cos\theta_{2}\right)}{C^{(\lambda)}_{n}% \left(1\right)}=\frac{\Gamma\left(\lambda+\frac{1}{2}\right)}{{\pi}^{\frac{1}{% 2}}\Gamma\left(\lambda\right)}\*\int_{0}^{\pi}\frac{C^{(\lambda)}_{n}\left(% \cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}\cos\phi\right)}{C^{(% \lambda)}_{n}\left(1\right)}(\sin\phi)^{2\lambda-1}\,\mathrm{d}\phi,$
$\lambda>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial and $n$ : nonnegative integer Referenced by: §18.17(ii), §18.17(ii), §18.18(ii) Permalink: http://dlmf.nist.gov/18.17.E5 Encodings: TeX, pMML, png See also: Annotations for §18.17(ii), §18.17(ii), §18.17 and Ch.18

Legendre

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Keywords:: Legendre polynomials, classical orthogonal polynomials, for products, integral representations
See also:: Annotations for §18.17(ii), §18.17 and Ch.18

18.17.6		$P_{n}\left(\cos\theta_{1}\right)P_{n}\left(\cos\theta_{2}\right)=\frac{1}{\pi}% \int_{0}^{\pi}P_{n}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta% _{2}\cos\phi\right)\,\mathrm{d}\phi.$
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function and $n$ : nonnegative integer Referenced by: §18.17(ii), §18.17(ii), §18.18(ii) Permalink: http://dlmf.nist.gov/18.17.E6 Encodings: TeX, pMML, png See also: Annotations for §18.17(ii), §18.17(ii), §18.17 and Ch.18

For formulas for Jacobi and Laguerre polynomials analogous to (18.17.5) and (18.17.6), see Koornwinder (1974, 1977). For addition formulas corresponding to (18.17.5) and (18.17.6) see (18.18.8) and (18.18.9), respectively.

§18.17(iii) Nicholson-Type Integrals

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Notes:: See Durand (1975).
Permalink:: http://dlmf.nist.gov/18.17.iii
See also:: Annotations for §18.17 and Ch.18

Legendre

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Keywords:: Legendre polynomials, Nicholson-type, integrals
See also:: Annotations for §18.17(iii), §18.17 and Ch.18

18.17.7		$\left(P_{n}\left(x\right)\right)^{2}+4{\pi}^{-2}\left(\mathsf{Q}_{n}\left(x% \right)\right)^{2}=4{\pi}^{-2}\*\int_{1}^{\infty}Q_{n}\left(x^{2}+(1-x^{2})t% \right)(t^{2}-1)^{-\frac{1}{2}}\,\mathrm{d}t,$
$-1<x<1$ .
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind, $Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)$ : Legendre function of the second kind, $t$ : real variable, $n$ : nonnegative integer and $x$ : real variable Permalink: http://dlmf.nist.gov/18.17.E7 Encodings: TeX, pMML, png See also: Annotations for §18.17(iii), §18.17(iii), §18.17 and Ch.18

For the Ferrers function $\mathsf{Q}_{n}\left(x\right)$ and Legendre function $Q_{n}\left(x\right)$ see §§14.3(i) and 14.3(ii), with $\mu=0$ and $\nu=n$ .

Hermite

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Keywords:: Hermite polynomials, Nicholson-type, integrals
See also:: Annotations for §18.17(iii), §18.17 and Ch.18

18.17.8		$\left(H_{n}\left(x\right)\right)^{2}+2^{n}(n!)^{2}{\mathrm{e}}^{x^{2}}\left(V% \left(-n-\tfrac{1}{2},2^{\frac{1}{2}}x\right)\right)^{2}=\frac{2^{n+\frac{3}{2% }}n!\,{\mathrm{e}}^{x^{2}}}{\pi}\int_{0}^{\infty}\frac{{\mathrm{e}}^{-(2n+1)t+% x^{2}\tanh t}}{(\sinh 2t)^{\frac{1}{2}}}\,\mathrm{d}t.$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\sinh\NVar{z}$ : hyperbolic sine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $\int$ : integral, $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $t$ : real variable, $n$ : nonnegative integer and $x$ : real variable Permalink: http://dlmf.nist.gov/18.17.E8 Encodings: TeX, pMML, png See also: Annotations for §18.17(iii), §18.17(iii), §18.17 and Ch.18

For the parabolic cylinder function $V\left(a,z\right)$ see §12.2. For similar formulas for ultraspherical polynomials see Durand (1975), and for Jacobi and Laguerre polynomials see Durand (1978).

§18.17(iv) Fractional Integrals

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Notes:: For (18.17.9) and (18.17.10) see Andrews et al. (1999, Theorem 6.7.2). For (18.17.11)–(18.17.15) see Askey and Fitch (1969).
Permalink:: http://dlmf.nist.gov/18.17.iv
See also:: Annotations for §18.17 and Ch.18

Jacobi

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Keywords:: Jacobi polynomials, fractional, integrals
See also:: Annotations for §18.17(iv), §18.17 and Ch.18

18.17.9		$\frac{(1-x)^{\alpha+\mu}P^{(\alpha+\mu,\beta-\mu)}_{n}\left(x\right)}{\Gamma% \left(\alpha+\mu+n+1\right)}=\int_{x}^{1}\frac{(1-y)^{\alpha}P^{(\alpha,\beta)% }_{n}\left(y\right)}{\Gamma\left(\alpha+n+1\right)}\frac{(y-x)^{\mu-1}}{\Gamma% \left(\mu\right)}\,\mathrm{d}y,$
$\mu>0$ , $-1<x<1$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §1.15(vi), §18.17(iv), §18.17(iv), §18.17(viii) Permalink: http://dlmf.nist.gov/18.17.E9 Encodings: TeX, pMML, png See also: Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18

18.17.10	$\displaystyle\frac{x^{\beta+\mu}(x+1)^{n}}{\Gamma\left(\beta+\mu+n+1\right)}P^% {(\alpha,\beta+\mu)}_{n}\left(\frac{x-1}{x+1}\right)$	$\displaystyle=\int_{0}^{x}\frac{y^{\beta}(y+1)^{n}}{\Gamma\left(\beta+n+1% \right)}P^{(\alpha,\beta)}_{n}\left(\frac{y-1}{y+1}\right)\*\frac{(x-y)^{\mu-1% }}{\Gamma\left(\mu\right)}\,\mathrm{d}y,$
$\mu>0$ , $x>0$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(iv), §18.17(iv) Permalink: http://dlmf.nist.gov/18.17.E10 Encodings: TeX, pMML, png See also: Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18
18.17.11	$\displaystyle\frac{\Gamma\left(n+\alpha+\beta-\mu+1\right)}{x^{n+\alpha+\beta-% \mu+1}}P^{(\alpha,\beta-\mu)}_{n}\left(1-2x^{-1}\right)$	$\displaystyle=\int_{x}^{\infty}\frac{\Gamma\left(n+\alpha+\beta+1\right)}{y^{n% +\alpha+\beta+1}}P^{(\alpha,\beta)}_{n}\left(1-2y^{-1}\right)\*\frac{(y-x)^{% \mu-1}}{\Gamma\left(\mu\right)}\,\mathrm{d}y,$
$\alpha+\beta+1>\mu>0$ , $x>1$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §1.15(vi), §18.17(iv), §18.17(iv) Permalink: http://dlmf.nist.gov/18.17.E11 Encodings: TeX, pMML, png See also: Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18

and three formulas similar to (18.17.9)–(18.17.11) by symmetry; compare the second row in Table 18.6.1. Formula (18.17.9), after substitution of (18.5.7), is a special case of (15.6.8). Formulas (18.17.9), (18.17.10) and (18.17.11) are fractional generalizations of $n$ -th derivative formulas which are, after substitution of (18.5.7), special cases of (15.5.4), (15.5.5) and (15.5.3), respectively.

Ultraspherical

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Keywords:: integrals, ultraspherical polynomials
See also:: Annotations for §18.17(iv), §18.17 and Ch.18

18.17.12	$\displaystyle\frac{\Gamma\left(\lambda-\mu\right)C^{(\lambda-\mu)}_{n}\left(x^% {-\frac{1}{2}}\right)}{x^{\lambda-\mu+\frac{1}{2}n}}$	$\displaystyle=\int_{x}^{\infty}\frac{\Gamma\left(\lambda\right)C^{(\lambda)}_{% n}\left(y^{-\frac{1}{2}}\right)}{y^{\lambda+\frac{1}{2}n}}\frac{(y-x)^{\mu-1}}% {\Gamma\left(\mu\right)}\,\mathrm{d}y,$
$\lambda>\mu>0$ , $x>0$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(iv) Permalink: http://dlmf.nist.gov/18.17.E12 Encodings: TeX, pMML, png See also: Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18
18.17.13	$\displaystyle\frac{x^{\frac{1}{2}n}(x-1)^{\lambda+\mu-\frac{1}{2}}}{\Gamma% \left(\lambda+\mu+\tfrac{1}{2}\right)}\frac{C^{(\lambda+\mu)}_{n}\left(x^{-% \frac{1}{2}}\right)}{C^{(\lambda+\mu)}_{n}\left(1\right)}$	$\displaystyle=\int_{1}^{x}\frac{y^{\frac{1}{2}n}(y-1)^{\lambda-\frac{1}{2}}}{% \Gamma\left(\lambda+\tfrac{1}{2}\right)}\frac{C^{(\lambda)}_{n}\left(y^{-\frac% {1}{2}}\right)}{C^{(\lambda)}_{n}\left(1\right)}\frac{(x-y)^{\mu-1}}{\Gamma% \left(\mu\right)}\,\mathrm{d}y,$
$\mu>0$ , $x>1$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §1.15(vi), §18.17(iv) Permalink: http://dlmf.nist.gov/18.17.E13 Encodings: TeX, pMML, png See also: Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18

Formulas (18.17.12) and (18.17.13) are fractional generalizations of the differentiation formulas given in Erdélyi et al. (1953b, §10.9(15)).

Laguerre

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Keywords:: Laguerre polynomials, fractional, integrals
See also:: Annotations for §18.17(iv), §18.17 and Ch.18

18.17.14	$\displaystyle\frac{x^{\alpha+\mu}L^{(\alpha+\mu)}_{n}\left(x\right)}{\Gamma% \left(\alpha+\mu+n+1\right)}$	$\displaystyle=\int_{0}^{x}\frac{y^{\alpha}L^{(\alpha)}_{n}\left(y\right)}{% \Gamma\left(\alpha+n+1\right)}\frac{(x-y)^{\mu-1}}{\Gamma\left(\mu\right)}\,% \mathrm{d}y,$
$\mu>0$ , $x>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.13.13 Referenced by: §18.17(iv) Permalink: http://dlmf.nist.gov/18.17.E14 Encodings: TeX, pMML, png See also: Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18
18.17.15	$\displaystyle{\mathrm{e}}^{-x}L^{(\alpha)}_{n}\left(x\right)$	$\displaystyle=\int_{x}^{\infty}{\mathrm{e}}^{-y}L^{(\alpha+\mu)}_{n}\left(y% \right)\frac{(y-x)^{\mu-1}}{\Gamma\left(\mu\right)}\,\mathrm{d}y,$
$\mu>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(iv), §18.17(iv) Permalink: http://dlmf.nist.gov/18.17.E15 Encodings: TeX, pMML, png See also: Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18

Formulas (18.17.14) and (18.17.15) are fractional generalizations of $n$ -th derivative formulas which are, after substitution of (13.6.19), special cases of (13.3.18) and (13.3.20), respectively.

§18.17(v) Fourier Transforms

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Keywords:: Fourier transforms, classical orthogonal polynomials
Notes:: For (18.17.16) use (18.5.5), integrate by parts $n$ times, expand ${\mathrm{e}}^{-\mathrm{i}y(1-x)}$ in a Maclaurin series, and integrate term by term. For (18.17.17) use (18.5.5), integrate repeatedly by parts, expand $\cos\left(xy\right)$ in a Maclaurin series, and integrate term by term; the proofs of (18.17.18) and (18.17.19) are similar. See also Watson (1944, §3.32). For (18.17.20) expand $\cos\left(xy\right)$ in a Maclaurin series, make the change of integration variable $1-2x^{2}=t$ , apply (18.5.5), integrate by parts $n$ times, and use (10.8.3); the proof of (18.17.21) is similar. For (18.17.22) see Strichartz (1994, §7.6). For (18.17.23) use (18.12.16) and the fact that the Fourier transform of ${\mathrm{e}}^{-\frac{1}{2}x^{2}}$ is ${\mathrm{e}}^{-\frac{1}{2}y^{2}}$ ; the proofs of (18.17.24), (18.17.27), and (18.17.28) are similar, except that in the case of (18.17.24), (18.12.15) replaces (18.12.16). For (18.17.25) use (18.18.23); similarly for (18.17.26). For (18.17.28_5) take $z=-\mathrm{i}y$ in (18.17.34). For (18.17.29) take the inverse Fourier transform and apply (18.17.25). For (18.17.30) consider the Fourier transform of this function instead of the cosine transform, and replace $L^{(n-\frac{1}{2})}_{n}\left(\frac{1}{2}x^{2}\right)$ by its explicit form (18.5.12); then integrate term by term and rearrange the the consequential finite double sum into a single sum. For (18.17.31) use (18.5.5) and then integrate by parts; similarly for (18.17.32).
Referenced by:: §18.17(vi)
Permalink:: http://dlmf.nist.gov/18.17.v
Addition (effective with 1.2.0):: Equations (18.17.16_5), (18.17.21_1)–(18.17.21_3), (18.17.28_5) were added.
See also:: Annotations for §18.17 and Ch.18

Throughout this subsection we assume $y>0$ ; often however, this restriction can be eased by analytic continuation. In particular, in case of exponential Fourier transforms, we may assume $y\in\mathbb{R}$ .

Jacobi

ⓘ

Keywords:: Fourier transform, Jacobi polynomials
See also:: Annotations for §18.17(v), §18.17 and Ch.18

18.17.16		$\int_{-1}^{1}(1-x)^{\alpha}(1+x)^{\beta}P^{(\alpha,\beta)}_{n}\left(x\right){% \mathrm{e}}^{\mathrm{i}xy}\,\mathrm{d}x=\frac{(\mathrm{i}y)^{n}{\mathrm{e}}^{% \mathrm{i}y}}{n!}2^{n+\alpha+\beta+1}\mathrm{B}\left(n+\alpha+1,n+\beta+1% \right){{}_{1}F_{1}}\left(n+\alpha+1;2n+\alpha+\beta+2;-2\mathrm{i}y\right).$
ⓘ Symbols: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ : $=M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ notation for the Kummer confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v), §18.17(vi) Permalink: http://dlmf.nist.gov/18.17.E16 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

For the beta function $\mathrm{B}\left(a,b\right)$ see §5.12, and for the confluent hypergeometric function ${{}_{1}F_{1}}$ see (16.2.1) and Chapter 13.

Ultraspherical

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Keywords:: Bessel functions, Fourier transform of ultraspherical polynomials, Fourier transforms, ultraspherical polynomials
See also:: Annotations for §18.17(v), §18.17 and Ch.18

18.17.16_5		$\int_{-1}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda)}_{n}\left(x\right)\,{% \mathrm{e}}^{\mathrm{i}xy}\,\mathrm{d}x=\frac{2\pi\,{\mathrm{i}}^{n}\Gamma% \left(n+2\lambda\right)J_{n+\lambda}\left(y\right)}{n!\,\Gamma\left(\lambda% \right)\,(2y)^{\lambda}},$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $\int$ : integral, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v), Erratum (V1.2.0) §18.17 Permalink: http://dlmf.nist.gov/18.17.E16_5 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.17		$\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda)}_{2n}\left(x\right)% \cos\left(xy\right)\,\mathrm{d}x=\frac{(-1)^{n}\pi\Gamma\left(2n+2\lambda% \right)J_{\lambda+2n}\left(y\right)}{(2n)!\Gamma\left(\lambda\right)(2y)^{% \lambda}},$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E17 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.18		$\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda)}_{2n+1}\left(x\right)% \sin\left(xy\right)\,\mathrm{d}x=\frac{(-1)^{n}\pi\Gamma\left(2n+2\lambda+1% \right)J_{2n+\lambda+1}\left(y\right)}{(2n+1)!\Gamma\left(\lambda\right)(2y)^{% \lambda}}.$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $\sin\NVar{z}$ : sine function, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E18 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

For the Bessel function $J_{\nu}$ see §10.2(ii).

Legendre

ⓘ

Keywords:: Fourier transforms, Legendre polynomials
See also:: Annotations for §18.17(v), §18.17 and Ch.18

18.17.19		$\int_{-1}^{1}P_{n}\left(x\right){\mathrm{e}}^{\mathrm{i}xy}\,\mathrm{d}x={% \mathrm{i}}^{n}\sqrt{\frac{2\pi}{y}}J_{n+\frac{1}{2}}\left(y\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E19 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.20		$\int_{0}^{1}P_{n}\left(1-2x^{2}\right)\cos\left(xy\right)\,\mathrm{d}x=(-1)^{n% }\tfrac{1}{2}\pi J_{n+\frac{1}{2}}\left(\tfrac{1}{2}y\right)J_{-n-\frac{1}{2}}% \left(\tfrac{1}{2}y\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E20 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.21		$\int_{0}^{1}P_{n}\left(1-2x^{2}\right)\sin\left(xy\right)\,\mathrm{d}x=\tfrac{% 1}{2}\pi\left(J_{n+\frac{1}{2}}\left(\tfrac{1}{2}y\right)\right)^{2}.$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E21 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

Hermite

ⓘ

Keywords:: Fourier transforms, Hermite polynomials
See also:: Annotations for §18.17(v), §18.17 and Ch.18

18.17.21_1		$\frac{1}{\sqrt{2\pi c}}\int_{-\infty}^{\infty}{\mathrm{e}}^{-\frac{1}{2}\ifrac% {x^{2}}{c}}H_{n}\left(x\right)\,{\mathrm{e}}^{\mathrm{i}xy}\,\mathrm{d}x=\left% (\mathrm{i}\sqrt{2c-1}\,\right)^{n}{\mathrm{e}}^{-\frac{1}{2}cy^{2}}H_{n}\left% (\frac{cy}{\sqrt{2c-1}}\right),$
$\Re\left(c\right)>0$ , $c\neq\tfrac{1}{2}$ ,
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Sources: Erdélyi et al. (1954a, §1.10(8), §2.10(10)); Erdélyi et al. (1953b, §10.13(30)) Proof sketch: In the integrand of the left-hand side substitute (18.10.8) with data for $H_{n}\left(x\right)$ given by Table 18.10.1. Interchange integrals. Then the inner integral can be evaluated. The resulting integral can again be evaluated by (18.10.8) for Hermite. Referenced by: §18.17(v), §18.17(v), Erratum (V1.2.0) §18.17 Permalink: http://dlmf.nist.gov/18.17.E21_1 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.21_2		$\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}{\mathrm{e}}^{-x^{2}}H_{n}\left(x% \right)\,{\mathrm{e}}^{\mathrm{i}xy}\,\mathrm{d}x={\mathrm{i}}^{n}{\mathrm{e}}% ^{-\frac{1}{4}y^{2}}y^{n},$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E21_2 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.21_3		$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}{\mathrm{e}}^{-\frac{1}{2}x^{2}}H_% {n}\left(x\right)\,{\mathrm{e}}^{\mathrm{i}xy}\,\mathrm{d}x={\mathrm{i}}^{n}{% \mathrm{e}}^{-\frac{1}{2}y^{2}}H_{n}\left(y\right).$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v), §18.17(v), Erratum (V1.2.0) §18.17 Permalink: http://dlmf.nist.gov/18.17.E21_3 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

In (18.17.21_1) the branch choice of $\sqrt{2c-1}$ for $0<c<\frac{1}{2}$ is unimportant because on the right-hand side only even powers of $\sqrt{2c-1}$ occur after expansion of the Hermite polynomial by (18.5.13). Formulas (18.17.21_2) and (18.17.21_3) are respectively the limit case $c\to\tfrac{1}{2}$ and the special case $c=1$ of (18.17.21_1).

18.17.22		$\frac{1}{2\sqrt{\pi}}\int_{-\infty}^{\infty}{\mathrm{e}}^{-\frac{1}{4}x^{2}}% \mathit{He}_{n}\left(x\right){\mathrm{e}}^{\frac{1}{2}\mathrm{i}xy}\,\mathrm{d% }x={\mathrm{i}}^{n}{\mathrm{e}}^{-\frac{1}{4}y^{2}}\mathit{He}_{n}\left(y% \right),$
ⓘ Symbols: $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E22 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.23		$\int_{0}^{\infty}{\mathrm{e}}^{-\frac{1}{2}x^{2}}\mathit{He}_{2n}\left(x\right% )\cos\left(xy\right)\,\mathrm{d}x=(-1)^{n}\sqrt{\tfrac{1}{2}\pi}y^{2n}{\mathrm% {e}}^{-\frac{1}{2}y^{2}},$
ⓘ Symbols: $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E23 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.24		$\int_{0}^{\infty}{\mathrm{e}}^{-x^{2}}\mathit{He}_{2n}\left(2x\right)\cos\left% (xy\right)\,\mathrm{d}x=(-1)^{n}\tfrac{1}{2}\sqrt{\pi}{\mathrm{e}}^{-\frac{1}{% 4}y^{2}}\mathit{He}_{2n}\left(y\right).$
ⓘ Symbols: $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E24 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.25		$\int_{0}^{\infty}{\mathrm{e}}^{-\frac{1}{2}x^{2}}\mathit{He}_{n}\left(x\right)% \mathit{He}_{n+2m}\left(x\right)\cos\left(xy\right)\,\mathrm{d}x=(-1)^{m}\sqrt% {\tfrac{1}{2}\pi}n!\,y^{2m}{\mathrm{e}}^{-\frac{1}{2}y^{2}}L^{(2m)}_{n}\left(y% ^{2}\right),$
ⓘ Symbols: $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $y$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E25 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.26		$\int_{0}^{\infty}{\mathrm{e}}^{-\frac{1}{2}x^{2}}\mathit{He}_{n}\left(x\right)% \mathit{He}_{n+2m+1}\left(x\right)\sin\left(xy\right)\,\mathrm{d}x=(-1)^{m}% \sqrt{\tfrac{1}{2}\pi}n!\,y^{2m+1}{\mathrm{e}}^{-\frac{1}{2}y^{2}}L^{(2m+1)}_{% n}\left(y^{2}\right).$
ⓘ Symbols: $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $\sin\NVar{z}$ : sine function, $y$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E26 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.27		$\int_{0}^{\infty}{\mathrm{e}}^{-\frac{1}{2}x^{2}}\mathit{He}_{2n+1}\left(x% \right)\sin\left(xy\right)\,\mathrm{d}x=(-1)^{n}\sqrt{\tfrac{1}{2}\pi}y^{2n+1}% {\mathrm{e}}^{-\frac{1}{2}y^{2}},$
ⓘ Symbols: $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\sin\NVar{z}$ : sine function, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E27 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.28		$\int_{0}^{\infty}{\mathrm{e}}^{-x^{2}}\mathit{He}_{2n+1}\left(2x\right)\sin% \left(xy\right)\,\mathrm{d}x=(-1)^{n}\tfrac{1}{2}\sqrt{\pi}{\mathrm{e}}^{-% \frac{1}{4}y^{2}}\mathit{He}_{2n+1}\left(y\right).$
ⓘ Symbols: $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\sin\NVar{z}$ : sine function, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E28 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

Laguerre

ⓘ

Keywords:: Fourier transforms, Laguerre polynomials, classical orthogonal polynomials
See also:: Annotations for §18.17(v), §18.17 and Ch.18

18.17.28_5		$\int_{0}^{\infty}{\mathrm{e}}^{-x}x^{\alpha}L^{(\alpha)}_{n}\left(x\right){% \mathrm{e}}^{\mathrm{i}xy}\,\mathrm{d}x=\frac{\Gamma\left(\alpha+n+1\right)(-% \mathrm{i}y)^{n}}{n!\left(1-\mathrm{i}y\right)^{\alpha+n+1}},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v), Erratum (V1.2.0) §18.17 Permalink: http://dlmf.nist.gov/18.17.E28_5 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.29		$\int_{0}^{\infty}x^{2m}{\mathrm{e}}^{-\frac{1}{2}x^{2}}L^{(2m)}_{n}\left(x^{2}% \right)\cos\left(xy\right)\,\mathrm{d}x=(-1)^{m}\sqrt{\tfrac{1}{2}\pi}\frac{1}% {n!}{\mathrm{e}}^{-\frac{1}{2}y^{2}}\mathit{He}_{n}\left(y\right)\mathit{He}_{% n+2m}\left(y\right),$
ⓘ Symbols: $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $y$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E29 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.30		$\int_{0}^{\infty}x^{2n}{\mathrm{e}}^{-\frac{1}{2}x^{2}}L^{(n-\frac{1}{2})}_{n}% \left(\tfrac{1}{2}x^{2}\right)\cos\left(xy\right)\,\mathrm{d}x=\sqrt{\tfrac{1}% {2}\pi}y^{2n}{\mathrm{e}}^{-\frac{1}{2}y^{2}}L^{(n-\frac{1}{2})}_{n}\left(% \tfrac{1}{2}y^{2}\right),$
ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E30 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.31		$\int_{0}^{\infty}{\mathrm{e}}^{-ax}x^{\nu-2n}L^{(\nu-2n)}_{2n-1}\left(ax\right% )\cos\left(xy\right)\,\mathrm{d}x=i\frac{(-1)^{n}\Gamma\left(\nu\right)}{2(2n-% 1)!}y^{2n-1}\left((a+iy)^{-\nu}-(a-iy)^{-\nu}\right),$
$\nu>2n-1$ , $a>0$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E31 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.32		$\int_{0}^{\infty}{\mathrm{e}}^{-ax}x^{\nu-1-2n}L^{(\nu-1-2n)}_{2n}\left(ax% \right)\cos\left(xy\right)\,\mathrm{d}x=\frac{(-1)^{n}\Gamma\left(\nu\right)}{% 2(2n)!}y^{2n}\left((a+iy)^{-\nu}+(a-iy)^{-\nu}\right),$
$\nu>2n$ , $a>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E32 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

§18.17(vi) Laplace Transforms

ⓘ

Keywords:: Laplace transforms, classical orthogonal polynomials
Notes:: For (18.17.33) expand the exponential in the integral as a power series in $z$ and interchange integration and summation. The resulting integral can be evaluated by considering the term $\ell=0$ in (18.18.14). (18.17.33) may also be verified by applying Kummer’s transformation (13.2.39) to (18.17.16). (18.17.34) follows by substituting (18.5.12) into the integrand and performing termwise integration. (18.17.35) follows by use of (18.5.5) and integration by parts.
Permalink:: http://dlmf.nist.gov/18.17.vi
See also:: Annotations for §18.17 and Ch.18

Many of the Fourier transforms given in §18.17(v) have analytic continuations to Laplace transforms. Some of the resulting formulas are given below.

Jacobi

ⓘ

Keywords:: Jacobi polynomials, Laplace transform, integrals
See also:: Annotations for §18.17(vi), §18.17 and Ch.18

18.17.33		$\int_{-1}^{1}{\mathrm{e}}^{-(x+1)z}P^{(\alpha,\beta)}_{n}\left(x\right)(1-x)^{% \alpha}(1+x)^{\beta}\,\mathrm{d}x=\frac{(-1)^{n}2^{\alpha+\beta+n+1}\Gamma% \left(\alpha+n+1\right)\Gamma\left(\beta+n+1\right)}{\Gamma\left(\alpha+\beta+% 2n+2\right)n!}z^{n}{{}_{1}F_{1}}\left({\beta+n+1\atop\alpha+\beta+2n+2};-2z% \right),$
$z\in\mathbb{C}$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ : $=M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ notation for the Kummer confluent hypergeometric function, $\mathbb{C}$ : complex plane, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(vi) Permalink: http://dlmf.nist.gov/18.17.E33 Encodings: TeX, pMML, png See also: Annotations for §18.17(vi), §18.17(vi), §18.17 and Ch.18

For the confluent hypergeometric function ${{}_{1}F_{1}}$ see (16.2.1) and Chapter 13.

Laguerre

ⓘ

Keywords:: Laguerre polynomials, Laplace transform, integrals
See also:: Annotations for §18.17(vi), §18.17 and Ch.18

18.17.34		$\int_{0}^{\infty}{\mathrm{e}}^{-xz}L^{(\alpha)}_{n}\left(x\right){\mathrm{e}}^% {-x}x^{\alpha}\,\mathrm{d}x=\frac{\Gamma\left(\alpha+n+1\right)z^{n}}{n!(z+1)^% {\alpha+n+1}},$
$\Re z>-1$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $\Re$ : real part, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(vii), §18.17(i), §18.17(v), §18.17(vi), §18.17(viii) Permalink: http://dlmf.nist.gov/18.17.E34 Encodings: TeX, pMML, png See also: Annotations for §18.17(vi), §18.17(vi), §18.17 and Ch.18

18.17.34_5		$\int_{0}^{\infty}{\mathrm{e}}^{-xz}L^{(\alpha)}_{m}\left(x\right)L^{(\alpha)}_% {n}\left(x\right)\,{\mathrm{e}}^{-x}\,x^{\alpha}\,\mathrm{d}x=\frac{\Gamma% \left(\alpha+m+1\right)\Gamma\left(\alpha+n+1\right)}{\Gamma\left(\alpha+1% \right)\,m!\,n!}\frac{z^{m+n}}{(z+1)^{\alpha+m+n+1}}{{}_{2}F_{1}}\left({-m,-n% \atop\alpha+1};z^{-2}\right),$
$\Re z>-1$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $\Re$ : real part, $z$ : complex variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable Source: Erdélyi et al. (1954a, 4.11(35)) Proof sketch: First assume $z>-1$ . Make change of integration variable $x\to\ifrac{x}{(z+1)}$ . Twice substitute (18.18.12) with $\lambda=(z+1)^{-1}$ . Then use the orthogonality relation for the Laguerre polynomials, see Table 18.3.1. Finally perform analytic continuation with respect to $z$ . Referenced by: Erratum (V1.2.0) §18.17 Permalink: http://dlmf.nist.gov/18.17.E34_5 Encodings: TeX, pMML, png See also: Annotations for §18.17(vi), §18.17(vi), §18.17 and Ch.18

Hermite

ⓘ

Keywords:: Hermite polynomials, Laplace transform, integrals
See also:: Annotations for §18.17(vi), §18.17 and Ch.18

18.17.35		$\int_{-\infty}^{\infty}{\mathrm{e}}^{-xz}H_{n}\left(x\right){\mathrm{e}}^{-x^{% 2}}\,\mathrm{d}x={\pi}^{\frac{1}{2}}(-z)^{n}{\mathrm{e}}^{\frac{1}{4}z^{2}},$
$z\in\mathbb{C}$ .
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{C}$ : complex plane, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(vi), §18.17(viii) Permalink: http://dlmf.nist.gov/18.17.E35 Encodings: TeX, pMML, png See also: Annotations for §18.17(vi), §18.17(vi), §18.17 and Ch.18

§18.17(vii) Mellin Transforms

ⓘ

Keywords:: Mellin transforms, classical orthogonal polynomials
Notes:: For (18.17.36) use (18.5.7) and apply (16.4.3). For (18.17.37) use (18.5.5) and integrate by parts $n$ times. For (18.17.38) use the first equality in (18.5.10), with $\lambda=\frac{1}{2}$ and $n$ replaced by $2n$ , integrate term by term, then apply (16.4.3); the proof of (18.17.39) is similar. For (18.17.40) use (18.5.12), integrate term by term, then apply (15.8.6). For (18.17.41) use (18.5.13) and integrate term by term.
Permalink:: http://dlmf.nist.gov/18.17.vii
Addition (effective with 1.2.0):: Equation (18.17.41_5) was added.
See also:: Annotations for §18.17 and Ch.18

Jacobi

ⓘ

Keywords:: Jacobi polynomials, Mellin transform
See also:: Annotations for §18.17(vii), §18.17 and Ch.18

18.17.36		$\int_{-1}^{1}(1-x)^{z-1}(1+x)^{\beta}P^{(\alpha,\beta)}_{n}\left(x\right)\,% \mathrm{d}x=\frac{2^{\beta+z}\Gamma\left(z\right)\Gamma\left(1+\beta+n\right){% \left(1+\alpha-z\right)_{n}}}{n!\Gamma\left(1+\beta+z+n\right)},$
$\Re z>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $\Re$ : real part, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Keywords: Mellin transform Referenced by: §18.17(vii) Permalink: http://dlmf.nist.gov/18.17.E36 Encodings: TeX, pMML, png See also: Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

Ultraspherical

ⓘ

Keywords:: Mellin transform, ultraspherical polynomials
See also:: Annotations for §18.17(vii), §18.17 and Ch.18

18.17.37		$\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda)}_{n}\left(x\right)x^{z% -1}\,\mathrm{d}x=\frac{\pi\,2^{1-2\lambda-z}\Gamma\left(n+2\lambda\right)% \Gamma\left(z\right)}{n!\Gamma\left(\lambda\right)\Gamma\left(\frac{1}{2}+% \frac{1}{2}n+\lambda+\frac{1}{2}z\right)\Gamma\left(\frac{1}{2}+\frac{1}{2}z-% \frac{1}{2}n\right)},$
$\Re z>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $\Re$ : real part, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Keywords: Mellin transform Referenced by: §18.17(vii) Permalink: http://dlmf.nist.gov/18.17.E37 Encodings: TeX, pMML, png See also: Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

Legendre

ⓘ

Keywords:: Legendre polynomials, Mellin transforms
See also:: Annotations for §18.17(vii), §18.17 and Ch.18

18.17.38		$\int_{0}^{1}P_{2n}\left(x\right)x^{z-1}\,\mathrm{d}x=\frac{(-1)^{n}{\left(% \frac{1}{2}-\frac{1}{2}z\right)_{n}}}{2{\left(\frac{1}{2}z\right)_{n+1}}},$
$\Re z>0$ ,
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Keywords: Mellin transform Referenced by: §18.17(vii) Permalink: http://dlmf.nist.gov/18.17.E38 Encodings: TeX, pMML, png See also: Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

18.17.39		$\int_{0}^{1}P_{2n+1}\left(x\right)x^{z-1}\,\mathrm{d}x=\frac{(-1)^{n}{\left(1-% \frac{1}{2}z\right)_{n}}}{2{\left(\frac{1}{2}+\frac{1}{2}z\right)_{n+1}}},$
$\Re z>-1$ .
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Keywords: Mellin transform Referenced by: §18.17(vii) Permalink: http://dlmf.nist.gov/18.17.E39 Encodings: TeX, pMML, png See also: Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

Laguerre

ⓘ

Keywords:: Laguerre polynomials, Mellin transform
See also:: Annotations for §18.17(vii), §18.17 and Ch.18

18.17.40		$\int_{0}^{\infty}{\mathrm{e}}^{-ax}L^{(\alpha)}_{n}\left(bx\right)x^{z-1}\,% \mathrm{d}x=\frac{\Gamma\left(z+n\right)}{n!}\{(a-b)^{n}}a^{-n-z}\{{}_{2}F_{% 1}}\left({-n,1+\alpha-z\atop 1-n-z};\frac{a}{a-b}\right),$
$\Re a>0$ , $\Re z>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $\Re$ : real part, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Keywords: Laplace transform, Mellin transform Referenced by: §18.17(vii) Permalink: http://dlmf.nist.gov/18.17.E40 Encodings: TeX, pMML, png See also: Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

This generalizes (18.17.34). For the hypergeometric function ${{}_{2}F_{1}}$ see §§15.1 and 15.2(i).

Hermite

ⓘ

Keywords:: Hermite polynomials, Mellin transform
See also:: Annotations for §18.17(vii), §18.17 and Ch.18

18.17.41		$\int_{0}^{\infty}{\mathrm{e}}^{-ax}\mathit{He}_{n}\left(x\right)x^{z-1}\,% \mathrm{d}x=\Gamma\left(z+n\right)a^{-n-2}{{}_{2}F_{2}}\left({-\tfrac{1}{2}n,-% \tfrac{1}{2}n+\tfrac{1}{2}\atop-\tfrac{1}{2}z-\tfrac{1}{2}n,-\tfrac{1}{2}z-% \tfrac{1}{2}n+\tfrac{1}{2}};-\tfrac{1}{2}a^{2}\right),$
$\Re a>0$ . Also, $\Re z>0$ , $n$ even; $\Re z>-1$ , $n$ odd.
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$ generalized hypergeometric function, $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Keywords: Laplace transform, Mellin transform Referenced by: §18.17(vii) Permalink: http://dlmf.nist.gov/18.17.E41 Encodings: TeX, pMML, png See also: Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

For the generalized hypergeometric function ${{}_{2}F_{2}}$ see (16.2.1).

§18.17(viii) Other Integrals

ⓘ

Notes:: (18.17.42) and (18.17.43) follow from the case $\alpha=\beta=\pm\frac{1}{2}$ of Szegő (1975, Theorem 4.61.2), where the hypergeometric function on the right-hand side is rewritten as in Erdélyi et al. (1953b, §19.8(19)). For (18.17.44) see Tuck (1964). (18.17.46) is obtained from (18.17.9) for $\alpha=\beta=0$ , $\mu=\frac{1}{2}$ , together with (18.7.5), (18.7.3), and the second row of Table 18.6.1. (18.17.45) is obtained from (18.17.46) by symmetry; compare Rows 5 and 9 of Table 18.6.1. (18.17.47) and (18.17.48) follow by use of (18.17.34) and (18.17.35). For (18.17.49) see Andrews et al. (1999, p. 328).
Permalink:: http://dlmf.nist.gov/18.17.viii
See also:: Annotations for §18.17 and Ch.18

Ultraspherical

ⓘ

Addition (effective with 1.2.0):: paragraph paragraph was added.
See also:: Annotations for §18.17(viii), §18.17 and Ch.18

18.17.41_5		$\int_{-1}^{1}C^{(\lambda)}_{\ell}\left(x\right)C^{(\lambda)}_{m}\left(x\right)% C^{(\lambda)}_{n}\left(x\right)(1-x^{2})^{\lambda-\tfrac{1}{2}}\,\mathrm{d}x=% \frac{{\left(\lambda\right)_{\frac{1}{2}\ell+\frac{1}{2}m-\frac{1}{2}n}}{\left% (\lambda\right)_{\frac{1}{2}m+\frac{1}{2}n-\frac{1}{2}\ell}}{\left(\lambda% \right)_{\frac{1}{2}n+\frac{1}{2}\ell-\frac{1}{2}m}}{\left(2\lambda\right)_{% \frac{1}{2}\ell+\frac{1}{2}m+\frac{1}{2}n}}\Gamma\left(\lambda+\frac{1}{2}% \right)\sqrt{\pi}}{(\tfrac{1}{2}\ell+\tfrac{1}{2}m-\tfrac{1}{2}n)!\,(\tfrac{1}% {2}m+\tfrac{1}{2}n-\tfrac{1}{2}\ell)!\,(\tfrac{1}{2}n+\tfrac{1}{2}\ell-\tfrac{% 1}{2}m)!\,\Gamma\left(\lambda+\frac{1}{2}\ell+\frac{1}{2}m+\frac{1}{2}n+1% \right)},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $\ell$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable Proved: Andrews et al. (1999, Corollary 6.8.4); Straightforward from (18.18.22).(proved) Referenced by: §18.17(vii), Erratum (V1.2.0) §18.17 Permalink: http://dlmf.nist.gov/18.17.E41_5 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

provided that $\ell+m+n$ is even and the sum of any two of $\ell,m,n$ is not less than the third; otherwise the integral is zero.

Chebyshev

ⓘ

Keywords:: Chebyshev polynomials, integrals
See also:: Annotations for §18.17(viii), §18.17 and Ch.18

18.17.42		$\pvint_{-1}^{1}T_{n}\left(y\right)\frac{(1-y^{2})^{-\frac{1}{2}}}{y-x}\,% \mathrm{d}y=\pi U_{n-1}\left(x\right),$
ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\pvint_{\NVar{a}}^{\NVar{b}}$ : Cauchy principal value, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.13.3 Referenced by: §18.17(viii) Permalink: http://dlmf.nist.gov/18.17.E42 Encodings: TeX, pMML, png See also: Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

18.17.43		$\pvint_{-1}^{1}U_{n-1}\left(y\right)\frac{(1-y^{2})^{\frac{1}{2}}}{y-x}\,% \mathrm{d}y=-\pi T_{n}\left(x\right).$
ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\pvint_{\NVar{a}}^{\NVar{b}}$ : Cauchy principal value, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.13.4 Referenced by: §18.17(viii) Permalink: http://dlmf.nist.gov/18.17.E43 Encodings: TeX, pMML, png See also: Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

These integrals are Cauchy principal values (§1.4(v)).

Legendre

ⓘ

Keywords:: Legendre polynomials, integrals
See also:: Annotations for §18.17(viii), §18.17 and Ch.18

18.17.44		$\int_{-1}^{1}\frac{P_{n}\left(x\right)-P_{n}\left(t\right)}{\|x-t\|}\,\mathrm{d}% t=2\left(1+\tfrac{1}{2}+\dots+\tfrac{1}{n}\right)P_{n}\left(x\right),$
$-1\leq x\leq 1$ .
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $t$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(viii) Permalink: http://dlmf.nist.gov/18.17.E44 Encodings: TeX, pMML, png See also: Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

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

18.17.45		$(n+\tfrac{1}{2})(1+x)^{\frac{1}{2}}\int_{-1}^{x}(x-t)^{-\frac{1}{2}}P_{n}\left% (t\right)\,\mathrm{d}t=T_{n}\left(x\right)+T_{n+1}\left(x\right)=(1+x)V_{n}% \left(x\right),$
ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $V_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the third kind, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $t$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(viii), §18.17(viii), §18.18(v), Erratum (V1.2.0) for Equations (18.17.45), (18.17.46) Permalink: http://dlmf.nist.gov/18.17.E45 Encodings: TeX, pMML, png See also: Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

18.17.46		$(n+\tfrac{1}{2})(1-x)^{\frac{1}{2}}\int_{x}^{1}(t-x)^{-\frac{1}{2}}P_{n}\left(% t\right)\,\mathrm{d}t=T_{n}\left(x\right)-T_{n+1}\left(x\right)=(1-x)W_{n}% \left(x\right).$
ⓘ 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, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $t$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(viii), Erratum (V1.2.0) for Equations (18.17.45), (18.17.46) Permalink: http://dlmf.nist.gov/18.17.E46 Encodings: TeX, pMML, png See also: Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

Laguerre

ⓘ

See also:: Annotations for §18.17(viii), §18.17 and Ch.18

18.17.47		$\int_{0}^{x}t^{\alpha}\frac{L^{(\alpha)}_{m}\left(t\right)}{L^{(\alpha)}_{m}% \left(0\right)}(x-t)^{\beta}\frac{L^{(\beta)}_{n}\left(x-t\right)}{L^{(\beta)}% _{n}\left(0\right)}\,\mathrm{d}t=\frac{\Gamma\left(\alpha+1\right)\Gamma\left(% \beta+1\right)}{\Gamma\left(\alpha+\beta+2\right)}x^{\alpha+\beta+1}\frac{L^{(% \alpha+\beta+1)}_{m+n}\left(x\right)}{L^{(\alpha+\beta+1)}_{m+n}\left(0\right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $t$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(viii) Permalink: http://dlmf.nist.gov/18.17.E47 Encodings: TeX, pMML, png See also: Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

Hermite

ⓘ

Keywords:: Hermite polynomials, integrals
See also:: Annotations for §18.17(viii), §18.17 and Ch.18

18.17.48		$\int_{-\infty}^{\infty}H_{m}\left(y\right){\mathrm{e}}^{-y^{2}}H_{n}\left(x-y% \right){\mathrm{e}}^{-(x-y)^{2}}\,\mathrm{d}y={\pi}^{\frac{1}{2}}2^{-\frac{1}{% 2}(m+n+1)}H_{m+n}\left(2^{-\frac{1}{2}}x\right){\mathrm{e}}^{-\frac{1}{2}x^{2}}.$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $y$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(viii) Permalink: http://dlmf.nist.gov/18.17.E48 Encodings: TeX, pMML, png See also: Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

18.17.49		$\int_{-\infty}^{\infty}H_{\ell}\left(x\right)H_{m}\left(x\right)H_{n}\left(x% \right){\mathrm{e}}^{-x^{2}}\,\mathrm{d}x=\frac{2^{\frac{1}{2}(\ell+m+n)}\ell% \,!\,m\,!\,n\,!\,\sqrt{\pi}}{(\tfrac{1}{2}\ell+\tfrac{1}{2}m-\tfrac{1}{2}n)\,!% \,(\tfrac{1}{2}m+\tfrac{1}{2}n-\tfrac{1}{2}\ell\,)\,!\,(\tfrac{1}{2}n+\tfrac{1% }{2}\ell-\tfrac{1}{2}m\,)\,!},$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $\ell$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(viii), §18.17(viii), §18.18(v) Permalink: http://dlmf.nist.gov/18.17.E49 Encodings: TeX, pMML, png See also: Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

provided that $\ell+m+n$ is even and the sum of any two of $\ell,m,n$ is not less than the third; otherwise the integral is zero.

Formulas (18.17.45) and (18.17.49) are integrated forms of the linearization formulas (18.18.22) and (18.18.23), respectively.

§18.17(ix) Compendia

ⓘ

Keywords:: Hermite polynomials, Jacobi polynomials, classical orthogonal polynomials, compendia, integrals
Permalink:: http://dlmf.nist.gov/18.17.ix
See also:: Annotations for §18.17 and Ch.18

For further integrals, see Apelblat (1983, pp. 189–204), Erdélyi et al. (1954a, pp. 38–39, 94–95, 170–176, 259–261, 324), Erdélyi et al. (1954b, pp. 42–44, 271–294), Gradshteyn and Ryzhik (2015, §§7.3–7.4), Gröbner and Hofreiter (1950, pp. 23–30), Marichev (1983, pp. 216–247), Oberhettinger (1972, pp. 64–67), Oberhettinger (1974, pp. 83–92), Oberhettinger (1990, pp. 44–47 and 152–154), Oberhettinger and Badii (1973, pp. 103–112), Prudnikov et al. (1986b, pp. 420–617), Prudnikov et al. (1992a, pp. 419–476), and Prudnikov et al. (1992b, pp. 280–308).

18.16 Zeros 18.18 Sums

NIST

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§18.17 Integrals

Contents

§18.17(i) Indefinite Integrals

Jacobi

Laguerre

Hermite

§18.17(ii) Integral Representations for Products

Ultraspherical

Legendre

§18.17(iii) Nicholson-Type Integrals

Legendre

Hermite

§18.17(iv) Fractional Integrals

Jacobi

Ultraspherical

Laguerre

§18.17(v) Fourier Transforms

Jacobi

Ultraspherical

Legendre

Hermite

Laguerre

§18.17(vi) Laplace Transforms

Jacobi

Laguerre

Hermite

§18.17(vii) Mellin Transforms

Jacobi

Ultraspherical

Legendre

Laguerre

Hermite

§18.17(viii) Other Integrals

Ultraspherical

Chebyshev

Legendre

Laguerre

Hermite

§18.17(ix) Compendia