§18.17 Integrals

Keywords:: classical orthogonal polynomials, integrals
Referenced by:: ¶ ‣ §18.10(iv)
Permalink:: http://dlmf.nist.gov/18.17

§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

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

¶ Jacobi

Keywords:: Jacobi polynomials, integrals

18.17.1 $2n\int _{0}^{x}(1-y)^{{\alpha}}(1+y)^{{\beta}}\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(y\right)dy=\mathop{P^{{(\alpha+1,\beta+1)}}_{{n-1}}\/}\nolimits\!\left(0\right)-(1-x)^{{\alpha+1}}(1+x)^{{\beta+1}}\mathop{P^{{(\alpha+1,\beta+1)}}_{{n-1}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $dx$ : differential of $x$ , $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.17(i)
Permalink:: http://dlmf.nist.gov/18.17.E1
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¶ Laguerre

Keywords:: Laguerre polynomials, integrals

18.17.2 $\int _{0}^{x}\mathop{L_{{m}}\/}\nolimits\!\left(y\right)\mathop{L_{{n}}\/}\nolimits\!\left(x-y\right)dy=\int _{0}^{x}\mathop{L_{{m+n}}\/}\nolimits\!\left(y\right)dy=\mathop{L_{{m+n}}\/}\nolimits\!\left(x\right)-\mathop{L_{{m+n+1}}\/}\nolimits\!\left(x\right).$

Symbols:: $dx$ : differential of $x$ , $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\int$ : integral, $y$ : 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
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¶ Hermite

Keywords:: Hermite polynomials, integrals

18.17.3 $\int _{0}^{x}\mathop{H_{{n}}\/}\nolimits\!\left(y\right)dy=\frac{1}{2(n+1)}(\mathop{H_{{n+1}}\/}\nolimits\!\left(x\right)-\mathop{H_{{n+1}}\/}\nolimits\!\left(0\right)),$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $dx$ : differential of $x$ , $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.17(i)
Permalink:: http://dlmf.nist.gov/18.17.E3
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18.17.4 $\int _{0}^{x}e^{{-y^{2}}}\mathop{H_{{n}}\/}\nolimits\!\left(y\right)dy=\mathop{H_{{n-1}}\/}\nolimits\!\left(0\right)-e^{{-x^{2}}}\mathop{H_{{n-1}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.17(i)
Permalink:: http://dlmf.nist.gov/18.17.E4
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§18.17(ii) Integral Representations for Products

Notes:: For (18.17.5) see Ismail (2005, (9.6.2)). (18.17.6) is the case $\alpha=0$ of (18.17.5).
Keywords:: classical orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.17.ii

¶ Ultraspherical

Keywords:: ultraspherical polynomials

18.17.5 $\frac{\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta _{1}\right)}{\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(1\right)}\frac{\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta _{2}\right)}{\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(1\right)}=\frac{\mathop{\Gamma\/}\nolimits\!\left(\lambda+\frac{1}{2}\right)}{\pi^{{\frac{1}{2}}}\mathop{\Gamma\/}\nolimits\!\left(\lambda\right)}\*\int _{0}^{\pi}\frac{\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta _{1}\mathop{\cos\/}\nolimits\theta _{2}+\mathop{\sin\/}\nolimits\theta _{1}\mathop{\sin\/}\nolimits\theta _{2}\mathop{\cos\/}\nolimits\phi\right)}{\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(1\right)}(\mathop{\sin\/}\nolimits\phi)^{{2\lambda-1}}d\phi,$ $\lambda>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial and $n$ : nonnegative integer
Referenced by:: ¶ ‣ §18.17(ii), §18.17(ii)
Permalink:: http://dlmf.nist.gov/18.17.E5
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¶ Legendre

Keywords:: Legendre polynomials, classical orthogonal polynomials

18.17.6 $\mathop{P_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta _{1}\right)\mathop{P_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta _{2}\right)=\frac{1}{\pi}\int _{0}^{\pi}\mathop{P_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta _{1}\mathop{\cos\/}\nolimits\theta _{2}+\mathop{\sin\/}\nolimits\theta _{1}\mathop{\sin\/}\nolimits\theta _{2}\mathop{\cos\/}\nolimits\phi\right)d\phi.$

Symbols:: $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and $n$ : nonnegative integer
Referenced by:: ¶ ‣ §18.17(ii), §18.17(ii)
Permalink:: http://dlmf.nist.gov/18.17.E6
Encodings:: TeX, pMML, png

For formulas for Jacobi and Laguerre polynomials analogous to (18.17.5) and (18.17.6), see Koornwinder (1974, 1977).

§18.17(iii) Nicholson-Type Integrals

Notes:: See Durand (1975).
Permalink:: http://dlmf.nist.gov/18.17.iii

¶ Legendre

Keywords:: Legendre polynomials, integrals

18.17.7 $\left(\mathop{P_{{n}}\/}\nolimits\!\left(x\right)\right)^{2}+4\pi^{{-2}}\left(\mathop{\mathsf{Q}_{{n}}\/}\nolimits\!\left(x\right)\right)^{2}=4\pi^{{-2}}\*\int _{1}^{\infty}\mathop{Q_{{n}}\/}\nolimits\!\left(x^{2}+(1-x^{2})t\right)(t^{2}-1)^{{-\frac{1}{2}}}dt,$ $-1<x<1$ .

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $dx$ : differential of $x$ , $\int$ : integral, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.17.E7
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For the Ferrers function $\mathop{\mathsf{Q}_{{n}}\/}\nolimits\!\left(x\right)$ and Legendre function $\mathop{Q_{{n}}\/}\nolimits\!\left(x\right)$ see §§14.3(i) and 14.3(ii), with $\mu=0$ and $\nu=n$ .

¶ Hermite

Keywords:: Hermite polynomials, integrals

18.17.8 $\left(\mathop{H_{{n}}\/}\nolimits\!\left(x\right)\right)^{2}+2^{n}(n!)^{2}e^{{x^{2}}}\left(\mathop{V\/}\nolimits\!\left(-n-\tfrac{1}{2},2^{{\frac{1}{2}}}x\right)\right)^{2}=\frac{2^{{n+\frac{3}{2}}}n!\, e^{{x^{2}}}}{\pi}\int _{0}^{\infty}\frac{e^{{-(2n+1)t+x^{2}\mathop{\tanh\/}\nolimits t}}}{(\mathop{\sinh\/}\nolimits 2t)^{{\frac{1}{2}}}}dt.$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function, $\int$ : integral, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.17.E8
Encodings:: TeX, pMML, png

For the parabolic cylinder function $\mathop{V\/}\nolimits\!\left(a,z\right)$ 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

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

¶ Jacobi

Keywords:: Jacobi polynomials, integrals

18.17.9 $\frac{(1-x)^{{\alpha+\mu}}\mathop{P^{{(\alpha+\mu,\beta-\mu)}}_{{n}}\/}\nolimits\!\left(x\right)}{\mathop{\Gamma\/}\nolimits\!\left(\alpha+\mu+n+1\right)}=\int _{x}^{1}\frac{(1-y)^{\alpha}\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(y\right)}{\mathop{\Gamma\/}\nolimits\!\left(\alpha+n+1\right)}\frac{(y-x)^{{\mu-1}}}{\mathop{\Gamma\/}\nolimits\!\left(\mu\right)}dy,$ $\mu>0$ , $-1<x<1$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $dx$ : differential of $x$ , $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: ¶ ‣ §18.17(iv), §18.17(iv), §18.17(viii)
Permalink:: http://dlmf.nist.gov/18.17.E9
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18.17.10 $\frac{x^{{\beta+\mu}}(x+1)^{n}}{\mathop{\Gamma\/}\nolimits\!\left(\beta+\mu+n+1\right)}\mathop{P^{{(\alpha,\beta+\mu)}}_{{n}}\/}\nolimits\!\left(\frac{x-1}{x+1}\right)=\int _{0}^{x}\frac{y^{\beta}(y+1)^{n}}{\mathop{\Gamma\/}\nolimits\!\left(\beta+n+1\right)}\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(\frac{y-1}{y+1}\right)\*\frac{(x-y)^{{\mu-1}}}{\mathop{\Gamma\/}\nolimits\!\left(\mu\right)}dy,$ $\mu>0$ , $x>0$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $dx$ : differential of $x$ , $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.17(iv)
Permalink:: http://dlmf.nist.gov/18.17.E10
Encodings:: TeX, pMML, png

18.17.11 $\frac{\mathop{\Gamma\/}\nolimits\!\left(n+\alpha+\beta-\mu+1\right)}{x^{{n+\alpha+\beta-\mu+1}}}\mathop{P^{{(\alpha,\beta-\mu)}}_{{n}}\/}\nolimits\!\left(1-2x^{{-1}}\right)=\int _{x}^{\infty}\frac{\mathop{\Gamma\/}\nolimits\!\left(n+\alpha+\beta+1\right)}{y^{{n+\alpha+\beta+1}}}\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(1-2y^{{-1}}\right)\*\frac{(y-x)^{{\mu-1}}}{\mathop{\Gamma\/}\nolimits\!\left(\mu\right)}dy,$ $\alpha+\beta+1>\mu>0$ , $x>1$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $dx$ : 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.E11
Encodings:: TeX, pMML, png

and three formulas similar to (18.17.9)–(18.17.11) by symmetry; compare the second row in Table 18.6.1.

¶ Ultraspherical

Keywords:: integrals, ultraspherical polynomials

18.17.12 $\frac{\mathop{\Gamma\/}\nolimits\!\left(\lambda-\mu\right)\mathop{C^{{(\lambda-\mu)}}_{{n}}\/}\nolimits\!\left(x^{{-\frac{1}{2}}}\right)}{x^{{\lambda-\mu+\frac{1}{2}n}}}=\int _{x}^{\infty}\frac{\mathop{\Gamma\/}\nolimits\!\left(\lambda\right)\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(y^{{-\frac{1}{2}}}\right)}{y^{{\lambda+\frac{1}{2}n}}}\frac{(y-x)^{{\mu-1}}}{\mathop{\Gamma\/}\nolimits\!\left(\mu\right)}dy,$ $\lambda>\mu>0$ , $x>0$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.17.E12
Encodings:: TeX, pMML, png

18.17.13 $\frac{x^{{\frac{1}{2}n}}(x-1)^{{\lambda+\mu-\frac{1}{2}}}}{\mathop{\Gamma\/}\nolimits\!\left(\lambda+\mu+\tfrac{1}{2}\right)}\frac{\mathop{C^{{(\lambda+\mu)}}_{{n}}\/}\nolimits\!\left(x^{{-\frac{1}{2}}}\right)}{\mathop{C^{{(\lambda+\mu)}}_{{n}}\/}\nolimits\!\left(1\right)}=\int _{1}^{x}\frac{y^{{\frac{1}{2}n}}(y-1)^{{\lambda-\frac{1}{2}}}}{\mathop{\Gamma\/}\nolimits\!\left(\lambda+\tfrac{1}{2}\right)}\frac{\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(y^{{-\frac{1}{2}}}\right)}{\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(1\right)}\frac{(x-y)^{{\mu-1}}}{\mathop{\Gamma\/}\nolimits\!\left(\mu\right)}dy,$ $\mu>0$ , $x>1$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.17.E13
Encodings:: TeX, pMML, png

¶ Laguerre

Keywords:: Laguerre polynomials, integrals

18.17.14 $\frac{x^{{\alpha+\mu}}\mathop{L^{{(\alpha+\mu)}}_{{n}}\/}\nolimits\!\left(x\right)}{\mathop{\Gamma\/}\nolimits\!\left(\alpha+\mu+n+1\right)}=\int _{0}^{x}\frac{y^{{\alpha}}\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(y\right)}{\mathop{\Gamma\/}\nolimits\!\left(\alpha+n+1\right)}\frac{(x-y)^{{\mu-1}}}{\mathop{\Gamma\/}\nolimits\!\left(\mu\right)}dy,$ $\mu>0$ , $x>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.13.13
Permalink:: http://dlmf.nist.gov/18.17.E14
Encodings:: TeX, pMML, png

18.17.15 $e^{{-x}}\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)=\int _{x}^{\infty}e^{{-y}}\mathop{L^{{(\alpha+\mu)}}_{{n}}\/}\nolimits\!\left(y\right)\frac{(y-x)^{{\mu-1}}}{\mathop{\Gamma\/}\nolimits\!\left(\mu\right)}dy,$ $\mu>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\int$ : integral, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.17(iv)
Permalink:: http://dlmf.nist.gov/18.17.E15
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§18.17(v) Fourier Transforms

Notes:: For (18.17.16) use (18.5.5), integrate by parts $n$ times, expand $e^{{-iy(1-x)}}$ in a Maclaurin series, and integrate term by term. For (18.17.17) use (18.5.5), integrate repeatedly by parts, expand $\mathop{\cos\/}\nolimits\!\left(xy\right)$ 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 $\mathop{\cos\/}\nolimits\!\left(xy\right)$ 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 $e^{{-\frac{1}{2}x^{2}}}$ is $e^{{-\frac{1}{2}y^{2}}}$ ; 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 (18.17.29) 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 $\mathop{L^{{(n-\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(\frac{1}{2}x^{2}\right)$ 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).
Keywords:: classical orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.17.v

Throughout this subsection we assume $y>0$ ; sometimes however, this restriction can be eased by analytic continuation.

¶ Jacobi

Keywords:: Fourier transform, Jacobi polynomials

18.17.16 $\int _{{-1}}^{1}(1-x)^{{\alpha}}(1+x)^{{\beta}}\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)e^{{ixy}}dx=\frac{(iy)^{n}e^{{iy}}}{n!}2^{{n+\alpha+\beta+1}}\mathop{\mathrm{B}\/}\nolimits\!\left(n+\alpha+1,n+\beta+1\right)\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left(n+\alpha+1;2n+\alpha+\beta+2;-2iy\right).$

Symbols:: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\int$ : 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

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

¶ Ultraspherical

Keywords:: ultraspherical polynomials

18.17.17 $\int _{0}^{1}(1-x^{2})^{{\lambda-\frac{1}{2}}}\mathop{C^{{(\lambda)}}_{{2n}}\/}\nolimits\!\left(x\right)\mathop{\cos\/}\nolimits\!\left(xy\right)dx=\frac{(-1)^{n}\pi\mathop{\Gamma\/}\nolimits\!\left(2n+2\lambda\right)\mathop{J_{{\lambda+2n}}\/}\nolimits\!\left(y\right)}{(2n)!\mathop{\Gamma\/}\nolimits\!\left(\lambda\right)(2y)^{{\lambda}}},$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : 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

18.17.18 $\int _{0}^{1}(1-x^{2})^{{\lambda-\frac{1}{2}}}\mathop{C^{{(\lambda)}}_{{2n+1}}\/}\nolimits\!\left(x\right)\mathop{\sin\/}\nolimits\!\left(xy\right)dx=\frac{(-1)^{n}\pi\mathop{\Gamma\/}\nolimits\!\left(2n+2\lambda+1\right)\mathop{J_{{2n+\lambda+1}}\/}\nolimits\!\left(y\right)}{(2n+1)!\mathop{\Gamma\/}\nolimits\!\left(\lambda\right)(2y)^{{\lambda}}}.$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : 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

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

¶ Legendre

Keywords:: Legendre polynomials

18.17.19 $\int _{{-1}}^{1}\mathop{P_{{n}}\/}\nolimits\!\left(x\right)e^{{ixy}}dx=i^{n}\sqrt{\frac{2\pi}{y}}\mathop{J_{{n+\frac{1}{2}}}\/}\nolimits\!\left(y\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : 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

18.17.20 $\int _{0}^{1}\mathop{P_{{n}}\/}\nolimits\!\left(1-2x^{2}\right)\mathop{\cos\/}\nolimits\!\left(xy\right)dx=(-1)^{n}\tfrac{1}{2}\pi\mathop{J_{{n+\frac{1}{2}}}\/}\nolimits\!\left(\tfrac{1}{2}y\right)\mathop{J_{{-n-\frac{1}{2}}}\/}\nolimits\!\left(\tfrac{1}{2}y\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : 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

18.17.21 $\int _{0}^{1}\mathop{P_{{n}}\/}\nolimits\!\left(1-2x^{2}\right)\mathop{\sin\/}\nolimits\!\left(xy\right)dx=\tfrac{1}{2}\pi\left(\mathop{J_{{n+\frac{1}{2}}}\/}\nolimits\!\left(\tfrac{1}{2}y\right)\right)^{2}.$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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

¶ Hermite

Keywords:: Hermite polynomials

18.17.22 $\frac{1}{2\sqrt{\pi}}\int _{{-\infty}}^{{\infty}}e^{{-\frac{1}{4}x^{2}}}\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)e^{{\frac{1}{2}ixy}}dx=i^{n}e^{{-\frac{1}{4}y^{2}}}\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(y\right),$

Symbols:: $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : 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

18.17.23 $\int _{0}^{{\infty}}e^{{-\frac{1}{2}x^{2}}}\mathop{\mathit{He}_{{2n}}\/}\nolimits\!\left(x\right)\mathop{\cos\/}\nolimits\!\left(xy\right)dx=(-1)^{n}\sqrt{\tfrac{1}{2}\pi}y^{{2n}}e^{{-\frac{1}{2}y^{2}}},$

Symbols:: $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : 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

18.17.24 $\int _{0}^{{\infty}}e^{{-x^{2}}}\mathop{\mathit{He}_{{2n}}\/}\nolimits\!\left(2x\right)\mathop{\cos\/}\nolimits\!\left(xy\right)dx=(-1)^{n}\tfrac{1}{2}\sqrt{\pi}e^{{-\frac{1}{4}y^{2}}}\mathop{\mathit{He}_{{2n}}\/}\nolimits\!\left(y\right).$

Symbols:: $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : 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

18.17.25 $\int _{0}^{\infty}e^{{-\frac{1}{2}x^{2}}}\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)\mathop{\mathit{He}_{{n+2m}}\/}\nolimits\!\left(x\right)\mathop{\cos\/}\nolimits\!\left(xy\right)dx=(-1)^{m}\sqrt{\tfrac{1}{2}\pi}n!\, y^{{2m}}e^{{-\frac{1}{2}y^{2}}}\mathop{L^{{(2m)}}_{{n}}\/}\nolimits\!\left(y^{2}\right),$

Symbols:: $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\int$ : 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

18.17.26 $\int _{0}^{\infty}e^{{-\frac{1}{2}x^{2}}}\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)\mathop{\mathit{He}_{{n+2m+1}}\/}\nolimits\!\left(x\right)\mathop{\sin\/}\nolimits\!\left(xy\right)dx=(-1)^{m}\sqrt{\tfrac{1}{2}\pi}n!\, y^{{2m+1}}e^{{-\frac{1}{2}y^{2}}}\mathop{L^{{(2m+1)}}_{{n}}\/}\nolimits\!\left(y^{2}\right).$

Symbols:: $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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

18.17.27 $\int _{0}^{\infty}e^{{-\frac{1}{2}x^{2}}}\mathop{\mathit{He}_{{2n+1}}\/}\nolimits\!\left(x\right)\mathop{\sin\/}\nolimits\!\left(xy\right)dx=(-1)^{n}\sqrt{\tfrac{1}{2}\pi}y^{{2n+1}}e^{{-\frac{1}{2}y^{2}}},$

Symbols:: $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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

18.17.28 $\int _{0}^{\infty}e^{{-x^{2}}}\mathop{\mathit{He}_{{2n+1}}\/}\nolimits\!\left(2x\right)\mathop{\sin\/}\nolimits\!\left(xy\right)dx=(-1)^{n}\tfrac{1}{2}\sqrt{\pi}e^{{-\frac{1}{4}y^{2}}}\mathop{\mathit{He}_{{2n+1}}\/}\nolimits\!\left(y\right).$

Symbols:: $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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

¶ Laguerre

Keywords:: Laguerre polynomials, classical orthogonal polynomials

18.17.29 $\int _{0}^{\infty}x^{{2m}}e^{{-\frac{1}{2}x^{2}}}\mathop{L^{{(2m)}}_{{n}}\/}\nolimits\!\left(x^{2}\right)\mathop{\cos\/}\nolimits\!\left(xy\right)dx=(-1)^{m}\sqrt{\tfrac{1}{2}\pi}\frac{1}{n!}e^{{-\frac{1}{2}y^{2}}}\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(y\right)\mathop{\mathit{He}_{{n+2m}}\/}\nolimits\!\left(y\right).$

Symbols:: $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\int$ : 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

18.17.30 $\int _{0}^{\infty}x^{{2n}}e^{{-\frac{1}{2}x^{2}}}\mathop{L^{{(n-\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(\tfrac{1}{2}x^{2}\right)\mathop{\cos\/}\nolimits\!\left(xy\right)dx=\sqrt{\tfrac{1}{2}\pi}y^{{2n}}e^{{-\frac{1}{2}y^{2}}}\mathop{L^{{(n-\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(\tfrac{1}{2}y^{2}\right).$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\int$ : 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

18.17.31 $\int _{0}^{\infty}e^{{-ax}}x^{{\nu-2n}}\mathop{L^{{(\nu-2n)}}_{{2n-1}}\/}\nolimits\!\left(ax\right)\mathop{\cos\/}\nolimits\!\left(xy\right)dx=i\frac{(-1)^{n}\mathop{\Gamma\/}\nolimits\!\left(\nu\right)}{2(2n-1)!}y^{{2n-1}}\left((a+iy)^{{-\nu}}-(a-iy)^{{-\nu}}\right),$ $\nu>2n-1$ , $a>0$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\int$ : 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

18.17.32 $\int _{0}^{\infty}e^{{-ax}}x^{{\nu-1-2n}}\mathop{L^{{(\nu-1-2n)}}_{{2n}}\/}\nolimits\!\left(ax\right)\mathop{\cos\/}\nolimits\!\left(xy\right)dx=\frac{(-1)^{n}\mathop{\Gamma\/}\nolimits\!\left(\nu\right)}{2(2n)!}y^{{2n}}\left((a+iy)^{{-\nu}}+(a-iy)^{{-\nu}}\right),$ $\nu>2n$ , $a>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\int$ : 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

§18.17(vi) Laplace Transforms

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.
Keywords:: classical orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.17.vi

¶ Jacobi

Keywords:: Jacobi polynomials, Laplace transform, integrals

18.17.33 $\int _{{-1}}^{1}e^{{-(x+1)z}}\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)(1-x)^{\alpha}(1+x)^{\beta}dx=\frac{(-1)^{n}2^{{\alpha+\beta+n+1}}\mathop{\Gamma\/}\nolimits\!\left(\alpha+n+1\right)\mathop{\Gamma\/}\nolimits\!\left(\beta+n+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(\alpha+\beta+2n+2\right)n!}z^{n}\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left({\beta+n+1\atop\alpha+\beta+2n+2};-2z\right),$ $z\in\Complex$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\Complex$ : complex plane (excluding infinity), $dx$ : differential of $x$ , $\in$ : element of, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\int$ : 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

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

¶ Laguerre

Keywords:: Laguerre polynomials, Laplace transform, integrals

18.17.34 $\int _{0}^{\infty}e^{{-xz}}\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)e^{{-x}}x^{\alpha}dx=\frac{\mathop{\Gamma\/}\nolimits\!\left(\alpha+n+1\right)z^{n}}{n!(z+1)^{{\alpha+n+1}}},$ $\realpart{z}>-1$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\int$ : integral, $\realpart{}$ : real part, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.17(i), §18.17(vi), §18.17(viii)
Permalink:: http://dlmf.nist.gov/18.17.E34
Encodings:: TeX, pMML, png

¶ Hermite

Keywords:: Hermite polynomials, Laplace transform, integrals

18.17.35 $\int _{{-\infty}}^{\infty}e^{{-xz}}\mathop{H_{{n}}\/}\nolimits\!\left(x\right)e^{{-x^{2}}}dx=\pi^{{\frac{1}{2}}}(-z)^{n}e^{{\frac{1}{4}z^{2}}},$ $z\in\Complex$ .

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\Complex$ : complex plane (excluding infinity), $dx$ : differential of $x$ , $\in$ : element of, $e$ : base of exponential function, $\int$ : 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

§18.17(vii) Mellin Transforms

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.
Keywords:: classical orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.17.vii

¶ Jacobi

Keywords:: Jacobi polynomials, Mellin transform

18.17.36 $\int _{{-1}}^{1}(1-x)^{{z-1}}(1+x)^{{\beta}}\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)dx=\frac{2^{{\beta+z}}\mathop{\Gamma\/}\nolimits\!\left(z\right)\mathop{\Gamma\/}\nolimits\!\left(1+\beta+n\right)\left(1+\alpha-z\right)_{{n}}}{n!\mathop{\Gamma\/}\nolimits\!\left(1+\beta+z+n\right)},$ $\realpart{z}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral, $\realpart{}$ : real part, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.17(vii)
Permalink:: http://dlmf.nist.gov/18.17.E36
Encodings:: TeX, pMML, png

¶ Ultraspherical

Keywords:: Mellin transform, ultraspherical polynomials

18.17.37 $\int _{0}^{1}(1-x^{2})^{{\lambda-\frac{1}{2}}}\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)x^{{z-1}}dx=\frac{\pi\, 2^{{1-2\lambda-z}}\mathop{\Gamma\/}\nolimits\!\left(n+2\lambda\right)\mathop{\Gamma\/}\nolimits\!\left(z\right)}{n!\mathop{\Gamma\/}\nolimits\!\left(\lambda\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\frac{1}{2}n+\lambda+\frac{1}{2}z\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\frac{1}{2}z-\frac{1}{2}n\right)},$ $\realpart{z}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral, $\realpart{}$ : real part, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.17(vii)
Permalink:: http://dlmf.nist.gov/18.17.E37
Encodings:: TeX, pMML, png

¶ Legendre

Keywords:: Legendre polynomials

18.17.38 $\int _{0}^{1}\mathop{P_{{2n}}\/}\nolimits\!\left(x\right)x^{{z-1}}dx=\frac{(-1)^{n}\left(\frac{1}{2}-\frac{1}{2}z\right)_{{n}}}{2\left(\frac{1}{2}z\right)_{{n+1}}},$ $\realpart{z}>0$ ,

Symbols:: $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.17(vii)
Permalink:: http://dlmf.nist.gov/18.17.E38
Encodings:: TeX, pMML, png

18.17.39 $\int _{0}^{1}\mathop{P_{{2n+1}}\/}\nolimits\!\left(x\right)x^{{z-1}}dx=\frac{(-1)^{n}\left(1-\frac{1}{2}z\right)_{{n}}}{2\left(\frac{1}{2}+\frac{1}{2}z\right)_{{n+1}}},$ $\realpart{z}>-1$ .

Symbols:: $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.17(vii)
Permalink:: http://dlmf.nist.gov/18.17.E39
Encodings:: TeX, pMML, png

¶ Laguerre

Keywords:: Laguerre polynomials, Mellin transform

18.17.40 $\int _{0}^{\infty}e^{{-ax}}\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(bx\right)x^{{z-1}}dx=\frac{\mathop{\Gamma\/}\nolimits\!\left(z+n\right)}{n!}\*{(a-b)^{n}}a^{{-n-z}}\*\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({-n,1+\alpha-z\atop 1-n-z};\frac{a}{a-b}\right),$ $\realpart{a}>0$ , $\realpart{z}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\int$ : integral, $\realpart{}$ : real part, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.17(vii)
Permalink:: http://dlmf.nist.gov/18.17.E40
Encodings:: TeX, pMML, png

For the hypergeometric function $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits$ see §§15.1 and 15.2(i).

¶ Hermite

Keywords:: Hermite polynomials, Mellin transform

18.17.41 $\int _{0}^{\infty}e^{{-ax}}\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)x^{{z-1}}dx=\mathop{\Gamma\/}\nolimits\!\left(z+n\right)a^{{-n-2}}\mathop{{{}_{{2}}F_{{2}}}\/}\nolimits\!\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-\tfrac{1}{2}z-\tfrac{1}{2}n,-\tfrac{1}{2}z-\tfrac{1}{2}n+\tfrac{1}{2}};-\tfrac{1}{2}a^{2}\right),$ $\realpart{a}>0$ . Also, $\realpart{z}>0$ , $n$ even; $\realpart{z}>-1$ , $n$ odd.

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.17(vii)
Permalink:: http://dlmf.nist.gov/18.17.E41
Encodings:: TeX, pMML, png

For the generalized hypergeometric function $\mathop{{{}_{{2}}F_{{2}}}\/}\nolimits$ 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 Szegö (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

¶ Chebyshev

Keywords:: Chebyshev polynomials, integrals

18.17.42 $\pvint _{{-1}}^{1}\mathop{T_{{n}}\/}\nolimits\!\left(y\right)\frac{(1-y^{2})^{{-\frac{1}{2}}}}{y-x}dy=\pi\mathop{U_{{n-1}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $dx$ : differential of $x$ , $\pvint _{a}^{b}$ : 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

18.17.43 $\pvint _{{-1}}^{1}\mathop{U_{{n-1}}\/}\nolimits\!\left(y\right)\frac{(1-y^{2})^{{\frac{1}{2}}}}{y-x}dy=-\pi\mathop{T_{{n}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $dx$ : differential of $x$ , $\pvint _{a}^{b}$ : 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

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

¶ Legendre

Keywords:: Legendre polynomials, integrals

18.17.44 $\int _{{-1}}^{1}\frac{\mathop{P_{{n}}\/}\nolimits\!\left(x\right)-\mathop{P_{{n}}\/}\nolimits\!\left(t\right)}{|x-t|}dt=2\left(1+\tfrac{1}{2}+\dots+\tfrac{1}{n}\right)\mathop{P_{{n}}\/}\nolimits\!\left(x\right),$ $-1\leq x\leq 1$ .

Symbols:: $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $dx$ : differential of $x$ , $\int$ : integral, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.17(viii)
Permalink:: http://dlmf.nist.gov/18.17.E44
Encodings:: TeX, pMML, png

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

18.17.45 $(n+\tfrac{1}{2})(1+x)^{{\frac{1}{2}}}\int _{{-1}}^{x}(x-t)^{{-\frac{1}{2}}}\mathop{P_{{n}}\/}\nolimits\!\left(t\right)dt=\mathop{T_{{n}}\/}\nolimits\!\left(x\right)+\mathop{T_{{n+1}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $dx$ : differential of $x$ , $\int$ : integral, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.17(viii)
Permalink:: http://dlmf.nist.gov/18.17.E45
Encodings:: TeX, pMML, png

18.17.46 $(n+\tfrac{1}{2})(1-x)^{{\frac{1}{2}}}\int _{{x}}^{1}(t-x)^{{-\frac{1}{2}}}\mathop{P_{{n}}\/}\nolimits\!\left(t\right)dt=\mathop{T_{{n}}\/}\nolimits\!\left(x\right)-\mathop{T_{{n+1}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $dx$ : differential of $x$ , $\int$ : integral, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.17(viii)
Permalink:: http://dlmf.nist.gov/18.17.E46
Encodings:: TeX, pMML, png

¶ Laguerre

18.17.47 $\int _{0}^{x}t^{\alpha}\frac{\mathop{L^{{(\alpha)}}_{{m}}\/}\nolimits\!\left(t\right)}{\mathop{L^{{(\alpha)}}_{{m}}\/}\nolimits\!\left(0\right)}(x-t)^{\beta}\frac{\mathop{L^{{(\beta)}}_{{n}}\/}\nolimits\!\left(x-t\right)}{\mathop{L^{{(\beta)}}_{{n}}\/}\nolimits\!\left(0\right)}dt=\frac{\mathop{\Gamma\/}\nolimits\!\left(\alpha+1\right)\mathop{\Gamma\/}\nolimits\!\left(\beta+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(\alpha+\beta+2\right)}x^{{\alpha+\beta+1}}\frac{\mathop{L^{{(\alpha+\beta+1)}}_{{m+n}}\/}\nolimits\!\left(x\right)}{\mathop{L^{{(\alpha+\beta+1)}}_{{m+n}}\/}\nolimits\!\left(0\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\int$ : integral, $m$ : 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

¶ Hermite

Keywords:: Hermite polynomials, integrals

18.17.48 $\int _{{-\infty}}^{\infty}\mathop{H_{{m}}\/}\nolimits\!\left(y\right)e^{{-y^{2}}}\mathop{H_{{n}}\/}\nolimits\!\left(x-y\right)e^{{-(x-y)^{2}}}dy=\pi^{{\frac{1}{2}}}2^{{-\frac{1}{2}(m+n+1)}}\mathop{H_{{m+n}}\/}\nolimits\!\left(2^{{-\frac{1}{2}}}x\right)e^{{-\frac{1}{2}x^{2}}}.$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : 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

18.17.49 $\int _{{-\infty}}^{\infty}\mathop{H_{{\ell}}\/}\nolimits\!\left(x\right)\mathop{H_{{m}}\/}\nolimits\!\left(x\right)\mathop{H_{{n}}\/}\nolimits\!\left(x\right)e^{{-x^{2}}}dx=\frac{2^{{\frac{1}{2}(\ell+m+n)}}\ell\,!\, m\,!\, n\,!\,\sqrt{\pi}}{(\tfrac{1}{2}\ell+\tfrac{1}{2}m-\tfrac{1}{2}n)\,!\,(\tfrac{1}{2}m+\tfrac{1}{2}n-\tfrac{1}{2}\ell\,)\,!\,(\tfrac{1}{2}n+\tfrac{1}{2}\ell-\tfrac{1}{2}m\,)\,!},$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\int$ : integral, $\ell$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.17(viii)
Permalink:: http://dlmf.nist.gov/18.17.E49
Encodings:: TeX, pMML, png

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.

§18.17(ix) Compendia

Keywords:: Hermite polynomials, Jacobi polynomials, classical orthogonal polynomials, integrals
Permalink:: http://dlmf.nist.gov/18.17.ix

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 (2000, pp. 788–806), 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).

§18.17 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

¶ Chebyshev

¶ Legendre

¶ Laguerre

¶ Hermite

§18.17(ix) Compendia