§18.17 Integrals

§18.17(i) Indefinite Integrals

Jacobi

 18.17.1 $2n\int_{0}^{x}(1-y)^{\alpha}(1+y)^{\beta}P^{(\alpha,\beta)}_{n}\left(y\right)% \mathrm{d}y=P^{(\alpha+1,\beta+1)}_{n-1}\left(0\right)-(1-x)^{\alpha+1}(1+x)^{% \beta+1}P^{(\alpha+1,\beta+1)}_{n-1}\left(x\right).$

Laguerre

 18.17.2 $\int_{0}^{x}L_{m}\left(y\right)L_{n}\left(x-y\right)\mathrm{d}y=\int_{0}^{x}L_% {m+n}\left(y\right)\mathrm{d}y=L_{m+n}\left(x\right)-L_{m+n+1}\left(x\right).$

Hermite

 18.17.3 $\int_{0}^{x}H_{n}\left(y\right)\mathrm{d}y=\frac{1}{2(n+1)}(H_{n+1}\left(x% \right)-H_{n+1}\left(0\right)),$
 18.17.4 $\int_{0}^{x}e^{-y^{2}}H_{n}\left(y\right)\mathrm{d}y=H_{n-1}\left(0\right)-e^{% -x^{2}}H_{n-1}\left(x\right).$

§18.17(ii) Integral Representations for Products

Ultraspherical

 18.17.5 $\frac{C^{(\lambda)}_{n}\left(\cos\theta_{1}\right)}{C^{(\lambda)}_{n}\left(1% \right)}\frac{C^{(\lambda)}_{n}\left(\cos\theta_{2}\right)}{C^{(\lambda)}_{n}% \left(1\right)}=\frac{\Gamma\left(\lambda+\frac{1}{2}\right)}{\pi^{\frac{1}{2}% }\Gamma\left(\lambda\right)}\*\int_{0}^{\pi}\frac{C^{(\lambda)}_{n}\left(\cos% \theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}\cos\phi\right)}{C^{(% \lambda)}_{n}\left(1\right)}(\sin\phi)^{2\lambda-1}\mathrm{d}\phi,$ $\lambda>0$.

Legendre

 18.17.6 $P_{n}\left(\cos\theta_{1}\right)P_{n}\left(\cos\theta_{2}\right)=\frac{1}{\pi}% \int_{0}^{\pi}P_{n}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta% _{2}\cos\phi\right)\mathrm{d}\phi.$

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

Legendre

 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.

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

 18.17.8 $\left(H_{n}\left(x\right)\right)^{2}+2^{n}(n!)^{2}e^{x^{2}}\left(V\left(-n-% \tfrac{1}{2},2^{\frac{1}{2}}x\right)\right)^{2}=\frac{2^{n+\frac{3}{2}}n!\,e^{% x^{2}}}{\pi}\int_{0}^{\infty}\frac{e^{-(2n+1)t+x^{2}\tanh t}}{(\sinh 2t)^{% \frac{1}{2}}}\mathrm{d}t.$

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

Jacobi

 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,
 18.17.10 $\displaystyle\frac{x^{\beta+\mu}(x+1)^{n}}{\Gamma\left(\beta+\mu+n+1\right)}P^% {(\alpha,\beta+\mu)}_{n}\left(\frac{x-1}{x+1}\right)$ $\displaystyle=\int_{0}^{x}\frac{y^{\beta}(y+1)^{n}}{\Gamma\left(\beta+n+1% \right)}P^{(\alpha,\beta)}_{n}\left(\frac{y-1}{y+1}\right)\*\frac{(x-y)^{\mu-1% }}{\Gamma\left(\mu\right)}\mathrm{d}y,$ $\mu>0$, $x>0$, 18.17.11 $\displaystyle\frac{\Gamma\left(n+\alpha+\beta-\mu+1\right)}{x^{n+\alpha+\beta-% \mu+1}}P^{(\alpha,\beta-\mu)}_{n}\left(1-2x^{-1}\right)$ $\displaystyle=\int_{x}^{\infty}\frac{\Gamma\left(n+\alpha+\beta+1\right)}{y^{n% +\alpha+\beta+1}}P^{(\alpha,\beta)}_{n}\left(1-2y^{-1}\right)\*\frac{(y-x)^{% \mu-1}}{\Gamma\left(\mu\right)}\mathrm{d}y,$ $\alpha+\beta+1>\mu>0$, $x>1$,

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

Ultraspherical

 18.17.12 $\displaystyle\frac{\Gamma\left(\lambda-\mu\right)C^{(\lambda-\mu)}_{n}\left(x^% {-\frac{1}{2}}\right)}{x^{\lambda-\mu+\frac{1}{2}n}}$ $\displaystyle=\int_{x}^{\infty}\frac{\Gamma\left(\lambda\right)C^{(\lambda)}_{% n}\left(y^{-\frac{1}{2}}\right)}{y^{\lambda+\frac{1}{2}n}}\frac{(y-x)^{\mu-1}}% {\Gamma\left(\mu\right)}\mathrm{d}y,$ $\lambda>\mu>0$, $x>0$, 18.17.13 $\displaystyle\frac{x^{\frac{1}{2}n}(x-1)^{\lambda+\mu-\frac{1}{2}}}{\Gamma% \left(\lambda+\mu+\tfrac{1}{2}\right)}\frac{C^{(\lambda+\mu)}_{n}\left(x^{-% \frac{1}{2}}\right)}{C^{(\lambda+\mu)}_{n}\left(1\right)}$ $\displaystyle=\int_{1}^{x}\frac{y^{\frac{1}{2}n}(y-1)^{\lambda-\frac{1}{2}}}{% \Gamma\left(\lambda+\tfrac{1}{2}\right)}\frac{C^{(\lambda)}_{n}\left(y^{-\frac% {1}{2}}\right)}{C^{(\lambda)}_{n}\left(1\right)}\frac{(x-y)^{\mu-1}}{\Gamma% \left(\mu\right)}\mathrm{d}y,$ $\mu>0$, $x>1$.

Laguerre

 18.17.14 $\displaystyle\frac{x^{\alpha+\mu}L^{(\alpha+\mu)}_{n}\left(x\right)}{\Gamma% \left(\alpha+\mu+n+1\right)}$ $\displaystyle=\int_{0}^{x}\frac{y^{\alpha}L^{(\alpha)}_{n}\left(y\right)}{% \Gamma\left(\alpha+n+1\right)}\frac{(x-y)^{\mu-1}}{\Gamma\left(\mu\right)}% \mathrm{d}y,$ $\mu>0$, $x>0$. 18.17.15 $\displaystyle e^{-x}L^{(\alpha)}_{n}\left(x\right)$ $\displaystyle=\int_{x}^{\infty}e^{-y}L^{(\alpha+\mu)}_{n}\left(y\right)\frac{(% y-x)^{\mu-1}}{\Gamma\left(\mu\right)}\mathrm{d}y,$ $\mu>0$.

§18.17(v) Fourier Transforms

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

Jacobi

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

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

 18.17.17 $\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda)}_{2n}\left(x\right)% \cos\left(xy\right)\mathrm{d}x=\frac{(-1)^{n}\pi\Gamma\left(2n+2\lambda\right)% J_{\lambda+2n}\left(y\right)}{(2n)!\Gamma\left(\lambda\right)(2y)^{\lambda}},$
 18.17.18 $\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda)}_{2n+1}\left(x\right)% \sin\left(xy\right)\mathrm{d}x=\frac{(-1)^{n}\pi\Gamma\left(2n+2\lambda+1% \right)J_{2n+\lambda+1}\left(y\right)}{(2n+1)!\Gamma\left(\lambda\right)(2y)^{% \lambda}}.$

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

Legendre

 18.17.19 $\int_{-1}^{1}P_{n}\left(x\right)e^{ixy}\mathrm{d}x=i^{n}\sqrt{\frac{2\pi}{y}}J% _{n+\frac{1}{2}}\left(y\right),$
 18.17.20 $\int_{0}^{1}P_{n}\left(1-2x^{2}\right)\cos\left(xy\right)\mathrm{d}x=(-1)^{n}% \tfrac{1}{2}\pi J_{n+\frac{1}{2}}\left(\tfrac{1}{2}y\right)J_{-n-\frac{1}{2}}% \left(\tfrac{1}{2}y\right),$
 18.17.21 $\int_{0}^{1}P_{n}\left(1-2x^{2}\right)\sin\left(xy\right)\mathrm{d}x=\tfrac{1}% {2}\pi\left(J_{n+\frac{1}{2}}\left(\tfrac{1}{2}y\right)\right)^{2}.$

Hermite

 18.17.22 $\frac{1}{2\sqrt{\pi}}\int_{-\infty}^{\infty}e^{-\frac{1}{4}x^{2}}\mathit{He}_{% n}\left(x\right)e^{\frac{1}{2}\mathrm{i}xy}\mathrm{d}x={\mathrm{i}^{n}}e^{-% \frac{1}{4}y^{2}}\mathit{He}_{n}\left(y\right),$
 18.17.23 $\int_{0}^{\infty}e^{-\frac{1}{2}x^{2}}\mathit{He}_{2n}\left(x\right)\cos\left(% xy\right)\mathrm{d}x=(-1)^{n}\sqrt{\tfrac{1}{2}\pi}y^{2n}e^{-\frac{1}{2}y^{2}},$
 18.17.24 $\int_{0}^{\infty}e^{-x^{2}}\mathit{He}_{2n}\left(2x\right)\cos\left(xy\right)% \mathrm{d}x=(-1)^{n}\tfrac{1}{2}\sqrt{\pi}e^{-\frac{1}{4}y^{2}}\mathit{He}_{2n% }\left(y\right).$
 18.17.25 $\int_{0}^{\infty}e^{-\frac{1}{2}x^{2}}\mathit{He}_{n}\left(x\right)\mathit{He}% _{n+2m}\left(x\right)\cos\left(xy\right)\mathrm{d}x=(-1)^{m}\sqrt{\tfrac{1}{2}% \pi}n!\,y^{2m}e^{-\frac{1}{2}y^{2}}L^{(2m)}_{n}\left(y^{2}\right),$
 18.17.26 $\int_{0}^{\infty}e^{-\frac{1}{2}x^{2}}\mathit{He}_{n}\left(x\right)\mathit{He}% _{n+2m+1}\left(x\right)\sin\left(xy\right)\mathrm{d}x=(-1)^{m}\sqrt{\tfrac{1}{% 2}\pi}n!\,y^{2m+1}e^{-\frac{1}{2}y^{2}}L^{(2m+1)}_{n}\left(y^{2}\right).$
 18.17.27 $\int_{0}^{\infty}e^{-\frac{1}{2}x^{2}}\mathit{He}_{2n+1}\left(x\right)\sin% \left(xy\right)\mathrm{d}x=(-1)^{n}\sqrt{\tfrac{1}{2}\pi}y^{2n+1}e^{-\frac{1}{% 2}y^{2}},$
 18.17.28 $\int_{0}^{\infty}e^{-x^{2}}\mathit{He}_{2n+1}\left(2x\right)\sin\left(xy\right% )\mathrm{d}x=(-1)^{n}\tfrac{1}{2}\sqrt{\pi}e^{-\frac{1}{4}y^{2}}\mathit{He}_{2% n+1}\left(y\right).$

Laguerre

 18.17.29 $\int_{0}^{\infty}x^{2m}e^{-\frac{1}{2}x^{2}}L^{(2m)}_{n}\left(x^{2}\right)\cos% \left(xy\right)\mathrm{d}x=(-1)^{m}\sqrt{\tfrac{1}{2}\pi}\frac{1}{n!}e^{-\frac% {1}{2}y^{2}}\mathit{He}_{n}\left(y\right)\mathit{He}_{n+2m}\left(y\right).$
 18.17.30 $\int_{0}^{\infty}x^{2n}e^{-\frac{1}{2}x^{2}}L^{(n-\frac{1}{2})}_{n}\left(% \tfrac{1}{2}x^{2}\right)\cos\left(xy\right)\mathrm{d}x=\sqrt{\tfrac{1}{2}\pi}y% ^{2n}e^{-\frac{1}{2}y^{2}}L^{(n-\frac{1}{2})}_{n}\left(\tfrac{1}{2}y^{2}\right).$
 18.17.31 $\int_{0}^{\infty}e^{-ax}x^{\nu-2n}L^{(\nu-2n)}_{2n-1}\left(ax\right)\cos\left(% xy\right)\mathrm{d}x=i\frac{(-1)^{n}\Gamma\left(\nu\right)}{2(2n-1)!}y^{2n-1}% \left((a+iy)^{-\nu}-(a-iy)^{-\nu}\right),$ $\nu>2n-1$, $a>0$,
 18.17.32 $\int_{0}^{\infty}e^{-ax}x^{\nu-1-2n}L^{(\nu-1-2n)}_{2n}\left(ax\right)\cos% \left(xy\right)\mathrm{d}x=\frac{(-1)^{n}\Gamma\left(\nu\right)}{2(2n)!}y^{2n}% \left((a+iy)^{-\nu}+(a-iy)^{-\nu}\right),$ $\nu>2n$, $a>0$.

§18.17(vi) Laplace Transforms

Jacobi

 18.17.33 $\int_{-1}^{1}e^{-(x+1)z}P^{(\alpha,\beta)}_{n}\left(x\right)(1-x)^{\alpha}(1+x% )^{\beta}\mathrm{d}x=\frac{(-1)^{n}2^{\alpha+\beta+n+1}\Gamma\left(\alpha+n+1% \right)\Gamma\left(\beta+n+1\right)}{\Gamma\left(\alpha+\beta+2n+2\right)n!}z^% {n}{{}_{1}F_{1}}\left({\beta+n+1\atop\alpha+\beta+2n+2};-2z\right),$ $z\in\mathbb{C}$.

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

Laguerre

 18.17.34 $\int_{0}^{\infty}e^{-xz}L^{(\alpha)}_{n}\left(x\right)e^{-x}x^{\alpha}\mathrm{% d}x=\frac{\Gamma\left(\alpha+n+1\right)z^{n}}{n!(z+1)^{\alpha+n+1}},$ $\Re z>-1$.

Hermite

 18.17.35 $\int_{-\infty}^{\infty}e^{-xz}H_{n}\left(x\right)e^{-x^{2}}\mathrm{d}x=\pi^{% \frac{1}{2}}(-z)^{n}e^{\frac{1}{4}z^{2}},$ $z\in\mathbb{C}$.

§18.17(vii) Mellin Transforms

Jacobi

 18.17.36 $\int_{-1}^{1}(1-x)^{z-1}(1+x)^{\beta}P^{(\alpha,\beta)}_{n}\left(x\right)% \mathrm{d}x=\frac{2^{\beta+z}\Gamma\left(z\right)\Gamma\left(1+\beta+n\right){% \left(1+\alpha-z\right)_{n}}}{n!\Gamma\left(1+\beta+z+n\right)},$ $\Re z>0$.

Ultraspherical

 18.17.37 $\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda)}_{n}\left(x\right)x^{z% -1}\mathrm{d}x=\frac{\pi\,2^{1-2\lambda-z}\Gamma\left(n+2\lambda\right)\Gamma% \left(z\right)}{n!\Gamma\left(\lambda\right)\Gamma\left(\frac{1}{2}+\frac{1}{2% }n+\lambda+\frac{1}{2}z\right)\Gamma\left(\frac{1}{2}+\frac{1}{2}z-\frac{1}{2}% n\right)},$ $\Re z>0$.

Legendre

 18.17.38 $\int_{0}^{1}P_{2n}\left(x\right)x^{z-1}\mathrm{d}x=\frac{(-1)^{n}{\left(\frac{% 1}{2}-\frac{1}{2}z\right)_{n}}}{2{\left(\frac{1}{2}z\right)_{n+1}}},$ $\Re z>0$,
 18.17.39 $\int_{0}^{1}P_{2n+1}\left(x\right)x^{z-1}\mathrm{d}x=\frac{(-1)^{n}{\left(1-% \frac{1}{2}z\right)_{n}}}{2{\left(\frac{1}{2}+\frac{1}{2}z\right)_{n+1}}},$ $\Re z>-1$.

Laguerre

 18.17.40 $\int_{0}^{\infty}e^{-ax}L^{(\alpha)}_{n}\left(bx\right)x^{z-1}\mathrm{d}x=% \frac{\Gamma\left(z+n\right)}{n!}\*{(a-b)^{n}}a^{-n-z}\*{{}_{2}F_{1}}\left({-n% ,1+\alpha-z\atop 1-n-z};\frac{a}{a-b}\right),$ $\Re a>0$, $\Re z>0$.

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

Hermite

 18.17.41 $\int_{0}^{\infty}e^{-ax}\mathit{He}_{n}\left(x\right)x^{z-1}\mathrm{d}x=\Gamma% \left(z+n\right)a^{-n-2}{{}_{2}F_{2}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+% \tfrac{1}{2}\atop-\tfrac{1}{2}z-\tfrac{1}{2}n,-\tfrac{1}{2}z-\tfrac{1}{2}n+% \tfrac{1}{2}};-\tfrac{1}{2}a^{2}\right),$ $\Re a>0$. Also, $\Re z>0$, $n$ even; $\Re z>-1$, $n$ odd.

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

§18.17(viii) Other Integrals

Chebyshev

 18.17.42 $\pvint_{-1}^{1}T_{n}\left(y\right)\frac{(1-y^{2})^{-\frac{1}{2}}}{y-x}\mathrm{% d}y=\pi U_{n-1}\left(x\right),$
 18.17.43 $\pvint_{-1}^{1}U_{n-1}\left(y\right)\frac{(1-y^{2})^{\frac{1}{2}}}{y-x}\mathrm% {d}y=-\pi T_{n}\left(x\right).$

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

Legendre

 18.17.44 $\int_{-1}^{1}\frac{P_{n}\left(x\right)-P_{n}\left(t\right)}{|x-t|}\mathrm{d}t=% 2\left(1+\tfrac{1}{2}+\dots+\tfrac{1}{n}\right)P_{n}\left(x\right),$ $-1\leq x\leq 1$.

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

 18.17.45 $(n+\tfrac{1}{2})(1+x)^{\frac{1}{2}}\int_{-1}^{x}(x-t)^{-\frac{1}{2}}P_{n}\left% (t\right)\mathrm{d}t=T_{n}\left(x\right)+T_{n+1}\left(x\right),$
 18.17.46 $(n+\tfrac{1}{2})(1-x)^{\frac{1}{2}}\int_{x}^{1}(t-x)^{-\frac{1}{2}}P_{n}\left(% t\right)\mathrm{d}t=T_{n}\left(x\right)-T_{n+1}\left(x\right).$

Laguerre

 18.17.47 $\int_{0}^{x}t^{\alpha}\frac{L^{(\alpha)}_{m}\left(t\right)}{L^{(\alpha)}_{m}% \left(0\right)}(x-t)^{\beta}\frac{L^{(\beta)}_{n}\left(x-t\right)}{L^{(\beta)}% _{n}\left(0\right)}\mathrm{d}t=\frac{\Gamma\left(\alpha+1\right)\Gamma\left(% \beta+1\right)}{\Gamma\left(\alpha+\beta+2\right)}x^{\alpha+\beta+1}\frac{L^{(% \alpha+\beta+1)}_{m+n}\left(x\right)}{L^{(\alpha+\beta+1)}_{m+n}\left(0\right)}.$

Hermite

 18.17.48 $\int_{-\infty}^{\infty}H_{m}\left(y\right)e^{-y^{2}}H_{n}\left(x-y\right)e^{-(% x-y)^{2}}\mathrm{d}y=\pi^{\frac{1}{2}}2^{-\frac{1}{2}(m+n+1)}H_{m+n}\left(2^{-% \frac{1}{2}}x\right)e^{-\frac{1}{2}x^{2}}.$
 18.17.49 $\int_{-\infty}^{\infty}H_{\ell}\left(x\right)H_{m}\left(x\right)H_{n}\left(x% \right)e^{-x^{2}}\mathrm{d}x=\frac{2^{\frac{1}{2}(\ell+m+n)}\ell\,!\,m\,!\,n\,% !\,\sqrt{\pi}}{(\tfrac{1}{2}\ell+\tfrac{1}{2}m-\tfrac{1}{2}n)\,!\,(\tfrac{1}{2% }m+\tfrac{1}{2}n-\tfrac{1}{2}\ell\,)\,!\,(\tfrac{1}{2}n+\tfrac{1}{2}\ell-% \tfrac{1}{2}m\,)\,!},$

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

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).