# §18.17 Integrals

## §18.17(i) Indefinite Integrals

### Jacobi

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

### Laguerre

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

### Hermite

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

## §18.17(ii) Integral Representations for Products

### Ultraspherical

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

### Legendre

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

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

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

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

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

### Jacobi

 18.17.9 $\frac{(1-x)^{\alpha+\mu}\mathop{P^{(\alpha+\mu,\beta-\mu)}_{n}\/}\nolimits\!% \left(x\right)}{\mathop{\Gamma\/}\nolimits\!\left(\alpha+\mu+n+1\right)}=\int_% {x}^{1}\frac{(1-y)^{\alpha}\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left(y% \right)}{\mathop{\Gamma\/}\nolimits\!\left(\alpha+n+1\right)}\frac{(y-x)^{\mu-% 1}}{\mathop{\Gamma\/}\nolimits\!\left(\mu\right)}\mathrm{d}y,$ $\mu>0$, $-1,
 18.17.10 $\displaystyle\frac{x^{\beta+\mu}(x+1)^{n}}{\mathop{\Gamma\/}\nolimits\!\left(% \beta+\mu+n+1\right)}\mathop{P^{(\alpha,\beta+\mu)}_{n}\/}\nolimits\!\left(% \frac{x-1}{x+1}\right)$ $\displaystyle=\int_{0}^{x}\frac{y^{\beta}(y+1)^{n}}{\mathop{\Gamma\/}\nolimits% \!\left(\beta+n+1\right)}\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left(% \frac{y-1}{y+1}\right)\*\frac{(x-y)^{\mu-1}}{\mathop{\Gamma\/}\nolimits\!\left% (\mu\right)}\mathrm{d}y,$ $\mu>0$, $x>0$, 18.17.11 $\displaystyle\frac{\mathop{\Gamma\/}\nolimits\!\left(n+\alpha+\beta-\mu+1% \right)}{x^{n+\alpha+\beta-\mu+1}}\mathop{P^{(\alpha,\beta-\mu)}_{n}\/}% \nolimits\!\left(1-2x^{-1}\right)$ $\displaystyle=\int_{x}^{\infty}\frac{\mathop{\Gamma\/}\nolimits\!\left(n+% \alpha+\beta+1\right)}{y^{n+\alpha+\beta+1}}\mathop{P^{(\alpha,\beta)}_{n}\/}% \nolimits\!\left(1-2y^{-1}\right)\*\frac{(y-x)^{\mu-1}}{\mathop{\Gamma\/}% \nolimits\!\left(\mu\right)}\mathrm{d}y,$ $\alpha+\beta+1>\mu>0$, $x>1$,

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

### Laguerre

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

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

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

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

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

### Legendre

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

### Hermite

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

### Laguerre

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

## §18.17(vi) Laplace Transforms

### Jacobi

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

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

### Laguerre

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

### Hermite

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

## §18.17(vii) Mellin Transforms

### Jacobi

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

### Ultraspherical

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

### Legendre

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

### Laguerre

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

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

### Hermite

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

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

## §18.17(viii) Other Integrals

### Chebyshev

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

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

### Legendre

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

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

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

### Laguerre

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

### Hermite

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

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).