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

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

### 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). 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

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

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. 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

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

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

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

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

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

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

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

 18.17.16_5 $\int_{-1}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda)}_{n}\left(x\right)\,{% \mathrm{e}}^{\mathrm{i}xy}\,\mathrm{d}x=\frac{2\pi\,{\mathrm{i}}^{n}\Gamma% \left(n+2\lambda\right)J_{n+\lambda}\left(y\right)}{n!\,\Gamma\left(\lambda% \right)\,(2y)^{\lambda}},$
 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 $\int_{-1}^{1}P_{n}\left(x\right){\mathrm{e}}^{\mathrm{i}xy}\,\mathrm{d}x={% \mathrm{i}}^{n}\sqrt{\frac{2\pi}{y}}J_{n+\frac{1}{2}}\left(y\right),$
 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

 18.17.21_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 $\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}{\mathrm{e}}^{-x^{2}}H_{n}\left(x% \right)\,{\mathrm{e}}^{\mathrm{i}xy}\,\mathrm{d}x={\mathrm{i}}^{n}{\mathrm{e}}% ^{-\frac{1}{4}y^{2}}y^{n},$
 18.17.21_3 $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}{\mathrm{e}}^{-\frac{1}{2}x^{2}}H_% {n}\left(x\right)\,{\mathrm{e}}^{\mathrm{i}xy}\,\mathrm{d}x={\mathrm{i}}^{n}{% \mathrm{e}}^{-\frac{1}{2}y^{2}}H_{n}\left(y\right).$

In (18.17.21_1) the branch choice of $\sqrt{2c-1}$ for $0 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 $\frac{1}{2\sqrt{\pi}}\int_{-\infty}^{\infty}{\mathrm{e}}^{-\frac{1}{4}x^{2}}% \mathit{He}_{n}\left(x\right){\mathrm{e}}^{\frac{1}{2}\mathrm{i}xy}\,\mathrm{d% }x={\mathrm{i}}^{n}{\mathrm{e}}^{-\frac{1}{4}y^{2}}\mathit{He}_{n}\left(y% \right),$
 18.17.23 $\int_{0}^{\infty}{\mathrm{e}}^{-\frac{1}{2}x^{2}}\mathit{He}_{2n}\left(x\right% )\cos\left(xy\right)\,\mathrm{d}x=(-1)^{n}\sqrt{\tfrac{1}{2}\pi}y^{2n}{\mathrm% {e}}^{-\frac{1}{2}y^{2}},$
 18.17.24 $\int_{0}^{\infty}{\mathrm{e}}^{-x^{2}}\mathit{He}_{2n}\left(2x\right)\cos\left% (xy\right)\,\mathrm{d}x=(-1)^{n}\tfrac{1}{2}\sqrt{\pi}{\mathrm{e}}^{-\frac{1}{% 4}y^{2}}\mathit{He}_{2n}\left(y\right).$
 18.17.25 $\int_{0}^{\infty}{\mathrm{e}}^{-\frac{1}{2}x^{2}}\mathit{He}_{n}\left(x\right)% \mathit{He}_{n+2m}\left(x\right)\cos\left(xy\right)\,\mathrm{d}x=(-1)^{m}\sqrt% {\tfrac{1}{2}\pi}n!\,y^{2m}{\mathrm{e}}^{-\frac{1}{2}y^{2}}L^{(2m)}_{n}\left(y% ^{2}\right),$
 18.17.26 $\int_{0}^{\infty}{\mathrm{e}}^{-\frac{1}{2}x^{2}}\mathit{He}_{n}\left(x\right)% \mathit{He}_{n+2m+1}\left(x\right)\sin\left(xy\right)\,\mathrm{d}x=(-1)^{m}% \sqrt{\tfrac{1}{2}\pi}n!\,y^{2m+1}{\mathrm{e}}^{-\frac{1}{2}y^{2}}L^{(2m+1)}_{% n}\left(y^{2}\right).$
 18.17.27 $\int_{0}^{\infty}{\mathrm{e}}^{-\frac{1}{2}x^{2}}\mathit{He}_{2n+1}\left(x% \right)\sin\left(xy\right)\,\mathrm{d}x=(-1)^{n}\sqrt{\tfrac{1}{2}\pi}y^{2n+1}% {\mathrm{e}}^{-\frac{1}{2}y^{2}},$
 18.17.28 $\int_{0}^{\infty}{\mathrm{e}}^{-x^{2}}\mathit{He}_{2n+1}\left(2x\right)\sin% \left(xy\right)\,\mathrm{d}x=(-1)^{n}\tfrac{1}{2}\sqrt{\pi}{\mathrm{e}}^{-% \frac{1}{4}y^{2}}\mathit{He}_{2n+1}\left(y\right).$

### Laguerre

 18.17.28_5 $\int_{0}^{\infty}{\mathrm{e}}^{-x}x^{\alpha}L^{(\alpha)}_{n}\left(x\right){% \mathrm{e}}^{\mathrm{i}xy}\,\mathrm{d}x=\frac{\Gamma\left(\alpha+n+1\right)(-% \mathrm{i}y)^{n}}{n!\left(1-\mathrm{i}y\right)^{\alpha+n+1}},$
 18.17.29 $\int_{0}^{\infty}x^{2m}{\mathrm{e}}^{-\frac{1}{2}x^{2}}L^{(2m)}_{n}\left(x^{2}% \right)\cos\left(xy\right)\,\mathrm{d}x=(-1)^{m}\sqrt{\tfrac{1}{2}\pi}\frac{1}% {n!}{\mathrm{e}}^{-\frac{1}{2}y^{2}}\mathit{He}_{n}\left(y\right)\mathit{He}_{% n+2m}\left(y\right),$
 18.17.30 $\int_{0}^{\infty}x^{2n}{\mathrm{e}}^{-\frac{1}{2}x^{2}}L^{(n-\frac{1}{2})}_{n}% \left(\tfrac{1}{2}x^{2}\right)\cos\left(xy\right)\,\mathrm{d}x=\sqrt{\tfrac{1}% {2}\pi}y^{2n}{\mathrm{e}}^{-\frac{1}{2}y^{2}}L^{(n-\frac{1}{2})}_{n}\left(% \tfrac{1}{2}y^{2}\right),$
 18.17.31 $\int_{0}^{\infty}{\mathrm{e}}^{-ax}x^{\nu-2n}L^{(\nu-2n)}_{2n-1}\left(ax\right% )\cos\left(xy\right)\,\mathrm{d}x=i\frac{(-1)^{n}\Gamma\left(\nu\right)}{2(2n-% 1)!}y^{2n-1}\left((a+iy)^{-\nu}-(a-iy)^{-\nu}\right),$ $\nu>2n-1$, $a>0$,
 18.17.32 $\int_{0}^{\infty}{\mathrm{e}}^{-ax}x^{\nu-1-2n}L^{(\nu-1-2n)}_{2n}\left(ax% \right)\cos\left(xy\right)\,\mathrm{d}x=\frac{(-1)^{n}\Gamma\left(\nu\right)}{% 2(2n)!}y^{2n}\left((a+iy)^{-\nu}+(a-iy)^{-\nu}\right),$ $\nu>2n$, $a>0$.

## §18.17(vi) Laplace Transforms

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

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

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

### Laguerre

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

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

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

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

### Hermite

 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.

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

## §18.17(viii) Other Integrals

### Ultraspherical

 18.17.41_5 $\int_{-1}^{1}C^{(\lambda)}_{\ell}\left(x\right)C^{(\lambda)}_{m}\left(x\right)% C^{(\lambda)}_{n}\left(x\right)(1-x^{2})^{\lambda-\tfrac{1}{2}}\,\mathrm{d}x=% \frac{{\left(\lambda\right)_{\frac{1}{2}\ell+\frac{1}{2}m-\frac{1}{2}n}}{\left% (\lambda\right)_{\frac{1}{2}m+\frac{1}{2}n-\frac{1}{2}\ell}}{\left(\lambda% \right)_{\frac{1}{2}n+\frac{1}{2}\ell-\frac{1}{2}m}}{\left(2\lambda\right)_{% \frac{1}{2}\ell+\frac{1}{2}m+\frac{1}{2}n}}\Gamma\left(\lambda+\frac{1}{2}% \right)\sqrt{\pi}}{(\tfrac{1}{2}\ell+\tfrac{1}{2}m-\tfrac{1}{2}n)!\,(\tfrac{1}% {2}m+\tfrac{1}{2}n-\tfrac{1}{2}\ell)!\,(\tfrac{1}{2}n+\tfrac{1}{2}\ell-\tfrac{% 1}{2}m)!\,\Gamma\left(\lambda+\frac{1}{2}\ell+\frac{1}{2}m+\frac{1}{2}n+1% \right)},$

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

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

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

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

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).