Integral involving Bessel function and Laguerre polynomial for a Hankel transform

Question

I'm attempting to solve the Hankel transform \begin{align} \int_0^\infty x^\alpha e^{-x^2/2} L_n^\nu(x^2) J_\nu(x p) \sqrt{x p} \, dx \end{align} or the unmodified version (redefining $\alpha$) \begin{align} \int_0^\infty x^\alpha e^{-x^2/2} L_n^\nu(x^2) J_\nu(x p) \, dx \end{align} where $\alpha, \nu, p$ are real and positive and $J_\nu$ is a Bessel function of the first kind and $L_n^\nu$ is an associated Laguerre polynomial. I've seen similar integrals in a couple of books and in this forum, however I can't seem to find this version. There is a similar one in Reference for proof of an integral from the "Tables of Integral Transforms" involving a Gaussian and a Laguerre polynomial, but the exponent of $x$ is specifically $\nu+1/2$ there. It would work for me if it were $\nu$. Same in most of the Hankel transformation in Harry Bateman, Tables of Integral Transforms, Volume 2, pp. 42–43.

In Integrals and series volume by A.P. Prudnikov, Yu. A. Brychkov and O.I. Marichev (like in page 2-461 or 2-474) there are similar integrals (with a variable change) but the exponential argument can't be $- \frac{1}{2}x^2$ if Laguerre argument is $x^2$. So... do you guys know how to attempt this or rather some other references I can investigate? I really appretiate it. If not, maybe someone knows this integral? \begin{align} \int_0^{\infty} x^{\nu} e^{-x^2/2} L_n^\nu(x^2) J_\nu(x p) dx \end{align} Thanks! Even if it is just some other references I would be grateful.

Answer 1

Mathematica 14 finds it for some concrete values of the parameters. For example,

Table[Integrate[x^\[Nu]*Exp[-x^2/2]*LaguerreL[10, \[Nu], x^2]*
BesselJ[\[Nu], p*x], {x, 0, Infinity}, Assumptions -> p > 0], {\[Nu], 1, 4}]

{(1/7257600) E^(-(p^2/4)) p Sqrt[\[Pi]/ 2] ((9823275 - 49116375 p^2 + 83024775 p^4 - 62195400 p^6 + 25115625 p^8 - 5859495 p^10 + 820320 p^12 - 69246 p^14 + 3427 p^16 - 91 p^18 + p^20) BesselI[0, p^2/ 4] - (9823275 - 7058925 p^2 + 31802175 p^4 - 33805800 p^6 + 17017425 p^8 - 4562685 p^10 + 700332 p^12 - 62914 p^14 + 3251 p^16 - 89 p^18 + p^20) BesselI[1, p^2/4]), (1/7257600) E^(-(p^2/4)) Sqrt[\[Pi]/ 2] (p^2 (75311775 - 196465500 p^2 + 252566775 p^4 - 152731800 p^6 + 51533325 p^8 - 10300680 p^10 + 1258320 p^12 - 94032 p^14 + 4167 p^16 - 100 p^18 + p^20) BesselI[0, p^2/ 4] - (150623550 + 42567525 p^2 - 42353550 p^4 + 110836575 p^6 - 90193950 p^8 + 36671355 p^10 - 8259960 p^12 + 1092984 p^14 - 86274 p^16 + 3973 p^18 - 98 p^20 + p^22) BesselI[1, p^2/ 4]), (1/(7257600 p)) E^(-(p^2/4)) Sqrt[\[Pi]/ 2] (p^2 (383107725 + 340540200 p^2 - 638512875 p^4 + 664924275 p^6 - 337473225 p^8 + 97821450 p^10 - 17100720 p^12 + 1852500 p^14 - 124155 p^16 + 4980 p^18 - 109 p^20 + p^22) BesselI[0, p^2/ 4] - (3064861800 + 808782975 p^2 + 170270100 p^4 - 171816525 p^6 + 322508475 p^8 - 212060475 p^10 + 72296190 p^12 - 14034000 p^14 + 1631580 p^16 - 114825 p^18 + 4768 p^20 - 107 p^22 + p^24) BesselI[1, p^2/4]), (1/( 7257600 p^2)) E^(-(p^2/4)) Sqrt[\[Pi]/ 2] (p^2 (9012153150 + 2359457100 p^2 + 1204052850 p^4 - 1787836050 p^6 + 1568805525 p^8 - 686543760 p^10 + 174458970 p^12 - 27096600 p^14 + 2636370 p^16 - 160140 p^18 + 5866 p^20 - 118 p^22 + p^24) BesselI[0, p^2/ 4] - (72097225200 + 18875656800 p^2 + 3624320700 p^4 + 583783200 p^6 - 560353500 p^8 + 821832795 p^10 - 452855340 p^12 + 132885390 p^14 - 22657920 p^16 + 2348580 p^18 - 149092 p^20 + 5636 p^22 - 116 p^24 + p^26) BesselI[1, p^2/4])}

Also

n=7;\[Nu]=3/2;HankelTransform[x^(\[Nu]-1)Exp[-x^2/2]*LaguerreL[n,\[Nu],x^2],x,p]
//FunctionExpand

$\frac{1}{512} (-663) \Gamma \left(\frac{9}{4}\right) \left(-11 \sqrt[4]{2} L_{-\frac{5}{4}}\left(-\frac{p^2}{2}\right)+77 \sqrt[4]{2} L_{-\frac{9}{4}}\left(-\frac{p^2}{2}\right)-297 \sqrt[4]{2} L_{-\frac{13}{4}}\left(-\frac{p^2}{2}\right)+715 \sqrt[4]{2} L_{-\frac{17}{4}}\left(-\frac{p^2}{2}\right)-1105 \sqrt[4]{2} L_{-\frac{21}{4}}\left(-\frac{p^2}{2}\right)+1071 \sqrt[4]{2} L_{-\frac{25}{4}}\left(-\frac{p^2}{2}\right)-595 \sqrt[4]{2} L_{-\frac{29}{4}}\left(-\frac{p^2}{2}\right)+145 \sqrt[4]{2} L_{-\frac{33}{4}}\left(-\frac{p^2}{2}\right)\right) $

The above results are huge and the general formula is not clear.