indefinite Bessel function integral $\int_{0}^{\infty} t^{\nu+1} I_\nu(bt) \exp(-p^2 t^2 + q t) \, dt$

Question

Is there a solution to the indefinite Bessel function integral $$\int_{0}^{\infty} t^{\nu+1} I_\nu(bt) \exp(-p^2 t^2 + q t) \, dt$$ similar to (10.43.23) from DLMF?

Answer 1

This seems hard, I can convert it into a sum but it might not be very helpful. For $$ I(\nu,b,p,q)=\int_{0}^{\infty} t^{\nu+1} I_\nu(bt) \exp(-p^2 t^2 + q t) \, dt $$

If we take the Mellin transform of both sides with respect to $q\to s$ we have $$ \int_0^\infty q^{s-1} I(\nu,b,p,q) \;dq=(-1)^{s}\Gamma(s)\int_{0}^{\infty} t^{\nu+1-s} I_\nu(bt) \exp(-p^2 t^2) \, dt $$ where the right hand integral can now be related to the identity in the DLMF link by integration under the integral sign. Mathematica can evaluate the RHS giving $$ \int_0^\infty q^{s-1} I(\nu,b,p,q) \;dq= (-1)^s 2^{-\nu-1}b^{\nu}(p^2)^{-1-\nu+\frac{s}{2}}\frac{\Gamma(1+\nu-\frac{s}{2}) \Gamma(s)}{\Gamma(1+\nu)}\;_1F_1\left(1+\nu-\frac{s}{2},1+\nu,\frac{b^2}{4 p^2}\right) $$ using Ramanujan's master theorem we can the inverse Mellin transform of the RHS as a sum, giving $$ I(\nu,b,p,q) =\frac{2^{-\nu-1}b^{\nu}}{\Gamma(1+\nu)p^{2+2\nu}}\sum_{s=0}^\infty \frac{\Gamma(1+\nu+\frac{s}{2})}{\Gamma(s+1)}\;_1F_1\left(1+\nu+\frac{s}{2},1+\nu,\frac{b^2}{4 p^2}\right)\left(\frac{q}{p}\right)^s $$ which seems to hold numerically for values of $\nu,b,p,q$. This doesn't seem to reduce for the number of general parameters here.