An integral $\int_0^\infty P_s(x-1)\,e^{-x}\,dx$ involving Legendre functions Let $P_s(x)$ denote the Legendre functions of the $1^{st}$ kind, i.e. the Legendre polynomial generalized to an arbitrary (not necessarily integer) order $s$. It can be expressed using the hypergeometric function:
$$P_s(x)={_2F_1}\left(-s,s+1;\ 1;\ \frac{1-x}2\right).\tag1$$
Let's consider the definite integral
$$\mathcal{J}(s)=\int_0^\infty P_s(x-1)\,e^{-x}\,dx.\tag2$$
It evaluates to integer values when $s\in\mathbb{Z}^+$. Using a computer-assisted search for the general term formula, I discovered the following conjecture, that I haven't yet been able to prove:
$$\mathcal{J}(s)\stackrel?=\left(K_{3/2}(1)\cdot I_{s+1/2}(-1)-I_{3/2}(-1)\cdot K_{s+1/2}(1)\right)\sqrt{-1},\tag3$$
where $I_\nu(z)$ and $K_\nu(z)$ are the modified Bessel functions of the $1^{st}$ and $2^{nd}$ kind.
Unfortunately, it only seems to hold for $s\in\mathbb{Z}^+$, and does not generalize to non-integer values of $s$.


*

*Can we prove the conjecture $(3)$?


*Can we find a more general formula that holds not only for integer values of $s$?


*Can we find (or at least conjecture) a closed form for $\mathcal{J}(1/2)$?


Not a proper answer, but it is the closest one to your integral that I was able to find. Using formula 7.141.1 in Gradshteyn-Ryzhik, after some simplifications, one can get:
$$\int_0^\infty P_s(x+1)\,e^{-x}\,dx=\frac{e\,\sqrt2}{\sqrt\pi} K_{s+\frac{1}{2}}(1),$$
where $K_\nu(x)$ is the modified Bessel function of the 2nd kind. Note that this formula contains $(x+1)$ rather than $(x-1)$ that appears in your question. I'm still trying to find a proper answer...
