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 $\mathcal{J}(1/2)$?


For $s=0,1,2,\ldots$ we compute
$$
\mathcal{J}(s) = 1, 0, 1, 5, 36, 329, 3655, 47844, 721315, 12310199, 234615096,
\ldots
$$
[This was obtained with the gp code
F(p) = sum(k=0, poldegree(p), k!*polcoeff(p,k))
N = 10
vector(N+1,s,F(pollegendre(s-1,x-1)))

and can also be obtained from the expansion of $P_s(x-1)$ in powers of $x$, 
giving $(-1)^s \sum_{k=0}^s 2^{-k} k! {s \choose k} {-s-1 \choose k}$.]
This matches OEIS sequence 806
up to sign, indicating that we're dealing with $(-1)^s y_s(-1)$
where $y_s$ is the 
Bessel polynomial
of degree $s$.  There's a lot of information about these numbers
in that OEIS entry, including a recurrence equivalent to
$$
{\mathcal J}(s) = (2s-1) {\mathcal J}(s-1) + {\mathcal J}(s-2)
$$
and a simpler formula in terms of modified Bessel functions:
a(n) = BesselK[n+1/2,-1] / BesselK[5/2,-1]

contributed by 
Vaclav Kotesovec
on Aug 07 2013.
These formulas, together with known recurrences for modified Bessel functions,
should soon also yield Vladimir Reshetnikov's conjectural formula 
in terms of $I_{s+1/2}(-1)$ and $K_{s+1/2}(1)$.
