I am reading an equation required to solve a particular integral involving Bessel function it says: $$\int_0^\infty e^{-at}J_v(bt)J_v(ct)\text{d}t=\frac{1}{\pi\sqrt{bc}}Q_{v-\frac{1}{2}}\left (\frac{a^2+b^2+c^2}{2bc}\right )$$ where $J$ are the Bessel function of first kind. However, the text does not mention what is $Q_{v-\frac{1}{2}}$ is this also a bessel function. It doesn't look like to be the error function the only function that I am familiar with to have $Q$.
The book where the equation is given is: A Treatise on the Theory of Bessel Functions by G. N. Watson, pp. 389.
$Q_\nu$ is a type of Legendre function. If we define first the Legendre function (or "Ferrer's function of the first kind)
$$P^\mu_{\nu}(x)=\left(\frac{1+x}{1-x}\right)^{\mu/2}~{}_2F_1\left(\begin{matrix}1+\nu ,-\nu \\ 1-\mu\end{matrix}~\bigg| ~ \frac{1-x}{2}\right)$$
Where $${}_p F_q\left(\begin{matrix}a_1,\dots,a_p \\ b_1,\dots ,b_q\end{matrix}~\bigg|~z\right)=\sum_{k=0}^\infty \prod_{i=1}^p\frac{\Gamma(a_i+k)}{\Gamma(a_i)}\cdot \prod_{j=1}^q\frac{\Gamma(b_j)}{\Gamma(b_j+k)}\cdot \frac{z^k}{k!}$$ Is a generalized hypergeometric function, then we can define the complementary Legendre function (or "Ferrer's function of the second kind") $$Q^\mu_\nu(x)=\frac{\pi}{2}\cot(\mu\pi) P^\mu_\nu(x)-\frac{\pi}{2}\operatorname{csc}(\mu\pi)\frac{\Gamma(1+\nu+\mu)}{\Gamma(1+\nu-\mu)}\left(\frac{1+x}{1-x}\right)^{\mu}P^{-\mu}_{\nu}(x)$$
We then define $$Q_{\nu}(x):=Q^0_{\nu}(x)$$ Unfortunately the above definition for $Q^\mu_{\nu}$ breaks down when $\mu=0$ (because $\cot(0)$ is undefined) but Wikipedia offers another definition
$$Q^\mu_{\nu}(z)=\frac{\sqrt \pi~\Gamma(\nu+\mu+1)}{2^{\nu+1}\Gamma(\nu+3/2)}\frac{\mathrm e^{\mathrm i\mu\pi}(z^2-1)^{\mu/2}}{z^{\nu+\mu+1}}~{}_2F_1\left(\begin{matrix}\frac{\nu+\mu+1}{2},\frac{\nu+\mu+2}{2} \\ \nu+\frac{3}{2}\end{matrix}~\bigg|~\frac{1}{z^2}\right)$$
Though keep in mind that conventions differ across sources, i.e, Wiki's definition for $P^\mu_\nu$ includes a prefactor of $\frac{1}{\Gamma(1-\mu)}$. With this in mind it appears that $$Q_\nu(z):=Q^0_\nu(z)=\frac{\sqrt \pi~\Gamma(\nu+1)}{2^{\nu+1}\Gamma(\nu+3/2)}\frac{1}{z^{\nu+1}}~{}_2F_1\left(\begin{matrix}\frac{\nu+1}{2},\frac{\nu+2}{2} \\ \nu+\frac{3}{2}\end{matrix}~\bigg|~\frac{1}{z^2}\right)$$
In the case that $\nu=m$ is an integer Wiki provides the direct formula
$$Q_n(x)=\frac{n!}{1\cdot 3\cdots (2n+1)}\left(x^{-n-1}+\frac{(n+1)(n+2)}{2(2n+3)}x^{-n-3}+\frac{(n+1)(n+2)(n+3)(n+4)}{2\cdot 4\cdot (2n+3)(2n+5)}x^{-n-5}+\cdots\right)$$
Or Bonnet's recursion formula
$$Q_n(x)=\begin{cases}\frac{1}{2}\log\left(\frac{1+x}{1-x}\right) & n=0 \\ P_1(x)Q_0(x)-1 & n=1 \\ \frac{2n-1}{n}x~Q_{n-1}(x)-\frac{n-1}{n}Q_{n-2}(x) & n\geq 2\end{cases}$$
Unfortunately the documentation on these functions is overall not very good, with even Wolfram's usually very reliable reference sites being very poor, see
https://functions.wolfram.com/HypergeometricFunctions/LegendreQ2AbbreviatedGeneral/
https://mathworld.wolfram.com/LegendreFunctionoftheSecondKind.html
Hopefully this answer went some way to answer your question.