On this Wikipedia page on 'conjugate priors' the following notation is used for the posterior predictive of a normal random variable with known mean $\mu$ but unknown variance $\sigma^2$:

$$t_{2\alpha'}(\tilde{x}|\mu,\sigma^2 = \beta'/\alpha')$$

Here's my interpretation of the notation:

$\tilde{x}$ is a new i.i.d. variable the distribution of which is described by the t-distribution.

$\mu$ is the known mean and is the first parameter of the distribution

$\sigma^2$ is second parameter of the distribution and is the Bayesian variance estimate given by the ratio of the posterior $\beta$ and $\alpha$

$\alpha'$ and $\beta'$ are the posterior hyper-parameters of $\sigma^2$

$t_{2\alpha'}$ is a t-distribution with $\nu = n - 1$ degrees of freedom that has been shifted by $\mu$ and multiplied by $\sigma$.

My guess is that the pdf is given by:

$$f(x) = \left(\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{\!-\frac{\nu+1}{2}} + \mu\right)\sigma$$

Can anyone more knowledgeable clear this up for me?

$\tilde{x}$ is a new i.i.d. variable the distribution of which is described by the t-distribution.

Formally, $\tilde{x}$ is the argument of the posterior predictive pdf and hence a dummy. The posterior predictive pdf is the density of the distribution of a new and independent variate with same distribution as the observation.

μ is the known mean and is the first parameter of the distribution

μ is the known mean of both the observation and the predictive variate. It is the location parameter of the $t$ distribution

$t_{2α′}$ is a t-distribution with ν=n−1 degrees of freedom that has been shifted by μ and multiplied by σ.

$t_{2α′}$ is a $t$ distribution with $2α′$ degrees of freedom, not $n-1$, location parameter $\mu$ and scale parameter $\sigma$.

$σ^2$ is second parameter of the distribution and is the Bayesian variance estimate given by the ratio of the posterior β and α

$σ$ is the scale parameter of the $t$ distribution, provided by the ratio of the hyperparameters of the posterior distribution.

the pdf is given by $$f(x) = \left(\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{\!-\frac{\nu+1}{2}} + \mu\right)\sigma$$

No, the density of a location-scale $t_\nu(\cdot;\mu,\sigma)$ distribution is $$\frac{\Gamma(\frac{\nu+1}{2})}{\sigma\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})}\left(1+\frac{(x-\mu)^2}{\nu\sigma}\right)^{\!-\frac{\nu+1}{2}}$$