Parameters (notation from wikipedia)
PDF: \begin{align} f(x, \lambda, \alpha, \beta, \mu, \delta) &= \frac{\left(\frac{\sqrt{\alpha^2 - \beta^2}} {\delta} \right)^\lambda e^{\beta(x - \mu)}} {\sqrt{2\pi} K_\lambda(\delta \sqrt{\alpha^2 - \beta^2} )} \frac{K_{\lambda - \frac{1}{2}}\left(\alpha \sqrt{\delta^2 + (x - \mu)^2}\right)} {\left(\frac{\sqrt{\delta^2 + (x - \mu)^2}} {\alpha} \right)^{\frac{1}{2} - \lambda}} \\ &= \frac{\left(\frac{\sqrt{\alpha^2 - \beta^2}} {\delta} \right)^\lambda e^{\delta\beta\left(\frac{x - \mu}{\delta}\right)}} {\sqrt{2\pi} K_\lambda(\delta \sqrt{\alpha^2 - \beta^2} )} \frac{K_{\lambda - \frac{1}{2}}\left(\delta\alpha \sqrt{1 + \left(\frac{x - \mu}{\delta}\right)^2}\right)} {\left(\frac{\delta}{\alpha}\sqrt{1 + \left(\frac{x - \mu}{\delta}\right)^2} \right)^{\frac{1}{2} - \lambda}} \\[5pt] &= \frac{1}{\delta} \frac{\left(\frac{\sqrt{(\delta\alpha)^2 - (\delta\beta)^2}} {\delta\alpha}\right)^\lambda (\delta\alpha)^\frac{1}{2} e^{\delta\beta\left(\frac{x - \mu}{\delta}\right)}} {\sqrt{2\pi} K_\lambda\left(\sqrt{(\delta\alpha)^2 - (\delta\beta)^2}\right)} \frac{K_{\lambda - \frac{1}{2}}\left(\delta\alpha \sqrt{1 + \left(\frac{x - \mu}{\delta}\right)^2}\right)} {\left(\sqrt{1 + \left(\frac{x - \mu}{\delta}\right)^2} \right)^{\frac{1}{2} - \lambda}} \end{align} Note \(\alpha\) and \(\beta\) always occur as a product \(\delta\alpha\) and \(\delta\beta\), respectively. Let \begin{align} \hat{\alpha} &= \delta \alpha \\ \hat{\beta} & = \delta \beta \end{align} Then \[ f(x, \lambda, \hat{\alpha}, \hat{\beta}, \mu, \delta) = \frac{1}{\delta} \frac{\left(\frac{\sqrt{\hat{\alpha}^2 - \hat{\beta}^2}} {\hat{\alpha}}\right)^\lambda \hat{\alpha}^\frac{1}{2} e^{\hat{\beta}\left(\frac{x - \mu}{\delta}\right)}} {\sqrt{2\pi} K_\lambda\left(\sqrt{\hat{\alpha}^2 - \hat{\beta}^2}\right)} \frac{K_{\lambda - \frac{1}{2}}\left(\hat{\alpha} \sqrt{1 + \left(\frac{x - \mu}{\delta}\right)^2}\right)} {\left(\sqrt{1 + \left(\frac{x - \mu}{\delta}\right)^2} \right)^{\frac{1}{2} - \lambda}} \] This is now a true location-scale family, with location \(\mu\) and scale \(\delta\), but the definition of two of the shape parameters is not the same as in the wikipedia article.