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This paper introduces inter-order formulas for partial and complete momentsof a Student $t$ distribution with $n$ degrees of freedom. We show how thepartial moment of order $n - j$ about any real value $m$ can be expressed interms of the partial moment of order $j - 1$ for $j$ in $\{1,\dots, n \}$.Closed form expressions for the complete moments are also established. We thenfocus on $L_p$-quantiles, which represent a class of generalized quantilesdefined through an asymmetric $p$-power loss function. Based on the resultsobtained, we also show that for a Student $t$ distribution the$L_{n-j+1}$-quantile and the $L_j$-quantile coincide at any confidence level$\tau$ in $(0, 1)$.

Inter-order relations between moments of a Student $t$ distribution, with an application to $L_p$-quantiles