Beck-type identities for Euler pairs of order $r$

Partition identities are often statements asserting that the set $\mathcal P_X$ofpartitionsof$n$subjecttocondition$X$isequinumeroustotheset$\mathcal P_Y$ofpartitionsof$n$subjecttocondition$Y$. A Beck-type identity is a companion identity to $\mathrm{\mid }{\mathcal{P}}_X\mathrm{\mid }=\mathrm{\mid }{\mathcal{P}}_Y\mathrm{\mid }$ asserting that the difference $b(n)$ between the number of parts in all partitions in ${\mathcal{P}}_X$ and the number of parts in all partitions in ${\mathcal{P}}_Y$ equals a $c\mathrm{\mid }{\mathcal{P}}_{X^{\mathrm{\prime }}}\mathrm{\mid }$ and also $c\mathrm{\mid }{\mathcal{P}}_{Y^{\mathrm{\prime }}}\mathrm{\mid }$, where $c$ is some constant related to the original identity, and $X^{\mathrm{\prime }}$, respectively $Y^{\mathrm{\prime }}$, is a condition on partitions that is a very slight relaxation of condition $X$, respectively $Y$. A second Beck-type identity involves the difference $b^{\mathrm{\prime }}(n)$ between the total number of different parts in all partitions in ${\mathcal{P}}_X$ and the total number of different parts in all partitions in ${\mathcal{P}}_Y$. We extend these results to Beck-type identities accompanying all identities given by Euler pairs of order $r$ (for any $r\ge 2$). As a consequence, we obtain many families of new Beck-type identities. We give analytic and bijective proofs of our results.