论文标题
Euler对的贝克型身份$ r $
Beck-type identities for Euler pairs of order $r$
论文作者
论文摘要
分区身份通常是声明的声明，表明$ n $的分区的设置$ \ MATHCAL P_X $受条件$ x $的约束对于$ n $的$ \ Mathcal p_y $ $ n $的$ n $ $ y $。贝克类型的身份是$ | \ Mathcal P_x | = | = | \ Mathcal p_y | $断言所有分区中的差额$ b（n）$的同伴身份$ c | \ Mathcal P_ {y'} | $，其中$ c $与原始身份相关，分别是$ x'$，$ y'$，是分区的条件，分别是条件$ x $的略微放松，分别是$ x $ $ y $。第二个Beck-Type身份涉及$ \ Mathcal P_x $中所有分区中不同零件总数与所有分区中不同零件的总数之间的差异$ b'（n）$。我们将这些结果扩展到贝克型身份，并伴随着订单$ r $的Euler对给出的所有身份（对于任何$ r \ geq 2 $）。结果，我们获得了许多新贝克型身份的家庭。我们为结果提供了分析和徒的证明。
Partition identities are often statements asserting that the set $\mathcal P_X$ of partitions of $n$ subject to condition $X$ is equinumerous to the set $\mathcal P_Y$ of partitions of $n$ subject to condition $Y$. A Beck-type identity is a companion identity to $|\mathcal P_X|=|\mathcal P_Y|$ 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 $c|\mathcal P_{X'}|$ and also $c|\mathcal P_{Y'}|$, where $c$ is some constant related to the original identity, and $X'$, respectively $Y'$, is a condition on partitions that is a very slight relaxation of condition $X$, respectively $Y$. A second Beck-type identity involves the difference $b'(n)$ 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 $r\geq 2$). As a consequence, we obtain many families of new Beck-type identities. We give analytic and bijective proofs of our results.
