Beck-Type Identities for Euler Pairs of Order r

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