Let “\(a\)” denote the infinite series \(\sum^\infty_{n=0} a_n\). Write \[ \phi_n(a)= {p_n\over P_n P_{n-1}} \sum^n_{k=1} P_{k- 1} a_k\quad\text{for }n\geq 1. \]
\[ |\overline N_p|(M,r)= \Biggl\{a= (a_n): \sum_n \Biggl[M\Biggl({|\phi_n(a)|\over \rho}\Biggr)\Biggr]^{Y_{ra}}< \infty\text{ for some }\rho> 0\Biggr\}, \] where \(r= (r_n)\) is a bounded sequence of strictly positive real numbers.
The authors prove that \(|\overline N_p|(M,r)\) is a topological linear space. Some inclusion relations between \(|\overline N_p|(M, r)\) spaces are also studied. Here, \(M\) is an Orlicz function.