Lacunary statistical convergence of order $$\alpha, \beta$$ α , β for generalized vector-valued difference sequence spaces

In this paper, we defined space $$S_\theta ^{\alpha ,\beta } (\Delta _v^m,E,q)$$ of all vector-valued lacunary $$\Delta _v^m$$ -statistical convergent sequences of order $$(\alpha , \beta )$$ and space $$N_\theta ^{\alpha ,\beta } (\Delta _v^m,E,q,p)$$ of all vector-valued strongly $$\Delta _v^m$$ -lacunary summable sequences of order $$(\alpha ,\beta )$$ by taking sequence $$(E_k, q_k)$$ of normed spaces, where p is a positive real number and $$\alpha ,\beta$$ are real numbers with $$0<\alpha \leqslant \beta \leqslant 1$$ . Some inclusion relations between these spaces are obtained. We also studied space $$\omega _\theta ^{\alpha ,\beta } (\Delta _v^m,f,E,q,p)$$ of all vector-valued strongly $$\Delta _v^m$$ -lacunary summable sequences of order $$(\alpha , \beta )$$ with respect to modulus function f by taking a bounded sequence $$(p_k)$$ of strictly positive real numbers with $$\displaystyle \inf _k p_k>0$$ . The inclusion relations between spaces $$\omega _\theta ^{\alpha ,\beta } (\Delta _v^m,f,E,q,p)$$ and $$S_\theta ^{\alpha ,\beta } (\Delta _v^m,E,q)$$ are determined.