  Igarashi introduce the concept of $(\alpha, \beta)$-metric in Cartan space
$\ell^{n}$ analogously to one in Finsler space and obtained the basic important
geometric properties and also investigate the special class of the space with
$(\alpha, \beta)$-metric in $\ell^{n}$ in terms of $ 'invariants' $. In the
present paper we determine the $ 'invariants' $ in two different cases of
deformed infinite series metric which characterize the special classes of
Cartan spaces $\ell^{n}$.
