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In this paper we are dealing with the Reflection Equation algebra ${\calM}(R)$, associated with a $GL_N$ type Hecke symmetry $R$. In this algebra wedefine the $q$-analogs of the partial derivatives $\partial_j^i$ in generators$m_i^j$ of ${\cal M}(R)$. The product $\hat L = MD$ of two matrices$M=\|m_i^j\|$ and $D=\|\partial_i^j\|$ turns out to be a generating matrix of amodified Reflection Equation algebra $\hat{\cal L}(R)$ which is similar to theuniversal enveloping algebra $U(gl_N)$ in many aspects. Central elements of themodified Reflection Equation algebra give rise to $q$-Casimir operators in arepresentation of $\hat{\cal L}(R)$ in the algebra ${\cal M}(R)$.  We perform a spectral analysis of the first $q$-Casimir operator andformulate a conjecture about the spectrum of the higher ones. At last, wedefine the normal ordering for the $q$-differential operators and inroduce the$q$-cut-and-join operators. In several explicit examples we express some of$q$-cut-and-join operators via the $q$-Casimir ones by analogy with theclassical case.

q-Casimir and q-cut-and-join operators related to Reflection Equation Algebras