Is it a necessary condition to demand that the derivative makes the degree of the polynomial be decressed by one when we define a deformed derivative?
When we consider the arithmetric lattice { x, x +d, x+2d, ....}, we can consider the deformed derivative
D f(x) = [ f(x + d ) - f(x)]/d,
When we consider the geometric lattice { x, q x, q^2 x, ....}, we can consider the deformed derivative
D f(x) = [ f(qx ) - f(x)]/( q-1)x
These two derivative is related to a linear lattice.
What happen if we consider the nonlinear lattice such as
{ x, p x^2 , ( px^2)^2 , .....}
In this case we can define the derivative as
Df(x) = [ f( px^2) - f(x) ]/ [ px^2 - x ]
which makes the degree of the polynomial increase after differentiation.
In this case is it a kind of deformed derivatve?
What is a deformed derivative? I guess you are referring to a fractional derivative implying derivative of non integer order. If that is the case then then derivation does not necessarily reduce the order of the function by one degree. It can be arbitrary! And then for dimensional consistency of problem, one has to include necessary constant with the fractional derivative. See equation 10-22 in attached link.
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