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On the nth quantum derivative J. Marshall Ash, Stefan Catoiu, and Ricardo Rios-Collantes-de-Teran
 

Summary: On the nth quantum derivative
J. Marshall Ash, Stefan Catoiu, and Ricardo RŽios-Collantes-de-TerŽan
Abstract. The nth quantum derivative Dnf (x) of the real-valued function
f is defined for each real non-zero x as
lim
q1
n
k=0
(-1)k n
k q
q(k-1)k/2f qn-kx
q(n-1)n/2 (q - 1)n
xn
,
where n
k q
is the q-binomial coefficient. If the nth Peano derivative exists at x,
which is to say that if f can be approximated by an nth degree polynomial at
the point x, then it is not hard to see that Dnf (x) must also exist at that point.
Consideration of the function |1 - x| at x = 1 shows that the second quantum

  

Source: Ash, J. Marshall - Department of Mathematical Sciences, DePaul University

 

Collections: Mathematics