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On the n th Quantum Derivative
Author(s) -
Ash J. Marshall,
Catoiu Stefan,
Ríos-Collantes-De-Terán Ricardo
Publication year - 2002
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610702003198
Subject(s) - peano axioms , derivative (finance) , mathematics , function (biology) , combinatorics , generalizations of the derivative , polynomial , second derivative , discrete mathematics , quantum , binomial coefficient , mathematical analysis , physics , quantum mechanics , evolutionary biology , financial economics , economics , biology
The n th quantum derivative D n f ( x ) of the real‐valued function f is defined for each real non‐zero x aslim q → 1∑ k = 0 n( − 1 ) k[ ] q q ( k − 1 ) k / 2 f ( q n − k x )q ( n − 1 ) n / 2( q − 1 ) n x n,where[nk] qis the q ‐binomial coefficient. If the n th Peano derivative exists at x , which is to say that if f can be approximated by an n th degree polynomial at the point x , then it is not hard to see that D n f ( x ) must also exist at that point. Consideration of the function |1− x | at x = 1 shows that the second quantum derivative is more general than the second Peano derivative. However, it can be shown that the existence of the n th quantum derivative at each point of a set necessarily implies the existence of the n th Peano derivative at almost every point of that set.