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On the Study of Recurrence and Difference Means
September 12, 2019
1 Abstract
This article is a very early draft on extending means using the concept of means
defined in my previous work, which relates the mean to an operation on some
finite or infinite sequence. Instead of further extending the concept of general-
ization of means, this paper defines a class of means where the operation is a
recurrence relation or difference equation.
2 Introduction
When considering a general concept of means, the basic forms of means that one
might consider are the arithmetic and geometric means. In Goldman 2019(?), a
more general concept of a mean was developed. Specifically, given sets A and B,
and S(A), the set of all sequences of elements from A, along with some operation
O:S(A)→B, then for some X∈ S(A),¯
Xis a mean of X, iff replacing every
element of X with ¯
Xyields the same result as O(X).Two examples of such
a mean were provided. In this paper, I will attempt to consider another class
of generalizations, relying on the generalization of the operator, rather than
the definition of the mean itself. Specifically, summation, which is associated
with the arithmetic mean, and product, which is associated with the geometric
mean, can be defined in terms of recurrence relations. This paper considers
means associated with other recurrence relations.
3 Recurrence Relations
The summation function can be defined recursively as
b
P
i=k
g(i)=0,for b < k.
b
P
i=k
g(i) = g(b) +
b−1
P
i=k
g(i),for b≥k. [1]
This definition can be restated by letting an=
n
P
i=k
g(i),resulting in
1
an= 0, n < k.
an=b(n) + an−1
Therefore the summation function is really just a recurrence relation. The
product can be defined in a similar way. Then one might wish to consider other
recurrence relations and their associated means.
3.1 First Example
As a first example, consider a finite sequence of real numbers x1, x2, ..., xn, and
the recurrence relation an=xn+xn−1+an−1,for n > 1,where a0= 0.It can be
shown that an= 2x1+ 2x2+... + 2xn−1+xn.Substituting our desired constant
value m, as the potential mean, we have an=
n−1
P
i=1
2m+m= (2n−2)m+mor
an= (2n−1)m.
So, our mean value is m=an
2n−1. Note that for the arithmetic mean, the
mean value m is equal to bn
n,where bn=
n
P
i=1
xi,whereas an=
n−1
P
i=1
2xi+xn.
4 Difference Equations and Dynamical Systems
Recurrence relations and difference equations are closely related to one another.
One can consider a difference equation with state atthat is dependent on some
time indexed parameter p(t),which is just a sequence. In this way, we can see
that for a dynamical system which is defined by a difference equation involving a
time indexed parameter, a mean-value of that parameter, at time τ, is a constant
value for that parameter which will lead the system to the same result at time
τ.
5 Course Equivalence
An alternative way to think of the abstract mean developed in Goldman 2019 is
to consider the abstract mean in the following way. Consider sets A and B, along
with the set of all of its sequences, S(A).Given an operation O:S(A)→B,
partition S(A) s.t. ∀x, y in the partition, O(x) = O(y) and |x|=|y|.Then
m∈Ais a mean for all sequences in the partition iff the constant sequence (m)
is in that partition.
But what of the other sequences? All of these sequences yield the same re-
sult for the operation. We can view O as some time indexed operation, and
each sequence of length n as a course over which the operation occurs. Doing so
2
gives us a concept of course equivalence, which ties in nicely with the dynam-
ical system approach. Essentially, our partitions become all the time indexed
parameter courses that lead to the same end state at time τ.
References
[1] Wikipedia contributors. Summation — Wikipedia, The Free Encyclopedia.
https: //en.wikipedia.org /w /index.php? title= Summation&oldid=
912511235. [Online; accessed 11-September-2019]. 2019.
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