Indefinite sum of general analytic function (inverse forward difference)
mike3
I have an interesting problem I've toyed with for a while.
Consider the "indefinite sum" of a function, i.e. finding F such that F
(x+1) - F(x) = f(x) for a given f(x). It is to the forward difference
operator what the integral is to the derivative operator. The goal is
to solve this equation for functions of a real number.
Now it seems there are many possibilites for F(x). If we have one,
another one is F*(x) + theta(x) where theta(x) is any 1-periodic
function. In general, for a finite difference quotient (F(x+h) - F(x))/
h or a difference F(x+h) - F(x) with a different step h, adding a h-
periodic function will generate another solution, and as h approaches
zero, of course the period approaches zero which means the function
collapses to a constant in the limiting case, i.e. the derivative, and
that's the "+ C" you add at the end of an indefinite integral.
But for the sum, we have this non-uniqueness problem. So the question
is then whether there is a good "principal" solution to the equation.
One approach I've seen is for functions f(x) that decay to zero as x
approaches infinity and decay quickly enough that sum_{n=0...inf} f(n
+x) converges for all x.
Then one can define sum_{n=1...x} f(x) as sum_{n=1...inf} f(n) - f(n +
x). If we set f(x) = 1/(x^n) for n > 1, we get a convergent sum, which
can be summed in terms of the Riemann zeta function and the Hurwitz
zeta function, and then these can be analytically continued through
the complex plane to get the values at positive n, which for integer n
yields Faulhaber's formula. Then we can sum powers to real and even
complex "numbers of terms".
The Faulhaber's formula then suggests the possibility of defining a
sum for analytic functions via the Taylor series... however
Faulhaber's formula may not converge for all such functions. When one
does this, one obtains coefficients for the Taylor series of the
indefinite sum as infinite sums in terms of the power series
coefficients, binomial numbers, and Bernoulli numbers. However, they
seem to diverge for functions whose series have finite convergence
radius, and even worse, they also seem to diverge for certain
functions that are entire, too! We can even construct some such that
the coefficients of the Faulhaber-sum form divergent sums as diverse
as 1 + 2 + 3 + 4 + 5 + ... and the divergent Mercator series for log
(-1) = i pi. I suppose one could use some sort of divergent summation
technique, but they don't always assign the same sum to the same
series, leaving open the question of which sum we should consider
"canonical". Is there an answer to this problem? Is there a
"canonical" way to define the indefinite sum of a general analytic
function? I know difference equations are a known and studied topic,
so there should be something about this too, know? Is there some
divergent summation method that sums a general divergent sum "like it
was a coefficient of the Faulhaber-on-Taylor-series formula" or
something?