Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On The Indefinite Sum In Fractional Calculus   We present a theorem on taking the repeated indefinite summation of a
holomorphic function $\phi(z)$ in a vertical strip of $\mathbb{C}$ satisfying
exponential bounds as the imaginary part grows. We arrive at this result using
transforms from fractional calculus. This affords us the ability to
indefinitely sum more complicated functions than previously possible; such as
holomorphic functions of order $n \in \mathbb{N}$ that have decay at plus or
minus imaginary infinity. We then further investigate the indefinite summation
operator by restricting ourselves to a space of functions of exponential type.
We arrive at a second representation for the indefinite summation operator,
equivalent to the first presented, and show we have defined a unique operator
on this space. We develop a convolution using the indefinite sum that is
commutative, associative, and distributive over addition. We arrive at a
formula for the complex iterates of the indefinite sum (the differsum), using
this convolution, that resembles the Riemann-Liouville differintegral. We close
with a generalization of the Gamma function.

27
The categories of this paper are: comp-gas
23
The categories of this paper are: solv-int
16
The categories of this paper are: stat
10
The categories of this paper are: q-bio
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The categories of this paper are: acc-phys
21
The categories of this paper are: adap-org
37
The categories of this paper are: supr-con
12
The categories of this paper are: nucl-th
31
The categories of this paper are: ao-sci
14
The categories of this paper are: math-ph
28
The categories of this paper are: alg-geom
19
The categories of this paper are: eess
4
The categories of this paper are: cond-mat
3
The categories of this paper are: physics
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The categories of this paper are: dg-ga
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The categories of this paper are: econ