While reading Theory of Probability by Jaynes (p. 281), I came across this difference equation: $$ M(n,G) = \sum_{j = 1}^{m} M(n-1, G - g_j) $$ where $g_1, \dots, g_m \in \mathbb{R}$.
The first thing that struck me is that the author calls this formula a linear difference equation, even though the degree of this equation is not an integer, since the $g_j$ can be any real numbers, including negative ones. The second thing that is novel for me is that there is more than one variable. However, the author simply uses the ansatz:
$$ M(n,G) = \exp(\alpha n + \lambda G) $$
which yields: $$ \exp(\alpha) = \sum_{j = 1}^{m}\exp(-\lambda g_j) $$ which seems to work. So I guessed that both non-integer degree and multiple variables are no big deal, you just cleverly use the exponential of the linear combination of all variables and a solution is born.
But then something unexpected happens. Let's denote any ansatz that satisfied the above criterion $f(n,G)$. Then the author concludes that (I paraphrase and simplify the equation)
An arbitrary superposition of such elementary solutions $$ H(n,G) = \int f(n,G)h(\lambda) d\lambda $$ is, from linearity a formal solution of the given difference equation.
This is not at all clear to me. Why would such a thing be the formal solution? Where did the integral come from? What is $h$? Also, what is meant by a formal solution? Does he mean that any solution can be written in such a form?
I find it surprisingly hard to find something about linear difference equations of non-integer degree, so if you can point me to any resources, I would be very grateful.