# Exponential Series with a sequence [closed]

For a convergent sequence $$(a_n)_n \rightarrow a$$ consider the exponential series $$\begin{equation*} \exp_{(a_n)_n}(-x) := \sum_{n=0}^{\infty} \frac{(-x)^n a_n}{n!}. \end{equation*}$$

Can there be anything said about the resulting expression, more than convergence? I presume that, since e.g. changing only $$a_0$$ already changes the result, this is not expressable in terms of the limit $$a$$.

Now, if $$a_n$$ is of the form $$u^n$$ for some constant $$y$$, this ofcourse results in $$e^{-xy}$$. Can there be something similar be said if $$a_n$$ more generally is a "homogeneous polynomial" of degree $$n$$ in some variables? By that I mean something like $$\begin{equation*} a_n=b_{0,n} u^n + b_{1,n} u^{n-1} v + \dots + b_{n,n} v^n. \end{equation*}$$ This is trying to generalize $$\begin{equation*} a_n= {n\choose 0} u^n + {n\choose 1} u^{n-1} v + \dots + {n\choose n} v^n = (u+v)^n. \end{equation*}$$

The Appell Sheffer polynomial formalism can be used to deal with these types of polynomials for $$a_0 = b_{0,0} = 1$$.

In umbral notation for which, e.g., $$(c.)^n = c_n$$,

$$e^{c.t} \; e^{xt} = e^{(c.+ x)t} = e^{t \; p.(x)},$$

with the Appell Sheffer polynomials

$$p_n(x) = (c.+x)^n = \sum_{k=0}^n \; \binom{n}{k} \; c_{n-k} \; x^{k} \; .$$

Let $$x = v$$ and $$c_n = d_n u^n$$ to get polynomials of your type. Convergence is guaranteed if $$e^{c.t}$$ converges, but even if it doesn't converge, the series can be formally multiplicatively inverted degree by degree.

The coefficients for Appell sequences (e.g.f.s) can be regarded as formal classical moments (many are actual moments of distributions, e.g., the Bernoulli numbers) and then related to formal classical cumulants, and, if the sequence is re-normalized and viewed as coefficients of an o.g.f., the renormalized polynomials can be regraded as moments correlated with the free cumulants of free probability theory.

Appell, Turan, Jensen, and Polya, among others, investigated the zeros of Appell sequences. See, e.g., "Hyperbolicity of Appell Polynomials of Functions in the δ-Laguerre-Pólya Class" by Jonas Iskander and Vanshika.

One of the more recent refs on the topic is "General Bivariate Appell Polynomials via Matrix Calculus and Related Interpolation Hints" by Francesco Aldo Costabile, Maria Italia Gualtieri and Anna Napoli. Another, dealing with a general Sheffer sequences modified to be bivariate, is "Operatorial methods and two variable Laguerre polynomials" by Dattoli and Torre. The Bernoulli and Laguerre polynomials are very useful sequences in pure mathematics and mathematical physics. All Sheffer polynomials $$S_n(x)$$ have an e.g.f. of the form

$$A(t) \; e^{x \; B(t)} = e^{t \; S.(x)}$$

with $$A(t)$$ and $$B(t)$$ analytic about the origin, $$A(0) = 1$$, $$B(0) = 0$$, and $$B'(t) \neq 0$$ and can be modified to be of your type.

"Can there be anything said..." is not specific enough. There are many papers on the subject, how zeros of such functions can be distributed. Some recent references are https://arxiv.org/abs/1908.09161 and https://arxiv.org/abs/1409.2736, and they contain references on the previous work.
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