Difficult Laplace Transform Type of Integral

Question

Good afternoon. I have the following integral that I need help integrating;

$$ \mathrm{F}\left(x\right) = \int_{0}^{\infty}\mathrm{e}^{s\left(j - 1/x\right)}\, \left[\mathrm{T}_{N}\left(s\right)\right]^{k - j}\,\mathrm{d}s $$

where $\mathrm{T}_{N}\left(s\right)$ is the truncated exponential function $$ \mathrm{T}_{N}\left(s\right) = \sum_{n = 0}^{N}\frac{s^{n}}{n!} $$

and $j,k$ are whole numbers.

I figured that tackling this using integration by parts is the best choice, but naturally this could take a while given that my choice of $u_{1}=\mathrm{T}_{N}^{k - j}\left(s\right)$ yields a $\mathrm{d}u_{1}$ of $$ \mathrm{d}u_{1} = \left(k - j\right)\mathrm{T}_{N}^{k-j-1}\left(s\right) \mathrm{T}_{N - 1}\left(s\right)\,\mathrm{d}s $$ since $\mathrm{d}\mathrm{T}_{N}\left(s\right)/\mathrm{d}s = \mathrm{T}_{N - 1}\left(s\right)$. This does not seem to be simplifying the problem though as the next iteration would require me to then choose my $u_{2}$ to be $$ u_{2} = \frac{1}{k - j}\,\frac{\mathrm{d}u_{1}}{\mathrm{d}s} = \mathrm{T}_{N}^{k - j - 1}\left(s\right) \mathrm{T}_{N - 1}\left(s\right) $$

Are there any other approaches to this without this drawn out IBP method or without considering expanding the truncated exponential ?.

Answer 1

So long as $j < 1/x$ it is possible to write the function as a finite sum of terms (i.e., in closed form) by writing the power of the truncated exponential as a polynomial, and then turning the integral into a polynomial sum of gamma functions. If we apply the multinomial theorem we can write the power of the truncated exponential as a polynomial of degree $N(k-j)$ as follow:

$$\begin{equation} \begin{aligned} \left[ \sum_{n=0}^N \frac{s^n}{n!} \right]^{k-j} &= \sum_{r_0 + \cdots + r_N = k-j} {k-j \choose \mathbf{r}} \prod_{n=0}^N \Big( \frac{s^n}{n!} \Big)^{r_n} \\[6pt] &= \sum_{r_0 + \cdots + r_N = k-j} \frac{(k-j)!}{\prod_{n=0}^N (r_n!) (n!)^{r_n}} \prod_{n=0}^N s^{n \cdot r_n} \\[6pt] &= \sum_{r_0 + \cdots + r_N = k-j} \frac{(k-j)!}{\prod_{n=0}^N (r_n!) (n!)^{r_n}} s^{\sum_{n=0}^N n \cdot r_n} \\[6pt] &= \sum_{i=0}^{N(k-j)} a_i s^i, \\[6pt] \end{aligned} \end{equation}$$

where the coefficients $a_0,a_1,...,a_{N(k-j)}$ depend on $N$ and $k-j$, and are obtained from this multinomial summation. We then have:

$$\begin{equation} \begin{aligned} F(x) &= \int \limits_0^\infty e^{s (j-\frac{1}{x})} \left[ \sum_{n=0}^N \frac{s^n}{n!} \right]^{k-j} ds \\[6pt] &= \int \limits_0^\infty e^{s (j-\frac{1}{x})} \sum_{i=0}^{N(k-j)} a_i s^i \ ds \\[6pt] &= \sum_{i=0}^{N(k-j)} a_i \int \limits_0^\infty e^{s (j-\frac{1}{x})} s^i \ ds \\[6pt] \end{aligned} \end{equation}$$

Now, assume that $j < 1/x$ so that $(j-\tfrac{1}{x}) <0$. In this case we can use the change-of-variable $m = s |j-\frac{1}{x}|$ to get $dm = -j ds$ and we have:

$$\begin{equation} \begin{aligned} F(x) &= \sum_{i=0}^{N(k-j)} a_i \int \limits_0^\infty e^{-s |j-\frac{1}{x}|} s^i \ ds \\[6pt] &= \sum_{i=0}^{N(k-j)} a_i \cdot \frac{1}{(-j) |j - \frac{1}{x}|^i} \int \limits_0^\infty e^{-s |j-\frac{1}{x}|} (s |j-\tfrac{1}{x}|)^i \ (-j) \ ds \\[6pt] &= \sum_{i=0}^{N(k-j)} a_i \cdot \frac{1}{(-j) |j - \frac{1}{x}|^i} \int \limits_0^\infty e^{-m} m^i \ dm \\[6pt] &= \sum_{i=0}^{N(k-j)} a_i \cdot \frac{i!}{(-j) |j - \frac{1}{x}|^i}. \\[6pt] \end{aligned} \end{equation}$$

This form of the function is a finite sum, where the only serious difficulty (for large $N$ or $k-j$) is to calculate the polynomial coefficients $a_0,a_1,...,a_{N(k-j)}$. Calculation of these coefficients may be cumbersome for large degree, but once they are calculated the function is then amenable to calculation.

Answer 2

I have only a partial answer: it pretty much just consists of repeated integration by parts, so maybe this isn't what you're looking for.

Write $A(n_1, \dots, n_r) = \int_0^\infty e^{-as} \prod_i T_{n_i}(s) \,ds$. Then integration by parts gives \begin{align*} A(n_1, \dots, n_r) &= -\frac{1}{a}\bigg[e^{-as}\prod_i T_{n_i}(s)\bigg]\bigg|_0^\infty + \frac{1}{a}\sum_{j : n_j > 0} \int_0^\infty e^{-as} T_{n_j - 1}(s)\prod_{i \neq j} T_{n_i}(s) \,ds \\ &= \frac{1}{a} + \frac{1}{a} \sum_{j : n_j > 0} A(n_1, \dots, n_j - 1, \dots, n_r) \end{align*} when all $n_i \geq 0$.

To make things a little easier, let's take $T_n(s) = 0$ when $n < 0$ (this is consistent with the identity $T_n'(s) = T_{n-1}(s)$), so we'll have $A(n_1, \dots, n_r) = 0$ when some $n_i < 0$, and so we can write $$A(n_1, \dots, n_r) = \frac{1}{a}\bigg(\mathbb{1}_{(n_i)} + \sum_i A(n_1, \dots, n_i - 1, \dots, n_r)\bigg)$$ for any integers $n_1, \dots, n_r$, where $\mathbb{1}_{(n_i)} = 1$ if all $n_i \geq 0$ and is $0$ otherwise.

Letting $f(x_1, \dots, x_r) = \sum A(n_1, \dots, n_r) x_1^{n_1} \cdots x_r^{n_r}$ be the generating function of $A$, this identity gives $$af = \frac{1}{(1-x_1) \cdots (1-x_r)} + (x_1 + \cdots + x_r)f$$ hence \begin{align*} f &= \frac{1}{(1 - x_1) \cdots (1 - x_r)(a - x_1 - \cdots - x_r)} \\ &= \frac{1}{a} \sum_{l_1, \dots, l_r, m_1, \dots, m_r} a^{-m_1 - \cdots - m_r}\binom{m_1 + \cdots + m_r}{m_1, \dots, m_r} x_1^{l_1 + m_1} \cdots x_r^{l_r + m_r} \\ &= \frac{1}{a} \sum_{n_1, \dots, n_r} \sum_{0 \leq m_i \leq n_i} a^{-m_1 - \cdots - m_r}\binom{m_1 + \cdots + m_r}{m_1, \dots, m_r} x_1^{n_1} \cdots x_r^{n_r} \\ \end{align*} and finally, $$A(n_1, \dots, n_r) = \frac{1}{a} \sum_{0 \leq m_i \leq n_i} a^{-m_1 - \cdots - m_r}\binom{m_1 + \cdots + m_r}{m_1, \dots, m_r} $$ (this could alternatively be found by repeatedly expanding the recurrence).

The case relevant to the question is the one where $n_1 = \cdots = n_r = N$, $r = k-j$, $a = 1/x - j$, but I'm not sure how to simplify this sum. I've been thinking about it in terms of lattice paths in $r$ dimensions: in particular the coefficient of $a^{-t}$ is the number of lattice paths of length $t$ where we have at most $n_i$ steps in the $i$-th direction for each $i$, but that doesn't seem to help simplify any further.

As noted by DinosaurEgg, we're only interested in $$A(N, \dots, N) = \frac{1}{a} \sum_{0 \leq m_i \leq N} a^{-m_1 - \cdots - m_r}\binom{m_1 + \cdots + m_r}{m_1, \dots, m_r}$$ but this sum still seems difficult to simplify to me, because of the $m_i \leq N$ constraints.