# How to get sums like these in the form of the Cauchy Product?

I know that the question isn't very well-worded, please feel free to change it to something better.

I have this sum:

$$\sum_{k=0}^{\infty} \sum_{l=0}^{k} a_lx^ll!a_{k-l}x^{k-l+1}(k-l)!\frac{1}{(k+1)!}$$

This is the result of convoluting a polynomial of the form $\sum_{n=0}^{\infty} a_n x^n$ with itself, so we'd have $\int_{0}^{x} (\sum_{n=0}^{\infty} a_n t^n)(\sum_{n=0}^{\infty} a_n (x-t)^n)dt$, which can be rewritten as $\sum_{k=0}^{\infty}\sum_{l=0}^{k}a_la_{k-l}\int_{0}^{x}t^l{(x-t)}^{k-l}dt$ which evaluates to the sum above.

I would like to find a way to put in a form such that it can be written as a product of two sums, that is $$\sum_{j=0}^{\infty}\sum_{k=0}^{j} b_{k}c_{j-k}$$

Most of it is able to be written in this form, where $b_k=a_kx^kk!$ and $c_{j-k}=a_{j-k}x^{j-k+1}(j-k)!$. However, I still have that $\frac{1}{(j+1)!}.$ What I want to know is if there's any way I could algebraically manipulate that so that it can completely be in the form I want.

Thank You.

• Where did you come across this sum? Jan 31, 2018 at 18:48
• @CarlSchildkraut I didn't actually come across it. I edited the question and added some new information which hopefully answers your questions.
– Sam
Jan 31, 2018 at 19:35

Not sure if I correctly understood but this can be done as follows.

Start writing the series obtained convoluting as $$\sum_{k=0}^{\infty} \frac{1}{(k+1)!} \sum_{\ell=0}^{k} \left( a_{\ell}\ a_{k-\ell}\ \ell!\ (k-\ell)! \right) x^{k+1}$$

Note that this is a power series with coefficients $A_k = \dfrac{1}{(k+1)!} \sum_\limits{\ell=0}^{k} \left( a_{\ell}\ a_{k-\ell}\ \ell!\ (k-\ell)! \right)$.

We want to write $\sum_\limits{k=0}^{\infty} A_k\ x^{k+1} = \sum_\limits{k=0}^{\infty} \sum_\limits{\ell=0}^{k} b_l\ c_{k-l}\ x^k = \left( \sum_\limits{m=0}^{\infty} b_m\ x^m \right) \left( \sum_\limits{n=0}^{\infty} c_n\ x^n \right)$, which still also a power series with coefficients $P_k = \sum_\limits{\ell=0}^{k} b_l\ c_{k-l}$.

Since a power series uniquely determines a function, it follows that the coeffiecients of both must be equal. That is, $P_0 = 0$ and $P_{k+1} = A_k$. Also, because $P_0 = b_0 c_0$ we have that $b_0 = 0$ or $c_0 = 0$. Without loss of generality suppose that $b_0 \neq 0$; then $c_0 = 0$.

We gonna write $c_n$ as a function of $a_k$ and $b_m$.

For $k = 0$ $$A_0 = a_{0}^2 = P_1 = b_0 c_1 + b_1 c_0 = b_0 c_1 \implies c_1 = \frac{a_0^2}{b_0}$$

For $k = 1$ $$A_1 = \frac{1}{2} \left( a_{0} a_{1} + a_{1} a_{0} \right) = a_{0} a_{1} = P_2 = b_0 c_2 + b_1 c_1 + b_2 c_0 = b_0 c_2 + b_1 c_1\\ \implies c_2 = \frac{1}{b_0} \left( a_{0} a_{1} - b_1 c_1 \right)$$

For $k = 2$ $$A_2 = \frac{1}{3} \left( 4 a_0 a_2 + a_1^2 \right) = P_3 = b_0 c_3 + b_1 c_2 + b_2 c_1\\ \implies c_3 = \frac{1}{b_0} \left( \frac{1}{3} \left( 4 a_0 a_2 + a_1^2 \right) - b_1 c_2 - b_2 c_1 \right)$$

And so on... Hence $$c_{n+1} = \frac{1}{b_0} \left( A_n - \sum_{\ell=1}^{k-1} b_{\ell}\ c_{k-\ell} \right)$$

This means that $b_n$ is a sequence that we are free to determine as long as we choose $c_n$ as above. Thus we may choose a sequence which make its respective power series to have infinite radius of convergence to avoid any problem, say $b_m = \dfrac{1}{m!}$; so $\sum_\limits{m=0}^{\infty} b_i\ x^i = e^x$. We are done.

• But how to ensure that the series $\sum\limits_{n = 0}^\infty c_n x^n$ converges in a suitable range? Feb 3, 2018 at 13:42
• @AlexFrancisco This is guaranteed by construction. The radius of convergence of $\left( \sum_\limits{m=0}^{\infty} b_m\ x^m \right) \left( \sum_\limits{n=0}^{\infty} c_n\ x^n \right)$ is $\min\{R_b, R_c\} = \min\{\infty, R_c\} = R_c$ and $\sum_\limits{k=0}^{\infty} A_k\ x^{k+1} = \left( \sum_\limits{m=0}^{\infty} b_m\ x^m \right) \left( \sum_\limits{n=0}^{\infty} c_n\ x^n \right)$. Since both sides are power series, their radius of convegence must be equal, that is $R_A = R_c$. Hence its convergence depends on the convergence of the initial series presented by the OP. Feb 3, 2018 at 14:43
• To write the equality $\sum\limits_{n = 0}^\infty a_n x^n = \left( \sum\limits_{n = 0}^\infty b_n x^n \right) \left( \sum\limits_{n = 0}^\infty c_n x^n \right)$ in the first place seems to be cyclic deduction. Feb 3, 2018 at 15:15
• @AlexFrancisco This is what OP is trying to do. Also the sequence $a_n$ is already known. Feb 3, 2018 at 15:22
• You understood correctly. Apologies for not being able to word things well. Thank you for the insightful response.
– Sam
Feb 3, 2018 at 15:28

Put plainly, you want to express the Integral Convolution of a given polynomial with itself by means of the product of two polynomials to be determined.

Keeping general, that is unspecified, the coefficients $a_k$ of the auto-convolved polynomial, that would mean to be able to express $${{l!\left( {k - l} \right)!} \over {\left( {k + 1} \right)!}} = {1 \over {\left( {k + 1} \right)\left( \matrix{ k \cr l \cr} \right)}} = f(l)g(k - l)$$ which is not possible, since the binomial is not splittable in such a multiplicative way.
In fact, that would imply \eqalign{ & \left( \matrix{ k \cr l \cr} \right) = f(l)\,g(k - l)\quad \Rightarrow \quad \left( \matrix{ k \cr l \cr} \right) = \left( \matrix{ k \cr k - l \cr} \right) = f(l)\,f(k - l)\quad \Rightarrow \cr & \Rightarrow \quad \left( {n + m} \right)! = F(n)\,F(m) \cr} which cannot be for every $n$ and $m$.

That means that it is not assured that the polynomial resulting from the convolution be in general factorizable , except from having the constant term null and thus allowing to separate a $x$.

For example $$\int_0^x {\left( {1 + t} \right)\left( {1 + x - t} \right)dt} = {{x^{\,3} } \over 6} + x^{\,2} + x = x\left( {{{x^{\,2} } \over 6} + x + 1} \right)$$

• I see. ${1 \over {\left( {k + 1} \right)\left( \matrix{ k \cr l \cr} \right)}} = f(l)g(k - l)$ is what I was trying to find. Is there a more specific reason why it can't be split in such a way?
– Sam
Feb 3, 2018 at 15:21
• added some explanation Feb 3, 2018 at 17:33