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Basically I have been working with polynomials of the form:

$$P_n(x)=e^{-x^2}\left(\frac{d^n}{dx^n}e^{x^2}\right)$$

I do realize that an explicit form for $P_n(x)$ has been asked for on this site before (Find an expression for the $n$-th derivative of $f(x)=e^{x^2}$), and I do realize that these polynomials are very similar to the Hermite Polynomials, yet I have been having trouble attempting to get a relatively clean expression for $(P_n(x))^2$.

The closed form expression for $P_n(x)$ is: $$P_n(x) = \left( \sum_{j=0}^{\lfloor n/2 \rfloor} \frac{n!}{j!(n-2j)!}(2x)^{n-2j} \right)$$ However, I am wondering if there is a relatively clean expression for $(P_n(x))^2$

Does anyone know a solution or know how to attempt to find a solution?

EDIT: In case it helps anyone, the first few are:

$(P_1(x))^2=4x^2$

$(P_2(x))^2=4 + 16x^2 + 16x^4$

$(P_3(x))^2=144x^2 + 192x^4 + 64x^6$

$(P_4(x))^2=144 + 1152x^2 + 2688x^4 + 1536x^6 + 256x^8$

$(P_5(x))^2=14400x^2 + 38400x^4 + 33280x^6 + 10240x^8 + 1024x^{10}$

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1 Answers1

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It looks to me like the exponential generating function for $P_n(x)^2$ is

$$\sum_{n=0}^\infty P_n(x)^2 \dfrac{t^n}{n!} = \dfrac{\exp(4 t x^2/(1-2t))}{\sqrt{1-4t^2}}$$

Thus $$P_n(x)^2 = \left. \dfrac{\partial^n}{\partial t^n} \dfrac{\exp(4 t x^2/(1-2t))}{\sqrt{1-4t^2}} \right |_{t=0}$$

Robert Israel
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  • I'm sorry I'm not very familiar with generating functions, does this give me an explicit expression for (P_n(x))^2? – ASKASK Jan 19 '15 at 19:29
  • Sorry, I was looking for something more alone the lines of a definition involving one or more summations. My hope was to get rid of the differentiation operator. – ASKASK Jan 19 '15 at 19:36
  • If $P_n(x) = \sum_{i=0}^n a_i x^i$, then $P_n(x)^2 = \sum_{i=0}^{2n} b_i x^i$ where $b_i = \sum_{j= \max(0,i-n)}^{\min{n,i}} a_j a_{i-j}$. Is that what you're looking for? – Robert Israel Jan 19 '15 at 19:46