- #1
mnb96
- 715
- 5
Hello,
I have a sequence of polynomials defined in the following way:
[tex]P_k(x) = \frac{\partial^k}{\partial x^k}e^{s(x)}\vert_{x=0}[/tex]
Essentially the polynomial Pk is the k-th derivative of [itex]\exp(s(x))[/itex] evaluated at x=0. The function s(x) is a polynomial of 2nd degree in x.
In mathematica I define the polynomials with the following code:
D[Exp[s[x]], {x, k}]
For k=1..n one obtains a list of n polynomials P1(x),...,Pn(x).
My question is: is it possible to ask Mathematica to find a recurrent relation that expresses any Pk as a function of the previous polynomials Pk-1, Pk-2... ? (assuming it exists).
A similar well-known problem exists for http://en.wikipedia.org/wiki/Hermite_polynomials#Definition"
Thanks.
I have a sequence of polynomials defined in the following way:
[tex]P_k(x) = \frac{\partial^k}{\partial x^k}e^{s(x)}\vert_{x=0}[/tex]
Essentially the polynomial Pk is the k-th derivative of [itex]\exp(s(x))[/itex] evaluated at x=0. The function s(x) is a polynomial of 2nd degree in x.
In mathematica I define the polynomials with the following code:
D[Exp[s[x]], {x, k}]
For k=1..n one obtains a list of n polynomials P1(x),...,Pn(x).
My question is: is it possible to ask Mathematica to find a recurrent relation that expresses any Pk as a function of the previous polynomials Pk-1, Pk-2... ? (assuming it exists).
A similar well-known problem exists for http://en.wikipedia.org/wiki/Hermite_polynomials#Definition"
Thanks.
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