# Poly family

1. Jun 8, 2012

### Xitami

$x^2+2\\\\ \frac{2}{3} x^3 + \frac{13}{3} x\\\\ \frac{1}{3} x^4 + \frac{14}{3} x^2 + 2\\ \\ \frac{2}{15} x^5 + \frac{10}{3} x^3 + \frac{83}{15} x\\ \\ \frac{2}{45} x^6 + \frac{16}{9} x^4 + \frac{323}{45} x^2 + 2\\\\ \dots$
?

2. Jun 8, 2012

### DivisionByZro

What is your question? We can't just guess.

3. Jun 8, 2012

### Xitami

But please guess :-) how to continue?

4. Jun 8, 2012

### Muphrid

How are these polynomials supposed to be related?

5. Jun 8, 2012

### micromass

Apparently, the OP is not a native english speaker and can't describe the problem well.

I guess that he encountered these polynomials while working and he is asking if somebody recognizes them or sees an easy pattern in them.

Last edited: Jun 8, 2012
6. Jun 10, 2012

### Xitami

Thank you Micromass

7. Jun 10, 2012

### haruspex

Clearly the constant term alternates between 0 and 2. At the other end, you can get the leading term from the preceding line by multiplying by 2x/n. So the leading coefficient is 2n-1/n! So a natural thing to try is:
- multiply each line by n! (starting with n=2 in the first line) to form the poly sequence Pn(x)
- form a new sequence from this according to Qn(x) = Pn(x) - 2x*Pn-1(x)
The coefficients that result look a little friendlier. The highest prime that occurs in this sample is 19, a lot better than 83.