## Do these polynomials have names?

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skullturf
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### Do these polynomials have names?

(1) Is there an equivalent of the online encyclopedia of integer sequences for polynomials? In particular, for polynomials in one variable where the nth polynomial in the sequence has degree n?

(2) Failing that, does anyone happen to recognize the following sequences of polynomials, or the recurrences that define them? (In fact, the sequence that I denote by G_n is basically the Chebyshev polynomials of the second kind after a change of variable.)

I got the polynomials by considering a counting problem. If these polynomials happen to have a name or have arisen in other contexts then we might have an example of a neat little connection between apparently different topics.

G_0 = 1
G_1 = x

G_n = xG_{n-1} - G_{n-2}

so the next few are

G_2 = x^2 - 1
G_3 = x^3 - 2x
G_4 = x^4 - 3x^2 + 1
G_5 = x^5 - 4x^3 + 3x

---

H_0 = 1
H_1 = x

H_n = (x-1)(H_{n-1} + H_{n-2})

so the next few are

H_2 = x^2 - 1
H_3 = x^3 - 2x + 1
H_4 = x^4 - 3x^2 + 2x
H_5 = x^5 - 4x^3 + 3x^2 + x - 1

gorcee
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### Re: Do these polynomials have names?

This relates. I haven't formally checked to see if your sequences are Sheffer sequences or not, but you might dig through the related topics to get to where you want to be.
http://en.wikipedia.org/wiki/Sheffer_sequence

For orthogonal polynomials, there is a scheme that relates them. It's called the Askey Scheme, and it relates families of orthogonal polynomials. It's also deeply related to probability theory through something that can be generally called the Weiner-Askey Polynomial Chaos (this is a field in which I have done some work). For non-orthogonal, polynomials, I don't know if there are any such schemes, and I do not know if they're interesting at all.

Edit: Your G polynomials are very nearly the Hermite polynomials. In fact, all you need is to multiply by (n-1) on your second term on the RHS. With this term missing, I do believe that you blow orthogonality out of the water, but I think you might be able to rescale to correct (uncertain, and too tired to check).

Edit edit: To highlight the relationship between probability and orthogonal polynomials, check this shizzle out.
Look at the coefficients of the "Probabilist's" Hermite polynomials
http://en.wikipedia.org/wiki/Hermite_polynomials

The sequence of coefficients, when counting the 0s for the skipped orders of x, is given in OEIS here:
http://oeis.org/search?q=1%2C0%2C-28%2C ... &go=Search
" A triangle constructed from the coefficients of the n-th derivative of the normal probability distribution function. "

eta oin shrdlu
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### Re: Do these polynomials have names?

Your Gn are, as you note, basically just the Chebyshev polynomials of the second kind. It would be a little odd to have another well-known name for something so closely related; in any case I don't know of another name.

Your Hn can be simplified into a form very similar to the Gn. In particular, note that [imath](x-1)^{\lfloor{n/2}\rfloor}\mid H_n(x)[/imath]; write [imath]H_n(x)=(x-1)^{\lfloor{n/2}\rfloor}h_n(x)[/imath], rewrite the recursion in terms of hn (take odd and even cases), and eliminate the odd terms to find that h2n is basically just Gn with another change of variables. I'm not sure if that helps you though.

skeptical scientist
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### Re: Do these polynomials have names?

Neither sequence of polynomials is orthogonal, so I'm not really sure how the above remarks about orthogonal polynomials apply.
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skullturf
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### Re: Do these polynomials have names?

Since the G_n are essentially Chebyshev polynomials with a change of variable, wouldn't they actually be orthogonal if you choose the interval and the weight function correctly?

Edit: In general, if a sequence of polynomials is generated by the "right" kind of recurrence, must it be an orthogonal family? Can you get the interval and the weight function from the recurrence somehow? According to a brief look at Wikipedia's article on orthogonal polynomials, you can go the "other way": if you have an orthogonal sequence of polynomials, you can find the recurrence they must satisfy.

gorcee
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### Re: Do these polynomials have names?

skeptical scientist wrote:Neither sequence of polynomials is orthogonal, so I'm not really sure how the above remarks about orthogonal polynomials apply.

The Askey scheme orders polynomials through the increasing order of their hypergeometric function parameters. It's reasonable to suggest, without digging deeper into the generation of these polynomials, that such a relationship may exist (or could be derived).

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