Wikipedia:Reference desk/Archives/Mathematics/2008 December 20

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December 20

Boubaker polynomials (2)

To user Eric

The hint you gave is wonderful!

The trouble you evoked can be avoided by calculating the sum

$S=\sum _{n\geq 3}B_{n}(x){\frac {t^{n}}{n!}}$

So that :

$f(x,t)=B_{0}(x)+B_{1}(x)+{\frac {B_{2}(x)}{2}}+S$

By the way if any one can give the Chebyshev or Dikson polynomials exponential generating fonction it will help a lot.

Duvvuri.kapur (talk) 09:40, 20 December 2008 (UTC)

You would like to pass from the "ordinary generating function" say of the Chebyshev polynomials of the first kind, $\scriptstyle \sum _{n=0}^{\infty }T_{n}(x)t^{n}={\frac {1-tx}{1-2tx+t^{2}}}$, to an "exponential generating function", that is one of the form $\scriptstyle \sum _{n=0}^{\infty }T_{n}(x){\frac {t^{n}}{n!}}$. This involves applying a suitable linear operator on the OGF, that takes $t^{n}$ to $\scriptstyle {\frac {t^{n}}{n!}}$, for all n, and that is continuous enough to transform power series term by term. There are several functional settings where you can do that; here the simplest for your needs is the one of formal series in t (say with coefficients that are functions of x). You can then observe that your transformation also sends $\scriptstyle {\frac {1}{1-\lambda t}}$ to $\exp(\lambda t)$; here the OGF to transform can be written indeed as sum $\scriptstyle {\frac {a_{1}}{1-\lambda _{1}t}}+{\frac {a_{2}}{1-\lambda _{2}t}}$ (in case, have a look to the examples of partial fraction decomposition). The same method works as well with the $U_{n}$, with the Dickson polynomials and with the Boubaker's, for all of them have OGF that are rational function of t. Is it clear? PMajer(talk) 11:59, 20 December 2008 (UTC)

Yes it is clear. But one can not find in the existing litterature the already established Chebyshev and Dikson polynomials exponential generating fonctions, this is strange ..Duvvuri.kapur (talk) 12:50, 20 December 2008 (UTC)

Well, the existing mathematics literature is so huuuuge that it is often hard to find something. I have no references for the EGF of your polynomials, but I am as sure that it's there, as I'm sure that I'm alive (say in practice 100%). At least, in your case you know exactly where to look for it, and it shouldn't be that difficult. But there are situations in which, to get a clue for where to look for a result, you first have to give a proof of it: only then you'll understand which is its AMS collocation and which is the keyword to put in google. Research in mathematics is becoming a search engine of the Babele library, whose catalog is just a copy of it. Mathematicians are mostly Platonists, so in principle it should not make a great difference to do research in a real or in an ideal library :) PMajer(talk) 13:49, 20 December 2008 (UTC)

List of exact values of trig functions

It's possible to find exact (albeit irrational) values for e.g. sine of 3 degrees using identities as indicated on Exact trigonometric constants. It's also possible to find values for sine that is a multiple of 5 degrees, e.g. $\sin 50^{\circ }={\frac {1}{4{\sqrt[{3}]{\frac {-1-{\sqrt {3}}i}{16}}}}}+{\sqrt[{3}]{\frac {-1-{\sqrt {3}}i}{16}}}$ (Sorry if I messed up the math.) Is there some place where many of these are listed? Most webpages I've seen only have a few, like these: [1] [2] [3] I'm looking for a big collection of these. If one doesn't exist, I'd like to make it. :) Pointers would be appreciated. Thanks. 4.242.147.223 (talk) 05:27, 20 December 2008 (UTC)

If you allow nth roots of complex numbers like ${\sqrt[{3}]{\frac {-1-{\sqrt {3}}i}{16}}}$, then $\zeta =\cos[(k/n)(360^{\circ })]+i\sin[(k/n)(360^{\circ })]$ is one of the n complex nth roots of 1. Then you can get the cosine, for example, as (ζ + ζ^-1)/2. So really, you're going to have to specify what kinds of nth roots of complex numbers you're allowing. Otherwise any sine or cosine of a rational angle can be expressed easily in terms of ${\sqrt[{n}]{1}}$ for some n, and some choice of root. Joeldl (talk) 11:28, 20 December 2008 (UTC)

This thread from a couple of years ago may be of interest to you. --NorwegianBlue ^talk 15:46, 20 December 2008 (UTC)

OK, good. Thanks for the quick and excellent responses (that's basically what I was looking for). Am I correct in saying that it's impossible to get an exact value of sines of irrational angles? 4.242.108.230 (talk) 16:12, 20 December 2008 (UTC)

No. There's an angle θ, probably irrational, with sin θ = 0.123456, for example. Joeldl (talk) 16:30, 20 December 2008 (UTC)

Good point. However, is there an exact value for e.g. sine of (square root of 5)? 4.242.108.230 (talk) 16:59, 20 December 2008 (UTC)

Well, if your angle "square root of 5" is in radians, then the sine is a transcendental number by the Hermite-Lindemann theorem. Joeldl (talk) 17:10, 20 December 2008 (UTC)

Thanks. 4.242.108.187 (talk) 17:50, 20 December 2008 (UTC)

On the flip side of the coin, if a triangle has integer side lengths, the angles are irrational. So it is possible to sin for many irrational angles using Pythagorean triples. Thenub314 (talk) 20:28, 20 December 2008 (UTC)

Yes, irrational in radians. I'm making this point because in an earlier part of the discussion, I used "rational angle" to mean a rational multiple of π. Joeldl (talk) 05:01, 21 December 2008 (UTC)

On/Off Notation

In proving the solution to a problem algebraically, is there a standard way of showing whether something is on or off. The problem involves lights being on or off and the turning of them on or off, I solved it, but I'm stumped as to how to write on/off algebraically. Any ideas appreciated! Thanks. Harland1 (^t/_c) 22:24, 20 December 2008 (UTC)

There isn't a standard mathematical notation for "on" and "off". Many people are comfortable with using "1" and "0" or "true" and "false" as place-holders for "on" and "off", but if you choose to do that, you should clearly explain the meaning of your words so that the reader is not confused.

However, I'm not sure what you mean by "algebraically". Can you explain? If you are planning on performing some kind of algebraic manipulations with the states of lights, then you should use the notation from the appropriate algebra. For example, if you want to represent the expression "Light C is on exactly when either light A or light B is on", then in the notation of boolean algebra, you might write $c=a\lor b$, where c is true or false according to whether light C is on or off, etc. As another example, if each light had a brightness (either 0 or 1, or maybe any positive integer, etc.), and you wanted to represent the expression "The brightness of light C is the sum of the brightnesses of lights A and B", then the notation $c=a+b$ would be appropriate, where c is an integer denoting the brightness of light C, etc. As one last example, if you are dealing with statements like "Light C is on if Fred enters the room first", then you probably aren't performing algebraic manipulations at all, and there won't be a standard notation for the situation. Eric. 68.18.63.75 (talk) 22:47, 20 December 2008 (UTC)

If you were tackling the lights out puzzle, then it would be reasonable to use integers modulo 2 to represent whether each light is on or off. Then, adding "1" to the light toggles the state of that light. Pressing a button is represented by adding 1 to certain lights (which lights depends on which button). The usefulness of this representation, is that it makes two facts about the puzzle clear: it doesn't matter what order buttons are pressed (since addition is commutative), and pressing any one button twice has no effect (since 1 + 1 = 0 in modulo 2 arithmetic). Eric. 68.18.63.75 (talk) 22:59, 20 December 2008 (UTC)