#generating function

[finder-review-simplest-fay]

S-cube Tutorials: Your Career Generating Function

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[peeterjoot]

Hermite polynomial normalization constant

Hermite polynomial normalization constant

[Click here for a PDF of this post with nicer formatting]

Question: Hermite polynomial normalization constant ([1] pr. 2.21)

Derive the normalization constant \( c_n \) for the Harmonic oscillator solution

\begin{equation}\label{eqn:hermiteOrtho:20} u_n(x) = c_n H_n\lr{ x \sqrt{\frac{m\omega}{\Hbar}} } e^{-m \omega x^2/2 \Hbar}, \end{equation}

by deriving the orthogonality relationship using…

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[curiosamathematica]

A Magic Trick from Fibonacci

100⁄89 = 1.1 2 3 5 95505…

10000⁄9899 = 1.01 02 03 05 08 13 21 34 55 90463…

1000000⁄998999 = 1.001 002 003 005 008 013 021 034 055 089 144…

The decimal expansion of the first fraction generates the first five Fibonacci numbers before blurring into other digits. The second fraction generates the first ten, and the third fraction generates the first fifteen.

This paper explains the pattern by exploiting the generating function of the Fibonacci numbers:

1⁄(1−x−x2) = F1 + F2·x + F3·x2 + F4·x3 + …

[whatilearninmath]

all about generating functions

part 6 of combinatorics (toc)

I AM NOT DEAD, PEOPLE. I was on March Break! And while on break, I was working on my second major...you know how these things are. Anyway, as I noted in my previous pre-break combinatorics post, the next topic on the docket was generating functions. This stuff is summation-heavy, so I will do my best to get as much into the actual post before I resort to LaTeX. (But hey, at least I can do this stuff in LaTeX. I am currently struggling trying to figure out a way to put the homotopy stuff from topology onto this blog. Ideas, anyone?)

Here's the motivation (which would have worked better if I had the foresight to quietly slip it into my original post about permutations and combinations): instead of thinking about sets, let's think for a moment about multisets.

A multiset is similar to a plain old boring set, but repetitions are allowed. In set theory, the set {1, 1, 2} and {1, 2} are the same set: they contain the same elements. However, in terms of multisets, {1, 1, 2} and {1, 2} are not the same. Formally, a multiset consists of element types and repetition numbers for each element type. We write a multiset as {n1·a1, n2·a2, ..., nk·ak, ...}, where ni is the repetition number of ai (how many times ai appears in the multiset). Multisets can be infinite in either the repetition numbers, the item types, or both.

While multisets might appear unnatural to those who are more familiar with set theory, the truth is that multisets appear a lot in "real life" (the two dirty words, I know. I'm sorry!). You wouldn't consider a bag with one orange and one apple to be equivalent to a bag with two oranges and one apple.

Now, it is natural to ask about r-combinations or r-permutations of a multiset. For example, how many 3-combinations or 3-permutations are there of the multiset {1, 1, 1, 2, 2, 3, 4, 4, 4, 5, 5}? In general, this problem is hard to solve because the answer depends on the repetition numbers of each element. The way we solve these problems is by using generating functions!

This point is about how far the Tumblr road will take me. Click for the LaTeX continuation! (It's so weird--when I'm in LaTeX, I automatically switch to "formal math homework jargon", with suppose, observe, remark, we aim to show... ugh, so not the tone I want on my math blog!)