Here is an addendum: the polynomial α·(α - 1)·(α - 2)·•••·(α - k + 1) actually has a special name: it is called the kth falling factorial of α. This is a kth degree polynomial on α, and the coefficients are known as the Stirling coefficients of the first kind. The binomial coefficient choose(α, k) is this equal to falling(α, k)/k!. This generalization of the binomial coefficient also appears not only as part of the binomial series, but it appears naturally in other contexts. For example, one idea may be to generalize higher-order derivatives to fractional order, and a Newton series, with this generalized binomial coefficient, can be used to explore this idea.
@@geraldsnodd The layers of Pascal's Tetrahedron are the coefficients of (x+y+z)^n. If you consider, for example (x+y+z)^2, you have x(x+y+z)+y(x+y+z)+z(x+y+z) =x^2+xy+xz+xy+y^2+yz+zx+zy+z^2 =x^2+2xy+2xz+y^2+2yz+z^2. The last line can be arranged geometrically in to a triangle of Pascal's tetrahedron, but the line before that essentially consist of three triangles one where x is first, one where y is first and one where z is first. If you consider the values (x,y,z) arranged in a triangle, you can visually multiply two of these triangles to get the terms above (e.g. x*(x,y,z) would be x^2+xy+xz). The nine terms of the product before you "combine like terms" can naturally be arranged into an iterate of Sierpinski's triangle. For example xx xy xz yx zx yy yz zy zz You get part of Pascal's tetrahedron by the commutative property: xy=yz, xz=zx, yz=zy x^2 2xy 2xz y^2 2yz z^2 Sorry if these diagrams don't look very clear by the way, they are difficult to type out in the comments section of a youtube video. So, each layer of Pascal's Tetrahedron could be considered as a commutative version of Sierpinski's Triangle. The idea is that multiplication of these diagrams behaves similarly to how Sierpinski's triangle iterates, you replace each thing in one diagram by a scaled copy of things in the other one.
Did... did you just read my mind or something? These are the topics I've been working in four of my classes these weeks. They're advanced pre-SAT(well.. not in US, so sort of) classes, so I'm on binomial expansion and I always add the generalization with this exact notation of alpha choose k. haha
Extreemly extremly cool generalization. I loved also the way you paired up the binomial expansion to make itobvious why it is n choose k. Wonderful work!
The timing of this video being released was awesome coz just today I was thinking about how newton calculated pi with the binomial expansion of (1+x) ^1/2 nd then realised that the formula for binomial theorem can't take 1/2 as an input so this cleared alot for me.
Before seeing the whole title I was like this is gonna be multiminomial expansion , and it was not . But do that topic in another video please! The shirt is so cool !!
Cool fact: in the triangle's prime numbered rows, all terms excerpt the 1's are multiplies of that row's prime numbers. E.g. row 5 is 1-5-10-10-5-1, and indeed, 5 and 10 are multiples of 5. Sweet.
There is some sort of generalization you can come up with, using the Gamma function, but it just is not anywhere near as elegant, nor is it useful, since you never encounter series where the series index runs over the real numbers.
Hey, why are the bounds -1 < α < 1? Also if you extend this to x + c where c is some kind of constant, the result will just be the same just multiplied by c^(α-k) at each summand right?
I use the binomial theorem to prove the power rule for derivatives when the exponent is a nonnegative integer. But it seems like circular reasoning to then extend the power rule for real exponents to prove the binomial theorem for real exponents. I’m guessing there must be another way to prove the power rule over the reals.
I feel ashamed about thinking you miswrote "sequence" as series cus i didn't remember an infinite sum related to the bn coeficient even tough I took a full semester of combinatorics some years ago... I must say the density of the pascal triangle as shown by Derek in one of his videos could have been a good adition
