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--- pow:problem3s21 [2021/03/30 20:23] – mazur
+++ pow:problem3s21 [2021/03/31 02:15] (current) – mazur
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+<box 80% round orange| Problem 3 (due Monday, March 29)>
+Let $g$ be a smooth function, i.e. a function which has derivatives of all
+orders. Recall that $g^{(k)}$ denotes the $k$-th derivative of $g$.
+For non-negative integers $n,k$, define the function $T_{n,k}(x)$ as follows:
+\[T_{n,k}(x)=\sum_{j=0}^n {n\choose j} (-1)^{n-j}g^{n-j}(x) (g^j)^{(k)}(x)\]
+(we set $g^0=1$ and $g^{(0)}=g$). For example, $T_{3,3}(x)=3g^2(x)g^{(3)}(x)-3g(x)(g^2)^{(3)}(x)+(g^3)^{(3)}(x)$.
+a) Prove that $T_{n,k}(x)=0$ for any $n>k$ and any $x$.
+b) Find a simple explicit formula for $T_{n,n}(x)$ (for example, $T_{3,3}(x)=6(g'(x))^3$. Check this!).
+</box>
+Four solutions were received: from Paul Barber, Yuqiao Huang, Prof. Vladislav Kargin, and Ashton Keith.
+Prof. Kargin submitted a beautiful solution different from our original solution. The solutions
+by Paul Barber, Yuqiao Huang, and Ashton Keith follow essentially the same idea as our original solution.
+Detailed solutions and some very nice results related to the problem are discussed in the following
+link {{:pow:2021sproblem3.pdf|Solution}}.