In my combinatorics textbook, a question reads as follows:
Compute the value of the following sums. Your answer should be an expression involving one or two binomial coefficients:
In one of the subsections of the question we find the following intriguing series which is to be summed: problem 1
I tried using all the identities in the book but still failed to find a suitable solution (even google offered no help). This the link to my textbook (if you wish to go and find the identities given, please do navigate to the the combinatorics section and read the chapter on binomial coefficients): Combinatorics and Graph Theory by John Harris.pdf
Along with this problem, I am unable to solve another problem:problem 2. The notation for rising and falling factorial powers may be confusing but this is what they mean: notation for rising and falling powers
Although it is clear that this problem has to be solved by induction, I however am not able to arrive at the final result. The following is an identity which may come to your help: identity involving problem 2
If anyone could please solve these two problems it would be of great help. Also, do pardon me for using links instead of typing the equations here, I am new to StackExchange!
For the first one I think it is $u_m=(-1)^m \binom{n-1}m$. You can show it by induction.
For the second, doing an induction like for the classic binomial expension and using: $x+y+n=(x+k)+(y+(n-k))$, $x+y-n=(x-k)+(y-(n-k))$
for all number k should work.