Problems from analysis
1 数学分析wsj习题5.41
but finally it occurred to me that ......
2
I found there was no suitable tool for it
so I had to look into Concrete Mathematics -- Gosper!
现学
proof 1:
r(n,k)=-k(m+k+1)\\ (n+1-k)\beta_0(n)+(n+1)\beta_1(n)=(m+k+1)((n+1-k)s(n,k+1)+ks(n,k))\\ consider \ \ the \ \ coefficient \ \ of \ \ k\\ the \ \ degree \ \ of \ \ s \ \ must \ \ be \ \ 0\\ hence \ s(n,k)=1 \ \ \beta_0(n)=-n-1 \ \ \beta_1(n)=m+n+2\\ S(n+1)/S(n)=(n+1)/(m+n+2) $$$
proof 2
on the other hand we can expand
proof 3
then I observe the form
it is similar to a identity that haunted me for more than a year
it is about the connection between the recurrence and the inclusive-exclusive expression of stirling number of second kind
the left hand side is inclusive-exclusive
the right of obtained by the recurrence
it is easy(?) to see this is equivalent to the probem
thus we get a combinatorial demonstration
ps : whether x is an integer is not important since 考虑多项式在无限个点值为零则为零
proof 4
but in fact if we consider starting from the right hand side
Partial Fraction Expansion
convert it into partial fraction
then set x=m.
proof 5
the real solution:
分部积分法+递推