I want to prove by Maple if  -1<x<1 and n is positive integer then

sum(k^n*x^k, k = 1 .. infinity) = (sum([sum((-1)^(m+1)*(binomial(n+1, m-1))(r-m+1)^n, m = 1 .. r)]*x^r, r = 1 .. n))/(1-x)^(n+1)

I would appreciate for any help.

By Maple

convert(LegendreQ(1,x),hypergeom)

hypergeom([1, 3/2], [5/2], 1/x^2)/(3*x^2)

LegendreQ(1,0.)

-1.000000000-3.141592654*10^(-15)*I

The 2F1 form suggests for me that in x=0 LegendreQ is infinite or at least undetermined. So what is the truth?

I work with LegendreQ(1,x) where x is real.

Thanks,

                      Sandor

convert( (z)_n, GAMMA)  does not give the desired  GAMMA(z+n)/GAMMA(z).

Instead of Maple gives the answer   (z)_n.

How could I obtain the GAMMA form?

Thanks,

                   Sandor

In Maple11 odeadvisor does not know the Euler type differential equation.

For example, R(r)+ r R'(r)+r^2 R''(r)=0.

It is the same situation in Maple12?

Thanks, Sandor

I have the expression

p(x,a) w''(x) + q(x,a) w'(x) + r(x,a) w(x)

How could I obtain p(x,a), etc.?

Thanks,

              Sandor