Re: [Maxima-discuss] Maxima

[Robert D. <rob...@gm...>]

On Sun, Nov 5, 2023 at 6:38 PM Alexander P-sky <apo...@gm...> wrote:

> Sum[Binomial[n - 1, n - i] Sum[Binomial[k + i, i] Binomial[n - 1, n - k],
{k, 0, n}], {i, 0, n}]

Well, Maxima has a function called simplify_sum which can handle many
identities, but, for the record, it doesn't recognize this.

(%i2) sum (binomial (k + i, i)*binomial (n - 1, n - k), k, 0, n);
          n
         ====
         \
(%o2)     >    binomial(k + i, i) binomial(n - 1, n - k)
         /
         ====
         k = 0
(%i3) sum (binomial (n - 1, n - i) * %o2, i, 0, n);
       n      n
      ====   ====
      \      \
(%o3)  >    ( >    binomial(k + i, i) binomial(n - 1, n - k))
      /      /
      ====   ====
      i = 0  k = 0
                                          binomial(n - 1, n - i)

(%i5) load (simplify_sum) $
(%i6) simplify_sum (%o3);
errexp1 non-rational term ratio to nusum
       n     n
      ====  ====
      \     \
(%o6)  >     >    binomial(k + i, i) binomial(n - 1, n - i)
      /     /
      ====  ====
      i = 0 k = 0
                                         binomial(n - 1, n - k)
(%i7) simplify_sum (%o2);
Is i + 1 positive, negative or zero?

p;
          n
         ====
         \
(%o7)     >    binomial(k + i, i) binomial(n - 1, n - k)
         /
         ====
         k = 0


best,

Robert