#2938 Zeilberger -- Incorrect result

[Raymond Rogers]

Here is a wxmaxima and maxima capture. I am certainly not proficient with Zeilberger's algorithm but I do know that Zeilberger(f(x,i,n),i,n) is not supposed contain "i". xmaxima has a separate problem I will post separately; it had the same result;

Wxmaxima:
(%i47) load("zeilberger")$
test1:-z^n*x^n*(-1)^i*(factorial(d+n)/factorial(d))/((x)^i*(-c)^i*factorial(-i+n)*(factorial(b+n)/factorial(b))*c*factorial(n));
(%o48) -(b!*(-1)^i*(n+d)!*x^n-i*z^n)/((-c)^i*c*d!*n!*(n+b)!*(n-i)!)
(%i51) Zeilberger(test1,i,n);
(%o51) [[(c*(n+d+1)*x*z)/(n-i+1),[(n+d+1)*z,-c*(n+1)*(n+b+1)]]]

Maxima:
Since the copy and paste doesn't seem to work here the capture is attached.

---

wxMaxima 14.12.1
Ubuntu 14
(as above) Maxima 5.32.1
Sorry I can't make any suggestions about fixes now.
Ray

[Raymond Rogers]

If I can't edit it; I apologize for the caps. I neither know where they came from or how to fix it.

[Leo Butler]

Edited formatting only.

[Raymond Rogers]

Thanks!

[Andrej Vodopivec]

Why do you think the result is not correct?

The return value should be read as [R,[a0,a1]] where R is the certificate and a0 and a1 are Zeilberger coefficients. Both a0 and a1 do not contain i, but is is perfectly legal for the certificate to contain i.

It is hard to symbolically test correctness, but for specific values of coefficients b,c, and d the result is correct:

(%i1) load(zeilberger)$
(%i2) test1(i,n):=-z^n*x^n*(-1)^i*(factorial(d+n)/factorial(d))/((x)^i*(-c)^i*factorial(-i+n)*(factorial(b+n)/factorial(b))*c*factorial(n))$
(%i3) res: Zeilberger(test1(i,n), i, n);
(%o3) [[(c*(n+d+1)*x*z)/(n-i+1),[(n+d+1)*z,-c*(n+1)*(n+b+1)]]]
(%i4) define(R(i,n), part(res, 1, 1));
(%o4) R(i,n):=(c*(n+d+1)*x*z)/(n-i+1)
(%i5) a0:part(res, 1, 2, 1);
      a1:part(res, 1, 2, 2);
(%o5) (n+d+1)*z
(%o6) -c*(n+1)*(n+b+1)
(%i7) a0*test1(i,n) + a1*test1(i,n+1) - (R(i+1,n)*test1(i+1,n)-R(i,n)*test1(i,n))$
(%i8) subst([b=20, d=10, c=5], %)$
(%i9) factcomb(minfactorial(%));
(%o9) 0

Note that since %o9 is 0, the result is correct for b=20, d=10 and c=5.

[Raymond Rogers]

Thanks for pointing that out! Sorry for the late thanks.