Sum(binomial(n,m)*x**m, (m, 0, n)) is given an irrelevant convergence condition and there is a strange factorization when y's are present

[ghost]

>>> pprint(Sum(binomial(n,m)*x**m, (m, 0, n)).doit())
⎧         n               
⎪  (x + 1)     for │x│ ≤ 1
⎪                         
⎪  n                      
⎪ ____                    
⎪ ╲                       
⎨  ╲    m ⎛n⎞             
⎪   ╲  x ⋅⎜ ⎟             
⎪   ╱     ⎝m⎠   otherwise 
⎪  ╱                      
⎪ ╱                       
⎪ ‾‾‾‾                    
⎩m = 0
>>> pprint(Sum(binomial(n,m)*x**m*y**(n-m), (m, 0, n)).doit())
⎧              n                 
⎪     n ⎛x    ⎞           │x│    
⎪    y ⋅⎜─ + 1⎟       for │─│ ≤ 1
⎪       ⎝y    ⎠           │y│    
⎪                                
⎪  n                             
⎪ ____                           
⎨ ╲                              
⎪  ╲    m  -m  n ⎛n⎞             
⎪   ╲  x ⋅y  ⋅y ⋅⎜ ⎟             
⎪   ╱            ⎝m⎠   otherwise 
⎪  ╱                             
⎪ ╱                              
⎪ ‾‾‾‾                           
⎩m = 0

Closing this issue would close #5446

[ghost]

Could this be solved by calling gosper instead of whatever method is used at present

[asmeurer]

The method that gives a Piecewise with "irrelevant" convergence is the Meijer G algorithm (the same algorithm used for integrals). The convergence conditions aren't tight in general, though in practice they often are. Is it possible to clear the convergence question using the is_convergent algorithms?

What do you mean by "strange factorization"?

[ghost]

m and n are non-negative integers, should I have made that more explicit
They were declared to be such when I ran the computation

It's a finite sum, if a convergence condition is returned then something has gone wrong

The second sum is (x + y)**n, but it is expressed in the form (1 + t)**n

If Meijer G is causing problems then should the code be changed so that gosper is tried first

[PhoenixAlx]

That expression should return (x+y)^n , isn't?