Mathematica -difference of two Symbolic summations

[Osiris]

I wish to compute
Sum[f, {i, 0, m}] - Sum[f, {i, 0, j}]

but can't get the expected result
Sum[f, {i, j+1, m}]


I also tried
Sum[f, {i, 0, m}] - f[0]
but can't get the expected result
Sum[f, {i, 1, m}]


It seems that mathematica can't change the minimum/maximum value of the summation variable i above...

I also tried maple, but failed.

 

[Simon_Tyler]

Yep, Mma and Maple aren't very good at abstract manipulations on unevaluated sums and integrals. If you need to do these types of operations on sums/integrals it is best to work with an object (eg a List or a dummy sum command) containing both the summand/integrand and the range/limits, then plug it back into the integral/sum once you are done. Many operations like the one you have above can be automated with pattern recognition.

 

[Osiris]

Yep, Mma and Maple aren't very good at abstract manipulations on unevaluated sums and integrals. If you need to do these types of operations on sums/integrals it is best to work with an object (eg a List or a dummy sum command) containing both the summand/integrand and the range/limits, then plug it back into the integral/sum once you are done. Many operations like the one you have above can be automated with pattern recognition.

Thanks Simon. I am a newbie of Mathematica and not quite clear about using pattern recognition.

Could you please show an example for an operation above?

How about a more general case:
Sum[a*f, {i, 0, m}] - b*f[0]
I expect to get
Sum[a*f, {i, 1, m}]+(a-b)*f(0)
where a and b are positive integers.

 

[Simon_Tyler]

Here's a few rules and examples to get you started. I've defined a dummy sum command so that Mma does not try to evaluate the sum at each step.

The last rule is pretty ugly.
i.e. it will be slow in expressions with lots of terms, because it will get tested many times. If you have a more specific structure in mind, then you could make the rules more specific and thus faster.

In[1]:= sumRules={
sum[expr_,{i_,low_,up1_}]+sum[expr_,{i_,up1_,up2_}]:>sum[expr,{i,low,up2}],
sum[expr_,{i_,low_,up1_}]-sum[expr_,{i_,low_,up2_}]:>sum[expr,{i,up2,up1}],
sum[a_. expr_,{i_,low_,up_}]+b_. expr2_/;(expr2==expr/.i->low):>sum[expr,{i,low+1,up}]+(a+b)expr2};
In[2]:= sumToSum[expr_]:=expr/.sum->Sum

In[3]:= sum[f[i],{i,0,m}]+sum[f[i],{i,m,j}]/.sumRules
Out[3]= sum[f[i],{i,0,j}]
In[4]:= sum[f[i],{i,0,m}]-sum[f[i],{i,0,j}]/.sumRules
Out[4]= sum[f[i],{i,j,m}]

In[5]:= sum[7f[i],{i,0,m}]-2f[0]/.sumRules
%/.f[i_]:>i^2+1
%//sumToSum
Out[5]= 5 f[0]+sum[f[i],{i,1,m}]
Out[6]= 5+sum[1+i^2,{i,1,m}]
Out[7]= 5+1/6 (7 m+3 m^2+2 m^3)