Difficulties in patterns matching

• To: mathgroup at smc.vnet.net
• Subject: [mg3675] Difficulties in patterns matching
• Date: Fri, 5 Apr 1996 02:51:52 -0500
• Organization: Dept. of Mathematics, Chalmers, Sweden
• Sender: owner-wri-mathgroup at wolfram.com

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Hi,

I spent some time to get a grip on the following patterns
matching problem but to no avail.

Consider the following expression.

test = (2*Sum[(gamma*x[i])/(sigma^2*(1 - (gamma*x[i])/sigma)), {i, 1, n}])/
gamma^3 + (-1 + gamma^(-1))*
Sum[(2*gamma*x[i]^3)/(sigma^4*(1 - (gamma*x[i])/sigma)^3) +
(2*x[i]^2)/(sigma^3*(1 - (gamma*x[i])/sigma)^2), {i, 1, n}] -
(2*Sum[(gamma*x[i]^2)/(sigma^3*(1 - (gamma*x[i])/sigma)^2) +
x[i]/(sigma^2*(1 - (gamma*x[i])/sigma)), {i, 1, n}])/gamma^2

I would like to apply the following transformation rules to
the above expression.

1) Sum[x[i]/(1-gamma x[i]/sigma),{i,1,n}]-> n f1[gamma,sigma]

2) Sum[x[i]^2/(1-gamma x[i]/sigma)^2,{i,1,n}]-> n f2[gamma,sigma]

3) And generally for different m's:

Sum[x[i]^m/(1-gamma x[i]/sigma)^m,{i,1,n}]-> n fm[gamma,sigma]

There are lots of such expressions and the main problem is
that I can't define a general transformation rule which can handle for
example rule no 1 for all different coefficients of x[i] in the numerator of
expressions and also for different coefficients of (1-gamma
x[i]/sigma) in the denominator of expressions. This means that the
first sum in "test" should be transformed to

n gamma f1[gamma,sigma] /sigma^2

These rules should also handle sums like

Sum[x[i]+y[i],{i,n}]=Sum[x[i],{i,n}]+Sum[y[i],{i,n}].