# Method for variable substitution in multiple summation

I want to ask: is there any general method for variable substitution in multiple summation?

For example in the following equation a new variable $\lambda=n+m-2\mu$ is introduced to transform the LHS to the RHS $$\sum_{n=0}^\infty \sum_{m=0}^\infty \sum_{\mu=0}^{\left\lfloor \frac{m+n}{2}\right\rfloor}f(n,m,\mu,n+m-2\mu) = \sum_{\lambda=0}^\infty \sum_{\mu=0}^\infty \sum_{n=0}^{2\mu+\lambda}f(n,2\mu+\lambda-n,\mu,\lambda)$$

Another example, in which a new variable $\delta=m+n+2 p-2 k-2 \mu-2 \sigma$ is introduced

$$\sum _{n=0}^{\infty } \sum _{m=0}^{\infty } \sum _{p=0}^M \sum _{k=0}^p \sum _{\sigma =0}^{p-k}\quad \sum _{\mu =0}^{\left\lfloor \frac{m+n}{2}+p-k-\sigma \right\rfloor } f(n,m,\mu ,p,k,\sigma ,m+n+2 p-2 k-2 \mu-2 \sigma )$$ $$= \sum _{\delta =0}^{\infty } \sum _{\mu =0}^{\infty } \sum _{p=0}^M \quad\sum _{\beta =0}^{\min \left(p,\left\lfloor \frac{\delta }{2}+\mu \right\rfloor \right)}\quad \sum _{n=0}^{2 (\mu -\beta )+\delta }\quad \sum _{k=0}^{p-\beta }\;\; f(n,\delta +2 \mu-2 \beta -n,\mu ,p,k,p-\beta -k,\delta )$$

Additional remarks: my goal is using a new summation index, e.g. $\lambda$, to express a particular linear combination of the old indices, which is appointed by me, e.g. $n+m-2\mu$. So this is a linear coordinate transformation. My problem is how to determine all the lower and upper bounds of the new summation indices frame, as well as the summation steps which are possibly not $1$.

I wonder whether there is a systematical and efficient technology, so I may be able to do those transformations automatically by programming.

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Any reference books and articles will be grateful. – Silvia Jul 20 '11 at 14:42
I think you need to be clearer about what the substitution should achieve. Otherwise, the question is a bit too vague. In your examples, I don't see any great simplification resulting from the substitutions, which is what confuses me. – gowers Jul 20 '11 at 22:50
@gowers Sorry for confusing, I added some additional explanations, hope it help making my question more clear. – Silvia Jul 21 '11 at 5:25
– Will Jagy Jul 21 '11 at 5:40
Graham, Knuth, and Patashnik spend a good part of a chapter talking about notation for multiple sums and their manipulation in the book Concrete Mathematics. Chapter 2, if memory serves. Gerhard "Ask Me About System Design" Paseman, 2011.07.20 – Gerhard Paseman Jul 21 '11 at 6:20