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I asked this question on mathoverflow, someone suggested me to ask here as well. So I post it here. Thanks for helping.

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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    Yes they are:( I used o$n$e of the explicit sum formulas of Chebyshev poly$n$omial (http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html eq.15), which introduces those floor functions.2011-07-21

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There is no such "general method", the reason being the following:

When we have to compute an integral $\int_B f(x){\rm d}(x)$ over some simple body $B\subset{\mathbb R^n}$ there usually are many elementary maps $g: \ A\to B, \ u\mapsto g(u)$ which are essentially bijective and make $B$ the image of an even simpler body $A$, or let become $f$ a simpler expression of the new variable $u$.

On the other hand, an $r$-fold summation runs over some simple subset of ${\mathbb Z}^r$, say an octant or a triangle $\{(i,k)\ |\ 0\leq k\leq i\leq n \}$. Unless you resort to number-theoretical tricks there are almost no elementary maps mapping such a set to another such set. Patrick Da Silva has given an example in his comment: It is a linear map with ones in the main diagonal and one off-diagonal element nonzero.

As an aside: It is a famous unsolved problem to get an exact estimate on the number of lattice points in a disk of radius $R$.

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    After all the years learning and studying, I finally fully understand your answer! Thank you!2018-03-03