Method for variable substitution in multiple summation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:35:22Z http://mathoverflow.net/feeds/question/70812 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70812/method-for-variable-substitution-in-multiple-summation Method for variable substitution in multiple summation Silvia 2011-07-20T14:02:44Z 2011-07-21T21:44:20Z <p>I want to ask: is there any <strong><em>general method</em></strong> for variable substitution in multiple summation? </p> <p>For example in the following equation a new variable $\lambda=n+m-2\mu$ is introduced to transform the LHS to the RHS $$\sum_{n=0}^\infty \sum_{m=0}^\infty \sum_{\mu=0}^{\left\lfloor \frac{m+n}{2}\right\rfloor}f(n,m,\mu,n+m-2\mu) = \sum_{\lambda=0}^\infty \sum_{\mu=0}^\infty \sum_{n=0}^{2\mu+\lambda}f(n,2\mu+\lambda-n,\mu,\lambda)$$</p> <p>Another example, in which a new variable $\delta=m+n+2 p-2 k-2 \mu-2 \sigma$ is introduced</p> <p>$$\sum _{n=0}^{\infty } \sum _{m=0}^{\infty } \sum _{p=0}^M \sum _{k=0}^p \sum _{\sigma =0}^{p-k}\quad \sum _{\mu =0}^{\left\lfloor \frac{m+n}{2}+p-k-\sigma \right\rfloor } f(n,m,\mu ,p,k,\sigma ,m+n+2 p-2 k-2 \mu-2 \sigma )$$ $$= \sum _{\delta =0}^{\infty } \sum _{\mu =0}^{\infty } \sum _{p=0}^M \quad\sum _{\beta =0}^{\min \left(p,\left\lfloor \frac{\delta }{2}+\mu \right\rfloor \right)}\quad \sum _{n=0}^{2 (\mu -\beta )+\delta }\quad \sum _{k=0}^{p-\beta }\;\; f(n,\delta +2 \mu-2 \beta -n,\mu ,p,k,p-\beta -k,\delta )$$</p> <p><strong>Additional remarks:</strong> 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 $1$.</p> <p>I wonder whether there is a systematical and efficient technology, so I may be able to do those transformations automatically by programming.</p>