User silvia - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T06:53:49Z http://mathoverflow.net/feeds/user/16598 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> http://mathoverflow.net/questions/70812/method-for-variable-substitution-in-multiple-summation Comment by Silvia Silvia 2011-07-21T09:18:55Z 2011-07-21T09:18:55Z link to the same question on math.stackexchange: <a href="http://math.stackexchange.com/questions/52846/method-for-variable-substitution-in-multiple-summation" rel="nofollow" title="method for variable substitution in multiple summation">math.stackexchange.com/questions/52846/&hellip;</a> http://mathoverflow.net/questions/70812/method-for-variable-substitution-in-multiple-summation Comment by Silvia Silvia 2011-07-21T09:17:43Z 2011-07-21T09:17:43Z @Will Jagy: Thanks for suggestion. I've posted it there. @Gerhard Paseman: Great! I read the Chapt.2 and found the interesting Iverson's convention, which led me to the wikipedia page for &quot;Iverson bracket&quot; and Knuth's note on arxiv (<a href="http://arxiv.org/abs/math/9205211" rel="nofollow">arxiv.org/abs/math/9205211</a>). http://mathoverflow.net/questions/70812/method-for-variable-substitution-in-multiple-summation Comment by Silvia Silvia 2011-07-21T05:25:09Z 2011-07-21T05:25:09Z @gowers Sorry for confusing, I added some additional explanations, hope it help making my question more clear. http://mathoverflow.net/questions/70812/method-for-variable-substitution-in-multiple-summation Comment by Silvia Silvia 2011-07-20T14:42:56Z 2011-07-20T14:42:56Z Any reference books and articles will be grateful.