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How would one shift the summation index to combine like terms in this expression? I am finding this concept very confusing for some reason.

$\phi_{h}(z-z') + \sum^{\infty}_{n=0}[W^{n+1}_{TS}\phi_{h}(z+z'-2(n+1)h) + W^{n}_{TS}\phi_{h}(z+ z'+2(n+1)h)] + \sum^{\infty}_{n=1}W^n_{TS}[\phi_{h}(z-z'+2nh) + \phi_{h}(z-z'-2nh)].$

ConsistentC
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  • Could you just add and subtract the term for n=0 in the second summation; then combine the whole thing together? – M. McIlree May 01 '22 at 12:26
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    Or you can use $\sum_{n=1}^\infty f(n)=\sum_{n=0}^\infty f(n+1)$, which is valid for any function $f$. This seems particulary suited to your equation. – TonyK May 01 '22 at 12:30
  • @TonyK would that that still work though given the first term in the first summation has an additional $W_{TS}$ coefficient? – ConsistentC May 01 '22 at 15:54
  • As I said, my substitution is valid for any function $f$. Obviously. So you can use it to rephrase the second summation; the first summation can stay as it is. You are left with a single summation, with three terms. – TonyK May 01 '22 at 18:12

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