# Sums with arguments x_1...x_infinity

Posted 8 years ago
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 Dear community, I am trying to define and symbolically differentiate a function of the following form: $$G=\sum_{x_t,y_t=1}^T f(x_t,y_t)$$ Where $t$ denotes a certain period of time and $f (x_t, y_t)$ is some function such as $aln(x_t)+bln(y_t)$. Is there any way to implement this so that something like D[G,x2] or D[G,x(t+1)] would produce meaningful results?Note: I've tried to use the symbolize command as described by Bruce Miller in this thread: http://community.wolfram.com/groups/-/m/t/272608, but this only works for a finite number of arguments $x_1$ through $x_t$; I wish to implement this for a function with countably many x's.
 Let $t\in \mathbb{Q}$ to stay in the countable area. Then you consider $x[t]$ as variable different from $x[t']$ for $t\not=t'$? With other words, one cannot consider derivatives of $x[t]$ with respect to $t$ because $t$ is an index? This can be evaluated  In:= Clear[f] f[x_[t_], y_[t_]] := Sin[x[t]^2 + y[t]^3] + Cos[x[t]^3 - y[t]^2] In:= D[f[p[o], q[o]], p[o]] Out= 2 Cos[p[o]^2 + q[o]^3] p[o] - 3 p[o]^2 Sin[p[o]^3 - q[o]^2] but not In:= D[f[p[o], q[o]], p[2 o]] Out= 0 In:= D[f[p[o], q[o]], p[o + 1]] Out= 0 in the last two cases - I presume - you expect to get a part of $G$ differentiated where the variable $p[2 o]$ etc. appears once the $o$ is fixed. That you can only reach at - irony - if again $p[\xi[o]]$ is related to $p[o]$ - with other words, $p[\xi[o]]$ and $p[o]$ are not independent variables, but only $o$.