Suppose we have a bivariate power series of the form $$\sum_{i}\sum_j a_{i,j} t^i s^j,$$ where for every fixed value of $i$ the corresponding univariate power series in $s$ is a rational function. Are there criteria to decide, when the whole (i.e., bivariate) series can be written as a rational function?
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$\begingroup$ Each rational function has zero order. This implies necessary conditions on coefficients $a_{i,j}$. $\endgroup$ – user64494 Nov 27 '18 at 11:53

$\begingroup$ Do you know to what extent the usual theory of univariate rational generating functions carries over to the case where the series coefficients are functions (say, univariate rational)? Do you still have a linerar recurrence relation? $\endgroup$ – Dima Pasechnik Nov 27 '18 at 21:01

$\begingroup$ @DimaPasechnik of course you have a finite linear recurrence relation (over any field, to say the least). $\endgroup$ – Fedor Petrov Nov 28 '18 at 13:21

$\begingroup$ I was referring to Thm 4.1.1 in Stanley's book wwwmath.mit.edu/~rstan/ec/ec1.pdf (some parts of the proof of it use the fact that $\mathbb{C}$ is algebraically closed). $\endgroup$ – Dima Pasechnik Nov 28 '18 at 13:42

$\begingroup$ The condition " for every fixed value o f$ i$ the corresponding univariate power series in $s$ is a rational function" is not sufficient to this end as the Maclaurin expansion of $\frac 1 {1\exp(t)s}$ shows. $\endgroup$ – user64494 Nov 29 '18 at 7:32