# Differential finiteness of power series with hypergeometric coefficients

Let $f$ be a formal (or convergent near the origin, as you wish) power series with variables $x_1$, ... , $x_n$ and complex coefficients. $$f = \sum_I f_I \mathbf x^I$$

This power series is said to be differentially finite (or D-finite) if the $\mathbb C(\mathbf x)$-vector-space generated by all the derivatives $\partial^I f / \partial \mathbf x^I$ is finite dimensional.

Assume that $f_I$ is a product of integer power of factorial of integer valued linear forms of I (!) That is to say, there exists $\lambda_1$, ... , $\lambda_m$ linear forms over $\mathbb Q^n$ with integer coefficients, and $p_1$, ..., $p_m$, integers (possibly negative) such that $$f_I = \prod_{k=1}^m \lambda_k(I)!^{p_k}.$$ In particular, if $I' = I + (0, \dotsc, 0,1,0,\dotsc )$, then $f_{I'}/f_I$ is a rational function of $I$.

With such an hypothesis, can we conclude that the power series $f$ is D-finite ?

Of course, we can translate the fact that $f_{I'}/f_I$ is an explicit rational function of $I$ into a first order recurrence with polynomial coefficients, and this recurrence can be translated into a differential equation for $f$. But I have no proofs that these differential equations are enough to get the D-finiteness of $f$. Experimentally, a Gröbner basis computation shows that they are enough.

It might be related to GKZ hypergeometric systems, as presented in the book Gröbner deformation of hypergeometric differential equations, but so far I haven't been able to apply this theory. Any reference, idea, or solution is welcome !

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Yes! In the multivariate setting, such a series is called proper hypergeometric by several authors. The main ingredient in the proof is the fact that multivariable D-finite functions are closed under Hadamard product (taking products of coefficients termwise), and is proved in L. Lipshitz, D-finite power series, J. Algebra 122 (1989) 353-373.

This result allows you to reduce to the case of series with factorial coefficients or inverse factorial coefficients, which can then be dealt with separately. Anyway, you will find full proofs in the book A=B by Petkovsek, Wilf and Zeilberger or Zeilberger's "A holonomic systems approach to special functions identities", J. Comput. Appl. Math. 32 (1990) 321-368.

There was a conjecture that the only D-finite hypergeometric series are the proper ones, and a proof of the correct statement was given in "On the structure of multivariate hypergeometric terms" by Abramov and Petkovsek, Adv. in Appl. Math, 29 (2002) 386–411.

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Great answer, thank you ! –  Lierre May 4 '12 at 10:02