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Hermite-Padé approximants of generalized hypergeometric functions. (English. Russian original) Zbl 0849.11052
Russ. Acad. Sci., Sb., Math. 83, No. 1, 189-219 (1995); translation from Mat. Sb. 185, No. 10, 39-72 (1994).
Let $$\alpha_1, \alpha_2, \dots, \alpha_p$$, $$\beta_1, \dots, \beta_q$$ with $$p= q+1$$ be complex numbers which are neither 0 nor negative integers. Consider the family of contiguous hypergeometric functions \begin{aligned} f_0 (z) &= {}_pF_q \left( {{\alpha_1, \ldots, \alpha_p} \atop {\beta_1, \ldots, \beta_q}} \Biggl|z\right)= \sum^\infty_{n=0} {{(\alpha_1)_n \dots (\alpha_p)_n} \over {(\beta_1)_n \dots (\beta_q)_n}} \cdot {z^n \over n!}\\ f_i (z) &= {}_p F_q \left( {{\alpha_1 +1,\ldots, \alpha_i+1, \alpha_{i+1}, \ldots, \alpha_p} \atop {\beta_1 +1,\ldots, \beta_i+1, \beta_{i+1}, \ldots, \alpha_p}} \Biggr|z\right) \end{aligned} for $$1\leq i\leq q$$. In this paper, the author studies Hermite-Padé approximation of type I and II for the family $$f_1, f_2, \dots, f_q$$. He gives very explicit formulae for polynomials $$P_0 (z), \dots, P_q (z)\in \mathbb{C} [z]$$ satisfying the conditions $$\deg P_i (z)\leq n_i -1$$, $$i=0, 1, 2, \dots,q$$, $$\text{ord}_0 R(z)\geq \sigma= -1+ \sum^{q}_{i=0} n_i$$, where $$R(z)$$ is the remainder of the Hermite-Padé approximation, $$R(z)= \sum^{q}_{i=1} P_i(z) f_i(z)$$. These formulae generalize works of Riemann and Heine about exact expressions for denominator, numerator and remainder for the continued fraction of Gauss hypergeometric ratio.
As Riemann’s method, the author’s proof uses monodromy via the introduction of Meijer $$G$$-functions, which are analytic continuations of $$f_i (z)$$, $$1\leq i\leq q$$. For particular $$n_i$$, this paper gives perfect systems for the first and second kind approximations for systems $$f_i (z)$$, $$1\leq i\leq n$$, which permit the computations of a classical determinant, very important for applications in diophantine approximations.
A very interesting formula gives the recurrence satisfied by the sequences of functions linked with the remainder and polynomials which are of the type $$u_{n+p} (z)= a_n (z) u_{n+1} (z)+ b_n (z) u_n (z)$$ and generalizes the continued fraction of Gauss.

##### MSC:
 11J91 Transcendence theory of other special functions 41A21 Padé approximation 33C20 Generalized hypergeometric series, $${}_pF_q$$
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