# zbMATH — the first resource for mathematics

Type II Hermite-Padé approximations of generalized hypergeometric series. (English) Zbl 1236.41017
The author studies simultaneous Hermite-Padé approximation to generalized hypergeometric and $$q$$-hypergeometric series of the form $F(t)=\sum_{n=0}^{\infty}{\prod_{k=0}^{n-1}P(k)\over\prod_{k=0}^{n-1}Q(k)}t^n,\;F_q(t)=\sum_{n=0}^{\infty}{\prod_{k=0}^{n-1}P(q^k)\over\prod_{k=0}^{n-1}Q(q^k)}t^n,$ where $$P$$ and $$Q$$ are polynomials.
The main method used to prove the results is based on a paper by W. Maier [J. f. M. 156, 93–148 (1927; JFM 53.0340.02)] and a recent modification introduced by the author [J. Math. Soc. Japan 61, No. 1, 291–213 (2009; Zbl 1169.11031)].
The interest in the type of results given, originates from the study of simultaneous approximation in the context of algebraic independence and irrationality in the Theory of Numbers.
After a short introduction there are four sections dedicated to
A.
Hypergeometric functions $$F$$, with its main result the explicit form of the type II Hermite-Padé approximants in the variable $$t$$ for the $$d$$ series $$\theta^bF(t),0\leq b\leq d-1$$ at $$m$$ points (for $$d=1$$ the theorem gives the type II simultaneous approximants). NB. $$\theta=t {d\over dt}$$.
Furthermore, several applications are given (classical hypergeometric seris, exponential series - these are actually $${}_1F_1(1;a;z)$$ series - , logarithmic and polylogarithmic series)
B.
The remainder series technique (using the modified Maier technique), leading to diagonal Hermite-Padé approximants with a free parameter.
C.
$$q$$-hypergeometric functions $$F_q$$ as given above, giving a new proof of a theorem due to Th. Stihl [Math. Ann. 268, 21–41 (1984; Zbl 0519.10024)].
D.
A new proof (using the modified Stihl-Maier method) for a result due to the author in a previous paper [Zbl 1169.11031].
The paper concludes with an appendix with some results on Stirling numbers (needed for the proofs) and a list of 18 references.

##### MSC:
 41A21 Padé approximation 41A28 Simultaneous approximation 33C20 Generalized hypergeometric series, $${}_pF_q$$ 33D99 Basic hypergeometric functions
Full Text:
##### References:
 [1] Amou, M., Matala-aho, T., Väänänen, K.: On Siegel–Shidlovskii’s theory for q-difference equations. Acta Arith. 127, 309–335 (2007) · Zbl 1113.11042 · doi:10.4064/aa127-4-2 [2] Andrews, G., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999) · Zbl 0920.33001 [3] Baker, G.A., Graves-Morris, P.: Padé Approximants, 2nd edn. Encyclopedia of Mathematics and its Applications, vol. 59. Cambridge University Press, Cambridge (1996) · Zbl 0923.41001 [4] de Bruin, M.G.: Some convergence results in simultaneous rational approximation to the set of hypergeometric functions $$$\backslash$${{}_{1}F_{1}(1;c_{i};z)$$\backslash$$}\^{n}_{i=1}$ . In: Padé Approximation and Its Applications (Bad Honnef, 1983). Lecture Notes in Math., vol. 1071, pp. 12–33. Springer, Berlin (1984) [5] de Bruin, M.G.: Some explicit formulae in simultaneous Padé approximation. Linear Algebra Appl. 63, 271–281 (1984) · Zbl 0556.65006 · doi:10.1016/0024-3795(84)90149-6 [6] de Bruin, M.G.: Simultaneous rational approximation to some q-hypergeometric functions. In: Nonlinear Numerical Methods and Rational Approximation (Wilrijk, 1987). Math. Appl., vol. 43, pp. 135–142. Reidel, Dordrecht (1988) [7] de Bruin, M.G., Driver, K.A., Lubinsky, D.S.: Convergence of simultaneous Hermite–Padé approximants to the n-tuple of q-hypergeometric series $_{1}$$\backslash$$varPhi_{1}({1,1$$\backslash$$choose c,$$\backslash$$gamma_{j}};z)$$\backslash$$}\^{n}_{j=1}$ . Numer. Algorithms 3(1–4), 185–192 (1992) · Zbl 0786.33010 · doi:10.1007/BF02141927 [8] Chudnovsky, G.V.: Padé approximations to the generalized hypergeometric functions I. J. Math. Pures Appl. 58, 445–476 (1979) · Zbl 0434.10023 [9] Hata, M., Huttner, M.: Padé approximation to the logarithmic derivative of the Gauss hypergeometric function. In: Analytic Number Theory (Beijing/Kyoto, 1999). Dev. Math., vol. 6, pp. 157–172. Kluwer Acad., Dordrecht (2002) · Zbl 1114.33002 [10] Hermite, Ch.: Sur la fonction exponentielle. C. R. Acad. Sci. 77, 18–24, 74–79, 226–233, 285–293 (1873) · JFM 05.0248.01 [11] Huttner, M.: Constructible sets of linear differential equations and effective rational approximations of polylogarithmic functions. Isr. J. Math. 153, 1–43 (2006) · Zbl 1143.34057 · doi:10.1007/BF02771777 [12] Maier, W.: Potenzreihen irrationalen Grenzwertes. J. Reine Angew. Math. 156, 93–148 (1927) · JFM 53.0340.02 [13] Matala-aho, T.: On q-analogues of divergent and exponential series. J. Math. Soc. Jpn. 61, 291–313 (2009) · Zbl 1169.11031 · doi:10.2969/jmsj/06110291 [14] Prévost, M.: A new proof of the irrationality of $$\zeta$$(2) and $$\zeta$$(3) using Padé approximants. J. Comput. Appl. Math. 67, 219–235 (1996) · Zbl 0855.11037 · doi:10.1016/0377-0427(95)00019-4 [15] Prévost, M., Rivoal, T.: Remainder Padé approximants for the exponential function. Constr. Approx. 25, 109–123 (2007) · Zbl 1102.41016 · doi:10.1007/s00365-006-0635-6 [16] Rhin, G., Toffin, Ph.: Approximants de Padé simultanés de logarithmes. J. Number Theory 24, 284–297 (1986) · Zbl 0596.10033 · doi:10.1016/0022-314X(86)90036-3 [17] Stihl, Th.: Arithmetische Eigenschaften spezieller Heinescher Reihen. Math. Ann. 268, 21–41 (1984) · Zbl 0533.10031 · doi:10.1007/BF01463871 [18] Zudilin, W.: Ramanujan-type formulas and irrationality measures of some multiples of $$\pi$$. Mat. Sb. 196(7), 51–66 (2005). (Russian. Russian summary) translation in Sb. Math. 196(7–8), 983–998 (2005) · doi:10.4213/sm1376
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.