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January, 2009 On $q$ -analoques of divergent and exponential series
J. Math. Soc. Japan 61(1): 291-313 (January, 2009). DOI: 10.2969/jmsj/06110291


We shall consider linear independence measures for the values of the functions $D_a(z)$ and $E_a(z)$ given below, which can be regarded as $q$-analogues of Euler's divergent series and the usual exponential series. For the $q$-exponential function $E_q(z)$, our main result (Theorem 1) asserts the linear independence (over any number field) of the values 1 and $E_q(\alpha_j)(j = 1,...,m)$ together with its measure having the exponent $\mu = O(m)$, which sharpens the known exponent $\mu = O(m^2)$ obtained by a certain refined version of Siegel's lemma (cf. [$\mathbf{1}$]). Let p be a prime number. Then Theorem 1 further implies the linear independence of the $p$- adic numbers $\prod_{n=0}^\infty(1 + kp^n), (k = 0,1,...,p-1)$, over $\boldsymbol{Q}$ with its measure having the exponent $\mu < 2p$. Our proof is based on a modification of Maier's method which allows to construct explicit Padé type approximations (of the second kind) for certain $q$-hypergeometric series.


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Tapani MATALA-AHO. "On $q$ -analoques of divergent and exponential series." J. Math. Soc. Japan 61 (1) 291 - 313, January, 2009.


Published: January, 2009
First available in Project Euclid: 9 February 2009

zbMATH: 1169.11031
MathSciNet: MR2272880
Digital Object Identifier: 10.2969/jmsj/06110291

Primary: 11J82
Secondary: 11J72 , 33D15

Keywords: $q$-series , linear independence measure , Padé type approximation

Rights: Copyright © 2009 Mathematical Society of Japan


Vol.61 • No. 1 • January, 2009
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