## On the irrationality of $$\sum (1/(q^ n+r))$$.(English)Zbl 0718.11029

In “Old and new Problems and results in combinatorial number theory” [Enseign. Math. 28 (1980; Zbl 0434.10001)] P. Erdős and R. L. Graham claimed that the irrationality of $$\sum^{\infty}_{n=1}(2^ n-3)^{-1}$$ is unresolved. Using Padé approximation to the q-analogue of log, $L_ q(x)=\sum^{\infty}_{n=1}x/(q^ n-x),\quad | q| >1,\quad x\neq q^ m\quad \quad \quad (m\in {\mathbb{N}})$ it is proved: If q is an integer greater than one and r is a nonzero rational $$(r\neq -q^ m)$$ then $$\sum^{\infty}_{n=1}1/(q^ n+r)$$ is irrational and is not a Liouville number.

### MSC:

 11J72 Irrationality; linear independence over a field

Zbl 0434.10001
Full Text:

### References:

 [1] Askey, R.; Ismail, M.E.H., Recurrence relations, continued fractions and orthogonal polynomials, Mem. amer. math. soc., 49, No. 300, (1984) · Zbl 0548.33001 [2] Baker, G.A.; Graves-Morris, P., () [3] Borwein, J.M.; Borwein, P.B., () [4] Borwein, P.B., Rational approximations to Stieltjes transforms, Math. scand., 53, 114-124, (1983) · Zbl 0527.41010 [5] Borwein, P.B., Padé approximants for the q-elementary functions, Constr. approx., 4, 391-402, (1988) · Zbl 0685.41015 [6] Chudnovsky, D.V.; Chudnovsky, G.V., Padé and rational approximation to systems of functions and their arithmetic applications, () · Zbl 0536.10028 [7] Erdös, P., On arithmetical properties of Lambert series, J. Indian math. soc., 12, 63-66, (1948), (N.S.) · Zbl 0032.01701 [8] Erdős, P.; Graham, R.L., Old and new problems and results in combinatorial number theory, Enseign. math., No. 28, (1980) · Zbl 0434.10001 [9] Exton, M., () [10] Jackson, F.H., A basic sine and cosine with symbolic solutions of certain differential equations, Proc. Edinburgh math. soc., 22, 28-39, (1904) · JFM 35.0445.01 [11] Mahler, K., Zur approximation der exponentialfunktion und des logarithmus, J. reine angew. math., 166, 118-150, (1931) · JFM 57.0242.03 [12] Wallisser, R., Rationale approximation des q-analogons der exponentialfunktion und irrationalitätsaussagen für diese funktion, Arch. math., 44, 59-64, (1985) · Zbl 0558.33004 [13] Wynn, P., A general system of orthogonal polynomials, Quart. J. math., 18, 81-96, (1967) · Zbl 0185.30001
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.