منابع مشابه
Little q-Legendre polynomials and irrationality of certain Lambert series
Certain q-analogs hp(1) of the harmonic series, with p = 1/q an integer greater than one, were shown to be irrational by Erdős [9]. In 1991–1992 Peter Borwein [4] [5] used Padé approximation and complex analysis to prove the irrationality of these q-harmonic series and of q-analogs lnp(2) of the natural logarithm of 2. Recently Amdeberhan and Zeilberger [1] used the qEKHAD symbolic package to f...
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In this paper a new direct proof for the irrationality of Euler's number e = ∞ k=0 1 k! is presented. Furthermore, formulas for the base b digits are given which, however, are not computably effective. Finally we generalize our method and give a simple criterium for some fast converging series representing irrational numbers.
متن کاملIrrationality proof of certain Lambert series using little q-Jacobi polynomials
We apply the Padé technique to find rational approximations to h±(q1, q2) = ∞ ∑ k=1 q 1 1± q 2 , 0 < q1, q2 < 1, q1 ∈ Q, q2 = 1/p2, p2 ∈ N \ {1}. A separate section is dedicated to the special case qi = q ri , ri ∈ N, q = 1/p, p ∈ N \ {1}. In this construction we make use of little q-Jacobi polynomials. Our rational approximations are good enough to prove the irrationality of h(q1, q2) and give...
متن کاملCAN THE IRRATIONALITY OF CERTAIN SERIES By
is irrational for every t >2 . By more complicated arguments one can show that the condition (ni, ni)=1 is superfluous. We do not give the details since I do not think that the condition E 1 /n i < oo is very relevant . In fact it could be replaced i=1 by a weaker but more complicated condition . I would expect that the series (2) is always irrational if n 7, jl-nk -~'co (perhaps even n k./ k->...
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During my long life I have spent lots of time thinking about irrationality of series . The reader with a little maliciousness may say "spent and wasted" since I have never discovered any new general methods nor had any spectacular success like Apéry. Nevertheless I hope to convince the reader that not all of it was completely wasted . First of all, I state some of my previous results several of...
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ژورنال
عنوان ژورنال: Tokyo Journal of Mathematics
سال: 2004
ISSN: 0387-3870
DOI: 10.3836/tjm/1244208475