منابع مشابه
q-Apery Irrationality Proofs by q-WZ Pairs
w x Ž . In 1948, Paul Erdos E1 proved the irrationality of h 1 . Recently, ̋ 2 w x Peter Borwein used Pade approximation techniques B1 and some coḿ w x Ž . plex analysis methods B2 to prove the irrationality of both h 1 and q Ž . Ln 2 . Here we present a proof in the spirit of Apery’s magnificent proof ́ q Ž . w x of the irrationality of z 3 A , which was later delightfully accounted by w x Alf v...
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If somehow you have not yet seen that proof, please ask someone, for example me, to show it to you. (Or maybe look at [2]. Well, meanwhile you can also extract it as a special case (substitute a = b and then n = 1) of what appears below in Section 2.) I learned this proof from Yoram Sagher who had learned it from John Conway. While we can consider it merely as a geometric realization of one of ...
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Abstract. Motivated by the telescoping proofs of two identities of Andrews and Warnaar, we find that infinite q-shifted factorials can be incorporated into the implementation of the q-Zeilberger algorithm in the approach of Chen, Hou and Mu to prove nonterminating basic hypergeometric series identities. This observation enables us to extend the q-WZ method to identities on infinite series. As e...
متن کاملIrrationality of ζ q ( 1 ) and ζ q ( 2 ) ?
In this paper we show how one can obtain simultaneous rational approximants for ζq(1) and ζq(2) with a common denominator by means of Hermite-Padé approximation using multiple little q-Jacobi polynomials and we show that properties of these rational approximants prove that 1, ζq(1), ζq(2) are linearly independent over Q. In particular this implies that ζq(1) and ζq(2) are irrational. Furthermor...
متن کاملIRRATIONALITY MEASURES FOR CERTAIN q-MATHEMATICAL CONSTANTS
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ژورنال
عنوان ژورنال: Advances in Applied Mathematics
سال: 1998
ISSN: 0196-8858
DOI: 10.1006/aama.1997.0565