Irrationality Results for Values of Generalized Tschakaloff Series

✍ Scribed by Masaaki Amou; Masanori Katsurada

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 137 KB
• Volume: 77
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1999.2376

✦ Synopsis

Arithmetical properties of values of the entire function T q (x)= n=0 x n Âq (1Â2) n(n+1) , where q is a parameter, | q | > 1, were first studied by L. Tschakaloff (1921, Math. Ann. 80, 62 74; 84, 100 114). In this paper we introduce a generalization of T q (x), given by (1.3), and prove the irrationality results for the values of (1.3) at rational points (see Theorem and Corollaries at the end of Section 1). One of the essential tools in the proof is a variant of Mahler's transcendence method, due to J. H. Loxton and A. J. van der Poorten (1977, in ``Transcendence Theory: Advances and Applications,'' pp. 211 226, Academic Press, San Diego).