Irrationality Results for Values of Generalized Tschakaloff Series, Original Research Article

Arithmetical properties of values of the entire functionTq(x)=∑∞n=0 xn/q(1/2) n(n+1), whereqis 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 ofTq(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 Mahlerʹs 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).