A Liouville-type estimate is proved for the irrationality measure of the quantities n=1; q; n; (1 q n; 2q (2) = (1 q n with q 2 Z n f0; 1g. The proof is based on the application of a q-analogue of the arithmetic method developed by Chudnovsky, Rukhadze, and Hata and of the transformation group for hypergeometric series -- the group-structure approach introduced by Rhin and Viola.

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. Furthermore we give an upper bound for the measure of irrationality.

In this paper we give analogues of the Ramanujan functions and nonlinear dierential equations for them. Investigating a modular structure of solutions for nonlinear dierential systems, we deduce new identities between the Ramanujan and hypergeometric functions. Another result of this paper is a solution of transcendence problems concerning nonlinear systems. In 1916 S. Ramanujan has proved [12] that the functions P (q) = 1 24 1 X n=1 1 (n)q n ; Q(q) = 1 + 240 1 X n=1 3 (n)q n ; R(q) = 1 504 1 X n=1 5 (n)q n ; (1) where k (n) = P djn d k , satisfy the system of nonlinear dierential equations q dP dq = 1 12 (P 2 Q); q dQ dq = 1 3 (PQ R); q dR dq = 1 2 (PR Q 2 ) (2) (see also [5, Chapter X, Sect. 5]). Note that Q and R are modular as functions of = 1 2i log q.

4> )(1 + q 2n ) 2 (where q 1=4 () = exp( i 4 )), and further = 2 3 4 = 16q Y n1 1 + q 2n 1 + q 2n 1 8 ; which function is related to the classic modular function j by j = 2 8 ( 2 + 1) 3 ( 2 ) 2 : Ramanujan's function can be dened by = 2 3 4 2 8 = q 2 Y<F7.344

In this work, we give a purely analytic introduction to the phenomenon of mirror symmetry for quintic threefolds via classical hypergeometric functions and dierential equations for them. Starting with a modular map and recent transcendence results for its values, we regard a mirror map z(q) as a concept generalizing the modular one. We give an alternative approach demonstrating the existence of non-linear dierential equations for the mirror map, and exploit both an elegant construction of Klemm{Lian{Roan{Yau and the Ax theorem to prove that the Yukawa coupling K(q) does not satisfy any algebraic dierential equation of order less than 7 with coecients from C (q). It is a classical question of transcendence number theory to investigate linear and algebraic independence of values of analytic functions satisfying both arithmetic conditions and functional (for instance, dierential) equations. This story has many dramatic and romantic episodes (like the solution of the 7th Hilbert problem), but it is far from an end. Skipping results about the transcendence of one-variable modular functions and their values, we concentrate on what has been done in 1996. The Mahler conjecture about the transcendence of at least one among the numbers q 2 C , 0 < jqj < 1, and J(q), where J(e 2i ) is a modular invariant, has been proved in [BDGP]. Yu. Nesterenko has generalized this result using the Ramanujan functions and dierential equations for them; in [Ne] he has proved that at least three among the numbers q, J(q), q J(q), and 2 q J(q), for q 2 C such that 0 < jqj < 1 and J(q) = 2 f0; 1728g, are algebraically independent over Q , where q = q d dq . It can be a subject of an independent paper to overview consequences of the results of [BDGP] and [Ne]; moreover, we believe t...