On Diophantine approximations of the solutions of q-functional equations

Abstract

Given a sequence of linear forms Rn = Pn;1 ¬ 1 + ¢ ¢ ¢ + Pn;m ¬ m ; Pn;1 ; : : : ; Pn;m 2 K; n 2 N; in m > 2 complex or p-adic numbers ¬ 1 ; : : : ; ¬ m 2 Kv with appropriate growth conditions, Nesterenko proved a lower bound for the dimension d of the vector space K¬ 1 + ¢ ¢ ¢ + K¬ m over K, when K = Q and v is the in¯nite place. We shall generalize Nesterenko' s dimension estimate over number¯elds K with appropriate places v, if the lower bound condition for jRn j is replaced by the determinant condition. For the q-series approximations also a linear independence measure is given for the d linearly independent numbers. As an application we prove that the initial values F (t); F (qt); : : : ; F (q m ¡ 1 t) of the linear homogeneous q-functional equationwhere N = N (q; t), Pi = Pi (q; t) 2 K[q; t] (i = 1; : : : ; m), generate a vector space of dimension d > 2 over K under some conditions for the coe± cient polynomials, the solution F (t) and t; q 2 K ¤ .

On Diophantine approximations of Ramanujan type q-continued fractions

“…Note that we obtained an improvement to [13] in the dependence on the first parameter of our upper bound term A. [14].)…”

“…Now, we apply Theorems 3.3 and 4.1 from [13] in the case (1). In the case (2), the proof uses a construction based on Thue-Siegel's lemma and again the iterates of Eq.…”

On irrationality measures of ∑l=0∞dl/∏j=1lMatala-aho1 2008Journal of Number TheorySelf Cite

“…Then there exists a polynomial P n (z) such that Q n (z)F (z) − P n (z) = R n (z), (9) where P n (z), Q n (z) ∈ Z[c, q, z] and deg z {P n (z), Q n (z)} n.…”

“…For other closely related results and transcendence questions of q-series we refer to the works [2,3,9,10,13,16,18]. …”

On Approximation Measures of certain q-continued fractions

“…(28) Now, we apply Theorems 3.3 and 4.1 from [5] in the case 1). In the case 2), the proof uses a construction based on Thue-Siegel's lemma and again the iterates of the equation (23), see [1].…”