International audience
In this paper, the algebraic independence of values of the functionG d (z) := h≥0 z d h /(1 − z d h), d > 1 a fixed integer, at non-zero algebraic points in the unit disk is studied. Whereas the case of multiplicatively independent points has been resolved some time ago, a particularly interesting case of multiplicatively dependent points is considered here, and similar results are obtained for more general functions. The main tool is Mahler's method reducing the investigation of the algebraic independence of numbers (over Q) to the one of functions (over the rational function field) if these satisfy certain types of functional equations.
Abstract
In this paper, we investigate the general solution and the generalized stability for the quartic, cubic and additive functional equation (briefly, QCA-functional equation)
for any k∈ℤ-{0,±1} in Menger probabilistic normed spaces.
The aim of this paper is to give some new identities and relations related to
the some families of special numbers such as the Bernoulli numbers, the
Euler numbers, the Stirling numbers of the first and second kinds, the
central factorial numbers and also the numbers y1(n,k,?) and y2(n,k,?)
which are given Simsek [31]. Our method is related to the functional
equations of the generating functions and the fermionic and bosonic p-adic
Volkenborn integral on Zp. Finally, we give remarks and comments on our
results.
AbstractIn this paper, we obtain Hyers-Ulam stability of the functional equationsf (x + y, z + w) + f (x − y, z − w) = 2f (x, z) + 2f (y, w),f (x + y, z − w) + f (x − y, z + w) = 2f (x, z) + 2f (y, w)andf (x + y, z − w) + f (x − y, z + w) = 2f (x, z) − 2f (y, w)in 2-Banach spaces. The quadratic forms ax2 + bxy + cy2, ax2 + by2 and axy are solutions of the above functional equations, respectively.