Niro Yanagihara

Wiman-Valiron method for difference equations

Let f(z) be an entire function of order less than 1/2. We consider an analogue of the Wiman-Valiron theory rewriting power series of f(z) into binomial series. As an application, it is shown that if a transcendental entire solution f(z) of a linear difference equation is of order χ 1/2, then we have log M (r, f) = Lr χ (1 + o(1)) with a constant L > 0.

On a nonlinear congruential pseudorandom number generator

A nonlinear congruential pseudorandom number generator with modulus M = 2 w is proposed, which may be viewed to comprise both linear as well as inversive congruential generators. The condition for it to generate sequences of maximal period length is obtained. It is akin to the inversive one and bears a remarkable resemblance to the latter.

Deficiency for meromorphic solutions of schröder equations

Eremenko and Sodin proved that meromorphic solution f (z) of the Schröder equation f (sz) = R (f (z)), |s| > 1, has no Valiron deficiency other than exceptional values of R(z). We consider transcendental meromorphic solutions of non-autonomous equation f (sz) =R (z, f (z)), |s| > 1. It is shown that there exists an equation of this form possessing a transcendental meromorphic solution, which has a Valiron deficiency other than a Nevanlinna deficiency. We also give some generalizations of the Eremenko and Sodin theorem for algebraic functions as targets.

The serial test for a nonlinear pseudorandom number generator

Let M = 2 ω , and G M = {1,3,...,M−1}. A sequence {y n }, y n ∈ G M , is obtained by the formula y n+1 = ay n + b + cy n mod M. The sequence {X n }, x n = y n /M, is a sequence of pseudorandom numbers of the maximal period length M/2 if and only if a + c = 1 (mod 4), b = 2 (mod 4). In this note, the uniformity is investigated by the 2-dimensional serial test for the sequence. We follow closely the method of papers by Eichenauer-Herrmann and Niederreiter.

Entire Functions of Small Order of Growth

Let f and F be transcendental entire functions. We are concerned with a growth estimate of F ∘ f when F satisfies the condition

Fine moduli spaces of infinite dimensional linear systems

\log M(r,F)=K(\log r)^p(1+O(1)),

Growth of meromorphic solutions of some functional equations I

where K is a positive constant and p > 1. It is shown that