Algebraic independence of certain entire functions of two variables generated by linear recurrences

Algebraic independence of reciprocal sums of certain Fibonacci-type numbers

The paper studies algebraic independence of certain reciprocal sums of Fibonacci and Lucas sequences. Also more general binary recurrences are considered. The main tool is Mahler's method reducing the investigation of the algebraic independence of function values to the one of functions if these satisfy certain types of functional equations.

Parameterized Telescoping Proves Algebraic Independence of Sums

Usually creative telescoping is used to derive recurrences for sums. In this article we show that the non-existence of a creative telescoping solution, and more generally, of a parameterized telescoping solution, proves algebraic independence of certain types of sums. Combining this fact with summation-theory shows transcendence of whole classes of sums. Moreover, this result throws new light on the question why, e.g., Zeilberger's algorithm fails to find a recurrence with minimal order.

On the Statistical Independence of Shift-Register Pseudorandom Multisequence over Part of the Period

In this paper we construct a pseudorandom multisequence $(x_{n_1,...,n_r})$ based on $k$th-order linear recurrences modulo $p$, such that the discrepancy of the $s$-dimensional multisequence $(x_{n_1+i_1,...,n_r+i_r})_{1 \leq i_j \leq s_j, 1 \leq j \leq r}$ $1 \leq n_j \leq N_j, 1 \leq j \leq r$ is equal to $O((N_1 ... N_r)^{-1/2} \ln^{s+3r}(N_1 ... N_r))$, where $s=s_1 ... s_r$, for all $N_1,...,N_r$ with $1 < N_1 ... N_r \leq p^k

沿线性递推数列的数字追加游走研究

On digit-appending walks along linear recurrences

该研究探讨基$b\boldsymbol{\u2265}2$下沿具单位圆外唯一单重主导根$\boldsymbol{\u03b1}$的整数线性递推数列的数字追加游走，证明$\boldsymbol{\u03b1}$与$b$乘法独立时，一致有界追加数字长度的无限游走不存在，无界时则存在。

Algebraic independence of the partial derivatives of certain functions with arbitrary number of variables

We construct a complex entire function with arbitrary number of variables which has the following property: The infinite set consisting of all the values of all its partial derivatives of any orders at all algebraic points, including zero components, is algebraically independent. In Section 2 of this paper, we develop a new technique involving linear isomorphisms and infinite products to replace the algebraic independence of the values of functions in question with that of functions easier to deal with. In Section 2 and 3, using the technique together with Mahler's method, we can reduce the al…

k-广义佩尔数列中乘法独立性研究

Multiplicative independence in the sequence of $k$-generalized Pell numbers

本文研究k-广义佩尔数列中项之间的乘法依赖关系，确定了所有满足条件的(m,n)对，并证明只有非常小的k,m,n时才存在解。