On some non-holonomic sequences
A sequence \(u:\mathbb{N}\to\mathbb{C}\) is called holonomic if it satisfies a homogeneous linear recurrence \(p_0(n)u(n)+p_1(n)u(n+1)+\cdots+p_d(n)u(n+d)=0\), \(n\geq 0\), where \(p_0,p_1,\dots,p_d\in \mathbb{C}[x]\), \(p_d\neq 0\). A formal power series \(f(z)=\sum^\infty_{n:0}u_n(z)z^n\in \mathbb{C}z\) is holonomic if it satisfies a homogeneous linear ordinary differential equation \(p_0(z)f(z)+p_1(z)f'(z)+\cdots+p_d(z)f^{(d)}(z)=0\) with polynomial coefficients. It is well known that a power series is holonomic if and only if its coefficient sequence is. The fundamental result of the paper is Theorem 1: Let \(a_1,\dots,a_p\), \(b_1,\dots,b_q\) be pairwise distinct positive integers. Define the sequence \(\{h(n), n\in \mathbb{N}\}\) by \(h(n)=\frac{(a_1)_n\dots (a_p)_n}{(b_1)_n\dots (b_q)_n}\), \(n\in\mathbb{N}\), where \((c_n)\) denotes the rising factorial \((c_n)_n=\prod^n_{i:1}(c+i-1)\), and let \(r\in\mathbb{Q}\setminus \mathbb{Z}\). Then the sequence \(\{h(n)^r, n\neq\mathbb{N}\}\) is not holonomic. Modulo some open Schanuel's conjecture about an algebraically independent finite set of real numbers, the author exhibits concrete examples of non-holonomic sequences such as \(\{\text{Log\,}n; n\in\mathbb{N}^*\}\) and \(\{(a+n)^{bn},n\in\mathbb{N}\}\) where \(a,b\in\mathbb{Q}\), \(b\neq 0\).
scientific article; zbMATH DE number 2161276