Study of Solutions to Higher Linear Difference Equations with Meromorphic Coefficients of Finite [p, q]-phi Order

Abstract

This paper examined the growth of meromorphic solutions to higher-order linear equations of the form.

$$ \begin{aligned}

& A_k(z) f\left(z+h_k\right)+\cdots+A_1(z) f\left(z+h_1\right)+A _1(z) f(z)=0, \\

& A_k(z) f\left(z+h_k\right)+\cdots+A_1(z) f\left(z+h_1\right)+A_{1}(z) f(z)=F(z)

\end{aligned}  $$

where $A_k(z), \ldots, A_0(z)$ are entire function and $h_j (j=1, \ldots, k)$ are non-zero distinct complex numbers.

Our search extend some earlier results proved by Ghosh and Khan and Bandypadhyay, Bela\"{\i}di and Benkarouba, by using the notion of $(p, q)-\varphi$ order.