## Hokkaido Mathematical Journal

- Hokkaido Math. J.
- Volume 42, Number 3 (2013), 357-383.

### On the order and hyper-order of meromorphic solutions of higher order linear differential equations

Maamar ANDASMAS and Benharrat BELAÏDI

#### Abstract

In this paper, we investigate the order of growth of solutions of the higher order linear differential equation

*f*^{(k)} + Σ^{k-1}_{j=0} (*h*_{j}*e*^{Pj(z)} + *d*_{j}) *f*^{(j)} = 0,

where *P*_{j}(*z*) (*j* = 0,1,…,*k*-1) are nonconstant polynomials such that deg *P*_{j} = *n* ≥ 1 and *h*_{j}(*z*), *d*_{j}(*z*) (*j* = 0,1,…,*k*-1) with *h*_{0} ≢ 0 are meromorphic functions of finite order such that max {ρ (*h*_{j}),ρ(*d*_{j}): *j* = 0,1,…,*k*-1} < *n*. We prove that every meromorphic solution *f* ≢ 0 of the above equation is of infinite order. Then, we use the exponent of convergence of zeros or the exponent of convergence of poles of solutions to obtain an estimation of the hyper-order of solutions.

#### Article information

**Source**

Hokkaido Math. J., Volume 42, Number 3 (2013), 357-383.

**Dates**

First available in Project Euclid: 12 November 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1384273387

**Digital Object Identifier**

doi:10.14492/hokmj/1384273387

**Mathematical Reviews number (MathSciNet)**

MR3137390

**Zentralblatt MATH identifier**

1291.34149

**Subjects**

Primary: 34M10: Oscillation, growth of solutions 30D35: Distribution of values, Nevanlinna theory

**Keywords**

Linear differential equations Meromorphic solutions Order of growth Hyper-order

#### Citation

ANDASMAS, Maamar; BELAÏDI, Benharrat. On the order and hyper-order of meromorphic solutions of higher order linear differential equations. Hokkaido Math. J. 42 (2013), no. 3, 357--383. doi:10.14492/hokmj/1384273387. https://projecteuclid.org/euclid.hokmj/1384273387