# Haut PDF Finding all hypergeometric solutions of linear differential equations ### Finding all hypergeometric solutions of linear differential equations

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r] ### SOME PROPERTIES OF TWO LINEARLY INDEPENDENT MEROMORPHIC SOLUTIONS OF SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS

ρ = ρ > . Let d j ( )( z j = 1, 2 ) be meromorphic functions that are not all vanishing identically such that max { ρ p ( ) ( ) d 1 , ρ p d 2 } < ρ p ( ) B . If f and 1 f are two nontrivial linearly 2 independent meromorphic solutions whose poles are of uniformly bounded multiplicity of (1.2), then the polynomial of solutions w = d f 1 1 + d f 2 2 satisfies i w ( ) = + , p 1 ### Growth of solutions and oscillation of differential polynomials generated by some complex linear differential equations

Theorem 1.4 Let P (z), Q(z), A 1 (z), A 0 (z), F satisfy the hypotheses of Theorem 1.2. Let d 0 (z), d 1 (z), d 2 (z) be entire functions that are not all equal to zero with ρ(d j ) < n (j = 0, 1, 2), ϕ(z) is an entire function with finite order such that ψ(z) is not a solution of equation (1.3). If f (z) is a solution of (1.3), then the differential polynomial g f (z) = d 2 f 00 + d 1 f 0 + d 0 f satisfies λ(g f − ϕ) = ∞ with at most one finite order solution f 0 . ### Growth of certain combinations of entire solutions of higher order linear differential equations

(ii) Let f (z) be an entire function with ρ (f ) = α ≤ ∞, and let L (f) = a k f (k) + a k −1 f (k −1) + · · · + a 0 f, where a 0 , a 1 , · · · , a k are entire functions which are not all equal zero and satisfy b = max {ρ (a j ) : j = 0, · · · , k} < α. Then ρ (L (f)) = α. Lemma 2.2.  For any given equation of the form (3), there must exists a solution ### On the growth of solutions of some second order linear differential equations with entire coefficients

satisfies σ 2 (f ) 6 σ. Lemma 2.5 () Let f (z) be a transcendental meromorphic function, and let α > 1 be a given constant. Then there exist a set E 8 ⊂ (1, ∞) with finite logarithmic measure and a constant B > 0 that depends only on α and i, j (0 6 i < j 6 k), such that for all z satisfying |z| = r / ∈ [0, 1] ∪ E 8 , we have ### On the [p, q]-order of meromorphic solutions of linear differential equations

(ii) If ρ [p+1,q] (F ) < α, then all meromorphic solutions f whose poles are of uniformly bounded multiplicities of equation (1.2) satisfy λ [p+1,q] (f ) = λ [p+1,q] (f ) = ρ [p+1,q] (f ) = α with at most one exceptional solution f 0 satisfying ρ [p+1,q] (f 0 ) < α. The main purpose of this paper is to consider the growth of meromorphic solutions of equations (1.1) and (1.2) with meromorphic coefficients of finite [p, q]-order in the complex plane. We obtain the following results which generalize and improve Theorem 9 and Theorem 10.
En savoir plus ### On the hyper-order of solutions of a class of higher order linear differential equations

exp {( 1 − ε ) ( 1 − ρ ) δ ( P d , θ ) r n } ≤ 2k. (3.25) This is absurd. Hence σ ( f ) ≥ n. Assume f is a transcendental meromorphic solution whose poles are of uni- formly bounded multiplicity of equation ( 1.1 ) . By Lemma 2.1, there exist a con- stant B > 0 and a set E 1 ⊂ ( 1, + ∞ ) having finite logarithmic measure such that for all z satisfying | z | = r / ∈ [ 0, 1 ] ∪ E 1 , we have ### Iterated order of solutions of linear differential equations with entire coefficients

order. Let n ≥ 2 be an integer and consider the linear differential equation A n (z) f (n) + A n−1 (z) f (n−1) + ... + A 1 (z) f ′ + A 0 (z) f = 0. (1.12) It is well-known that if A n ≡ 1, then all solutions of this equation are entire functions but when A n is a nonconstant entire function, equation (1.12) can possess meromorphic solutions. For instance the equation ### On the hyper-order of transcendental meromorphic solutions of certain higher order linear differential equations

λ (d) = σ(d) = λ(1/f) ≤ σ < σ(f) = σ(g) = +∞. For each sufficiently large |z| = r, let z r = re iθ r be a point satisfying |g(z r )| = M(r, g). By Lemma 2.6, there exist a constant δ r (> 0), a sequence {r m } m∈N , r m → +∞ and a set E 5 of finite logarithmic measure such that the estimation (3.3) holds for all z satisfying |z| = r m ∈ E / 5 , r m → +∞ and arg z = θ ∈ [θ r − δ r , θ r + δ r ]. Since |g(z)| is continuous in |z| = r, then there exists a constant λ r (> 0) such that for all z satisfying |z| = r sufficiently large and arg z = θ ∈ [θ r − λ r , θ r + λ r ], we have
En savoir plus ### On the (p, q)-order of solutions of some complex linear differential equations

A k (z) f (k) + A k−1 (z) f (k−1) + · · · + A 1 (z) f 0 + A 0 (z) f = 0 (1.4) and A k (z) f (k) + A k−1 (z) f (k−1) + · · · + A 1 (z) f 0 + A 0 (z) f = F(z), (1.5) where k ≥ 2, A j (z) ( j = 0, 1, · · · , k) , F(z) are entire functions with A 0 A k F 6≡ 0, many authors investigated the properties of their solutions and obtained some interesting results, (see e.g. , , , ). It well-known that if A k (z) ≡ 1, then all solutions of (1.4) and (1.5) are entire functions, but when A k (z) is a nonconstant entire function, then equation (1.4) or (1.5) can possess meromorphic solutions. For instance the equation
En savoir plus ### Growth of solutions of linear differential equations in the unit disc

+ a 0 (z) f = 0 (1.1) is well understood when the coefficients a j are polynomials [16, 17, 22, 27] or of finite (iterated) order of growth in the complex plane [7, 14, 15, 21]. In particular, Wittich  showed that all solutions of (1.1) are entire functions of finite order if and only if all coefficients are polynomials, and Gundersen, Steinbart and Wang  listed all possible orders of solutions of (1.1) in terms of the degrees of the polynomial coefficients. ### On the order and hyper-order of meromorphic solutions of higher order linear differential equations

Theorem 1.4 Let P j (z) = P n i=0 a i,j z i (j = 0, 1, . . . , k − 1) be noncon- stant polynomials, where a 0,j , a 1,j , . . . , a n,j (j = 0, 1, . . . , k − 1) are com- plex numbers such that a n,j a n,0 6= 0 (j = 1, 2, . . . , k − 1), let h j (z), d j (z) (j = 0, 1, . . . , k − 1) be meromorphic functions with h 0 6≡ 0. Suppose that arg an,j 6= arg a n,0 (j = 1, 2, . . . , k − 1), arg(an,1 + an,j ) 6= arg an,0 (j = 2, 3) or a n,j = cj a n,0 (0 < cj < 1) (j = 1, 2, . . . , k − 1) and ρ = max{ρ(h j ), ρ(d j ) : j = 0, 1, . . . , k − 1} < n. Then for any meromorphic solution f 6≡ 0 of equation (1.4), f , f 0 , f 00 all have infinitely many fixed points and satisfy
En savoir plus ### On the hyper order of solutions of a class of higher order linear differential equations

where (d 5 > 0, d 6 > 0) are some constants. This is a contradiction. Therefore, when max {c 1 , ..., c s−1 } < c 0 , every solution f ≡ 0 of (1.7) has inﬁnite order. Now we prove that ρ 2 (f ) = n. Put c = max {cj : j = s} , then 0 < c < 1. Since deg Ps > deg (P j − c j P s) (j = s) , by Lemma 2.5, there exist real numbers b > 0, λ, R 2 and θ 1 < θ 2 such that for all r ≥ R 2 and θ 1 ≤ θ ≤ θ 2 , ### Some properties of solutions of certain complex linear differential equations with meromorphic doefficients

of Theorem A. Let d 0 , d 1 , d 2 be complex constants that are not all equal to zero.If f (z) 6≡ 0 is any meromorphic solution of equation (1.2), then: (i) f, f 0 , f 00 all have infinitely many fixed points and satisfy λ (f − z) = λ  f 0 − z  = λ  f 00 − z  = ∞, (ii) the differential polynomial ### Differential polynomials generated by meromorphic solutions of [p, q]-order to complex linear differential equations

(i) max{ρ [p,q] (A i ) : i = 1, 2, · · · , k − 1} < ρ [p,q] (A 0 ) = ρ (0 < ρ < +∞) or that (ii) max{ρ [p,q] (A i ) : i = 1, 2, · · · , k − 1} ≤ ρ [p,q] (A 0 ) = ρ (0 < ρ < +∞) and max{τ M,[p,q] (A i ) : ρ [p,q] (A i ) = ρ [p,q] (A 0 )} < τ M,[p,q] (A 0 ) = τ (0 < τ < +∞) . Let d j (z) (j = 0, 1, · · · , k) be finite [p, q] −order entire functions that are not all vanishing identically such that h 6≡ 0. If f 6≡ 0 is a solution of (2.1), then the differential polynomial (1.2) satisfies ### On the growth of solutions of some non-homogeneous linear differential equations

(ii) There exists a set E 4 ⊂ [0, 2π) which has linear measure zero, such that if θ ∈ [0, 2π) \ E 4 , then there is a constant R = R (θ) > 0 such that (2.1) holds for all z satisfying arg z = θ and |z| ≥ R. Lemma 2.4 (). Let f (z) be an entire function and suppose that G(z) := log ### Relation between small functions with differential polynomials generated by meromorphic solutions of higher order linear differential equations

f (k) + A (z) f = 0 (k ≥ 2) , where A is a meromorphic function in the complex plane. 1. Introduction and main results In this paper, we shall assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions see [10, 17]. For the definition of the iterated order of a meromorphic function, we use the same definition as in  , (, p. 317), (, p. 129). For all r ∈ R, we define exp 1 r := e r and exp p+1 r := exp exp p r  , p ∈ N. We also define for all r sufficiently large log 1 r := log r and log p+1 r := log log p r  , p ∈ N. Moreover, we denote by exp 0 r := r, log 0 r := r, log −1 r := exp 1 r and exp −1 r := log 1 r. Definition 1.1. [11, 12] Let f be a meromorphic function. Then the iterated p−order ρ p (f ) of f is defined as
En savoir plus ### Growth of solutions of complex linear differential equations with entire coefficients of finite iterated order

Then i λ (f ) = i λ (f ) = i (f ) = p + 1 and λ p+1 (f ) = λ p+1 (f ) = σ p+1 (f ). To avoid some problems caused by the exceptional set, we recall the following lemma. Lemma 2.9  Let g : [0, +∞) → R and h : [0, +∞) → R be monotone non- decreasing functions such that g (r) 6 h (r) for all r / ∈ E 5 ∪ [0, 1], where E 5 ⊂ (1, +∞) is a set of finite logarithmic measure. Let α > 1 be a given constant. Then there exists an r 0 = r 0 (α) > 0 such that g (r) 6 h (αr) for all r > r 0 . ### Oscillation and fixed points of derivatives of solutions of some linear differential equations

Theorem 3  Let k > 2 and A (z) be a transcendental meromorphic function of finite iterated order ρ p (A) = ρ > 0 such that δ (∞, A) = δ > 0. Suppose, moreover, that either: (i) all poles of f are of uniformly bounded multiplicity or that (ii) δ (∞, f) > 0. If ϕ (z) 6≡ 0 is a meromorphic function with finite iterated p−order ρ p (ϕ) < +∞, then every meromorphic solution f (z) 6≡ 0 of (1) satisfies ### Infinite order solutions of complex linear differential equations

Abstract. In this paper we investigate the growth of solutions of the differential equation f .k/ C A k 1 .´/ f .k 1/ C    C A 1 .´/ f 0 C A 0 .´/ f D 0; where A 0 .´/ ; : : : ; A k 1 .´/ are entire func- tions with 0 <  .A 0 /  1=2: We will show that if there exists a real constant ˇ <  .A 0 / and a set E ˇ  .1; C1/ with log dens E ˇ D 1; such that for all r 2 E ˇ ; we have min j´jDr jA j .´/ j  exp .r ˇ / .j D 1; 2; : : : ; k 1/ ; then every solution f 6 0 of the above differential equation is of infinite order with hyper-order  2 .f /   .A 0 /. The paper extends previous results by the author and Hamani.
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