# ( p , q )-Growth of an Entire GASHE Function and the Coefficient βn = |an|/Γ(n+μ+1)

Authors: Mohammed Harfaoui; Abdellah Mourassil; Loubna Lakhmaili
Journal: International Journal of Mathematics and Mathematical Sciences
 (2018)

---

## Abstract

The main purpose of this paper is to extend the work concerning the measures of growth of an entire function solution of the generalized axially symmetric Helmholtz equation∂2u/∂x2+∂2u/∂y2+(2μ/y)(∂u/∂y)=0,  μ>0, by studying the general measures of growth ((p,q)-order, lower (p,q)-order, (p,q)-type, and lower (p,q)-type) in terms of coefficients |an|/Γ(n+μ+1) and the ratios of these successive coefficients.


---

## Body

## 1. Introduction

The partial differential equation(1)∂2u∂x2+∂2u∂y2+2μy∂u∂y+k2u=0is called generalized axially symmetric Helmholtz equation (GASHE) and the solutions of (1) are called GASHE functions. The GASHE function u is regular about the origin and has the following Bessel-Gegenbauer series expansion:(2)ur,θ=Γ2μkr-μ∑n=0∞ann!Γ2μ+nJμ+nkrCnμcos⁡θ,where x=rcos⁡(θ) and y=rsin⁡(θ), Jμ+n are Bessel functions of first kind, Cnμ are Gegenbauer polynomials, and Γ(2μ)=2μ-1(μ-1)!.When series (2) converges absolutely and uniformly on the compact subsets of the whole complex plane, then the GASHE function u is said to be entire. For u being entire, it is known [1, page 214] that (3)lim⁡supr→∞⁡anΓn+μ+11/n=0.The concept of orderρ(f) and lower order λ(f) of an entire function was introduced by R. P. Boas [2] as follows:(4)ρfλf=limr→+∞supinf⁡log⁡log⁡Mr,flogrand the concept of type T(f) and lower type t(f) has been introduced to give more precise description of growth of entire functions when they have the same nonzero finite order. An entire function, of order ρ, 0<ρ<+∞, is said to be of type T(f) and lower type t(f) if(5)Tftf=limr→+∞supinflog⁡Mr,frρf,0<ρf<∞where M(r,f) = max0≤θ≤2π|f(r,θ)|.Gilbert and Howard [3] have studied the order ρ(u) of an entire GASHE function u in terms of the coefficients an occurring in the series expansion (2) of u. McCoy [4] studied the rapid growth of entire function solution of Helmholtz equation using the concept of index. Kumar [5, 6] extended and improved this result and studied the growth using the concept of index pair. Khan and Ali [7] studied the generalized order and type of entire GASHE function. Kumar and Singh [8] have studied the lower order and lower type of entire GASHE function in terms of the coefficients in its Bessel-Gegenbauer series expansion (2) when the order of u is a finite nonzero number. But, for the class of order ρ(u)=0 and ρ(u)=∞, we cannot define a type of u. For this reason, numerous attempts have been made to refine the concept of order and type. Therefore, the (p,q)-order and (p,q)-type of an entire function have been defined [9, 10]. In this paper, we extend the work of Kumar and Singh [8] to this new classification of entire function.For(p,q)≠(1,1) and p≥q≥1, we define the (p,q)-order and lower (p,q)-order as (6)ρp,q,uλp,q,u=limr→+∞supinflogp⁡Mr,ulogqr,where p and q are integers such that b≤ρ(p,q,u)≤∞ where b=0 if p>q and b=1 if p=q.The(p,q)-type and lower (p,q)-type are defined as (7)Tp,q,utp,q,u=limr→+∞supinflogp-1Mr,ulogq-1rρp,q,u,and log[0](x)=x and logm(x)=logm-1log⁡(x) for m≥1 and we use the notations(8)Pα=Pα,p,q=αifp>q1+αifp=q=2max⁡1,αif3≤p=q<∞∞ifp=q=∞,and(9)Mα=Mα,p,q=1e.αifp,q=2,1α-1α-1ααifp,q=2,21ifp≥3,Nα=Nα,p,q=1αifp=21ifp≠2.We note that the smallest integerp is 2 (p≥2) since, for example, the order is given by ρ=inf{μ>0:|f(z)|=O(e|z|μ),|z|→+∞}.To prove thatP(α)=P(β), M(α)=M(β), or N(α)=N(β) we can prove that α=β for the different values of p and q. From [9], we define the relation between (p,q)-order, lower (p,q)-order, the coefficients of u, and the ratios of these successive coefficients as follows.Theorem 1.
Letu(z)=∑n=0∞anzn be an entire function of (p,q)-order ρ(p,q,u), and then(10)ρp,q,u=PLp,qwhere(11)Lp,q=lim⁡supn→+∞⁡logp-1nlogq-1-1/nlog⁡anTheorem 2.
Letu(z)=∑n=0∞anzn be an entire function of (p,q)-order ρ(p,q,u), and then(12)ρp,q,u=PL∗p,qwhere(13)L∗p,q=lim⁡supn→∞⁡logp-1nlogqan/an+1Theorem 3.
Letu(z)=∑n=0∞anzn be an entire function of (p,q)-order ρ(p,q,u) and (an/an+1)n a nondecreasing function of n for n>n0 and then(14)λp,q,u=Plp,q,where(15)lp,q=lim⁡infn→+∞⁡logp-1nlogq-1-1/nlog⁡an.Theorem 4.
Letu(z)=∑n=0∞anzn be an entire function of (p,q)-order ρ(p,q,u) and (an/an+1) a nondecreasing function of n for n>n0 and then(16)λp,q,u=Pl∗p,q,where(17)l∗p,q=lim⁡infn→∞⁡logp-1nlogqan/an+1.From [10], we define the relation between (p,q)-type, lower (p,q)-type, the coefficients of u, and the ratios of these successive coefficients as follows.Theorem 5.
Letu(z)=∑n=0∞anzn be an entire function. The function f(z) is of (p,q)-order ρ(p,q,u) and (p,q)-type T(p,q,u) if and only if T=MV, where b=1 if p=q and b=0 if p>q, and V is defined as(18)Vp,q,u=lim⁡supn→∞⁡logp-2nlogq-2-1/nlog⁡anρ-C⁡with C=1 if (p,q)=(2,2) and C=0 if (p,q)≠(2,2).Theorem 6.
Letu(z)=∑n=0∞anzn be an entire function of (p,q)-order ρ and lower (p,q)-type t(p,q,u) and (an/an+1)n a nondecreasing function of n for n>n0 and then t=Mv, where(19)vp,q,u=lim⁡infn→∞⁡logp-2nlogq-2-1/nlog⁡anρ-Cwith C=1 if (p,q)=(2,2) and C=0 if (p,q)≠(2,2).Theorem 7.
Letu(z)=∑n=0∞anzn be an entire function of (p,q)-order ρ(p,q,u) and lower (p,q)-type t(p,q,u), and an/an+1 forms a nondecreasing function of k for k>k0; then(20)N∗R≤t≤T≤N∗Q,where(21)Qp,q=lim⁡supn→∞⁡logp-2nlogq-1an/an+1ρ-C,and(22)Rp,q=lim⁡infn→∞⁡logp-2nlogq-1an/an+1ρ-C,with C=1 if (p,q)=(2,2) and C=0 if (p,q)≠(2,2).


## 2. Auxiliary Results

Letf and g be two functions defined as(23)fz=∑n=0+∞ann!k2nzn,gz=∑n=0+∞ann!k2r∗nzn.According to [3], we know that if u is an entire GASHE function, then f and g are also entire functions of the complex variable z, and(24)Γμ+1/2k∗π22-μmr,g≤Mr,u≤kMr,fwhere m(r,g)=maxn≥0an/n!k/2r∗n and M(r,f)=maxz≤rfz. In this section, we shall prove some auxiliary results which will be used in the sequel.Lemma 8.
Letf and g be entire functions of particular form defined above. Then the (p,q)-orders and the (p,q)-types of f and g, respectively, are identical.Proof.
LetΦz=∑n=0∞anzn be an entire function, and then, according to Theorem 1, the (p,q)-order of ϕ is given as(25)ρp,q,ϕ=lim⁡supn→+∞⁡logp-1nlogq-1-1/nlog⁡an,and the (p,q)-type is defined in view of Theorem 5 as(26)Tp,q,ϕ=lim⁡supn→+∞⁡logp-2nlogq-2-1/nlog⁡anρp,q,ϕ.
In the consequence of [3], we have(27)ρ2,1,f=ρ2,1,g,T2,1,f=T2,1,g.
Here we consider the case whenq≥2.
We have(28)1ρ2,1,f=lim⁡infn→+∞⁡-1/nlog⁡k/2nan/n!log⁡n=lim⁡infn→+∞⁡log⁡an-1+log⁡n!-log⁡k/2nnlog⁡n=lim⁡infn→+∞⁡log⁡an-1nlog⁡n+log⁡n!nlog⁡n-logk/2nnlog⁡n=lim⁡infn→+∞⁡log⁡an-1nlog⁡n+1,and then 1/ρ2,1,f≥1 and ρ2,1,f≤1.
This implies that we will necessarily haveρ2,1,f=0 to define ρp,q,f. And we have (29)ρ2,1,f=0⇒lim⁡infn→+∞⁡1ρ2,1,f=+∞⇒lim⁡infn→+∞⁡log⁡an-1nlog⁡n=+∞⇒lim⁡supn→+∞⁡nlog⁡nlog⁡an-1=0⇒limn→+∞⁡log⁡n!log⁡an-1=0,an1/n→0.
Hence, for the functionf we have (30)1ρp,q,f=lim⁡infn→+∞⁡logq-1-1/nlog⁡k/2nan/n!logp-1⁡n=lim⁡infn→+∞⁡logq-1log⁡an-1/n+log⁡n!/n-1/nlog⁡k/2nlogp-1n=lim⁡infn→+∞⁡1logp-1n∗logq-1log⁡an-1n1+log⁡n!log⁡an-1-1log⁡an-1log⁡k2n=lim⁡infn→+∞⁡logq-1-1/nlog⁡anlogp-1n+log⁡1+o1logp-1n=lim⁡infn→+∞⁡logq-1-1/nlog⁡anlogp-1nand (31)1ρp,q,g=lim⁡infn→+∞⁡logq-1-1/nlog⁡k/2r∗nan/n!logp-1n=lim⁡infn→+∞⁡logq-1logan-1/n+logn!/n-1/nlog⁡k/2r∗nlogp-1n=lim⁡infn→+∞⁡logq-1-1/nlog⁡anlogp-1n+log⁡1+o1logp-1n=lim⁡infn→+∞⁡logq-1-1/nlog⁡anlogp-1nSince f and g have the same (p,q)-order it follows that (32)ρp,q,f=ρp,q,g=ρ.
Now we will prove thatf and g have the same (p,q)-type for q=2. (33)1Tp,2,f=lim⁡infn→+∞⁡-1/nlog⁡k/2nan/n!ρlogp-2n=lim⁡infn→+∞⁡-1/nlog⁡an+logn!/n-1/nlog⁡k/2nρlogp-2n=lim⁡infn→+∞⁡-1/nlog⁡anρlogp-2n∗1+log⁡n!log⁡an-1-1log⁡an-1log⁡k2nρ=lim⁡infn→+∞⁡-1/nlog⁡anρlogp-2n1+o1ρ=lim⁡infn→+∞⁡-1/nlog⁡anρlogp-2n.
In the same way we prove that(34)1Tp,2,g=lim⁡infn→+∞-1/nlog⁡anρlogp-2n.
Now, for the caseq≥3, we have (35)1Tp,q,f=lim⁡infn→+∞logq-2-1/nlog⁡k/2nan/n!ρlogp-2n=lim⁡infn→+∞⁡logq-2-1/nlog⁡an+log⁡n!/n-1/nlog⁡k/2nρlogp-2n=lim⁡infn→+∞1logp-2n·logq-2-log⁡ann1+log⁡n!log⁡an-1-log⁡k/2nlog⁡an-1ρ=lim⁡infn→+∞logq-2-1/nlog⁡an1+o1ρlogp-2n=lim⁡infn→+∞⁡logq-2-1/nlog⁡an+log⁡1+o1ρlogp-2n=lim⁡infn→+∞logq-2-1/nlog⁡anρlogp-2n.
The same is true for(36)1Tp,q,g=lim⁡infn→+∞logq-2-1/nlog⁡an/n!ρlogp-2nsince f and g have identical (p,q)-order and (p,q)-type.Lemma 9.
For an entire GASHE functionu of (p,q)-order ρ(p,q,u), lower (p,q)-order λ(p,q,u), (p,q)-type T(p,q,u), and lower (p,q)-type t(p,q,u). If f and g are entire functions as defined above, then(37)ρp,q,f=ρp,q,u=ρp,q,g,(38)λp,q,f≤λp,q,u≤λp,q,g,(39)Tp,q,f=Tp,q,u=Tp,q,g,(40)tp,q,f≤tp,q,u≤tp,q,g.Proof.
Using (24), we get (41)logpmr,glogqr≤logpMr,ulogqr≤logpMr,flogqr.
From the above relation we obtain(42)ρp,q,f≤ρp,q,u≤ρp,q,g,λp,q,f≤λp,q,u≤λp,q,g,and since ρ(p,q,f)=ρ(p,q,g) it proves (37) and (38).
Denoting byρ the common value of (p,q)-order of f, g, and u, we have from (24) (43)logp-1mr,glogq-1rρ≤logp-1Mr,ulogq-1rρ≤logp-1Mr,flogq-1rρ.
This proves (39) and (40).Before we start the next section, let us defineβn=an/Γn+μ+1, γn=an/n!k/2n, and δn=an/n!k/2r∗n.It is known, according to [3], that if βn/βn+1 is a nondecreasing function of n then Γn/Γn+1 and δn/δn+1 also is a nondecreasing function of n.


## 3. Main Results

Theorem 10.
Letu be an entire GASHE function of (p,q)-order ρ(p,q,u) and (p,q)-type T(p,q,u). If βn/βn+1 is a nondecreasing function of n for n>n0, then(44)ρp,q,u=lim⁡supn→∞⁡logp-1nlogq-1-1/nlog⁡βn,(45)Tp,q,u=lim⁡supn→∞logp-2nlogq-2-1/nlog⁡βnρ.Proof.
For an entire functionϕz=∑0∞anzn and according to Theorem 1, we have(46)ρp,q,ϕ=lim⁡supn→∞logp-1nlogq-1-1/nlog⁡an.
We know that ifβn/βn+1 is a nondecreasing function of n for n>n0, and then also γn/γn+1 and δn/δn+1.
Applying (46) to f, we get (47)ρp,q,f=lim⁡supn→∞⁡logp-1nlogq-1-1/nlog⁡an/n!k/2n=lim⁡supn→∞⁡logp-1n·logq-1-1nlog⁡ann!Γn+μ+1Γn+μ+1k2n-1=lim⁡supn→∞⁡logp-1n·logq-1-1nlog⁡βnn+1μk2n-1=lim⁡supn→∞⁡logp-1n·logq-1-1nlog⁡βn-log⁡n+1μk/2nn-1=lim⁡supn→∞⁡logp-1nlogq-1-1/nlog⁡βn+o1=lim⁡supn→∞⁡logp-1nlogq-1-1/nlog⁡βn.
Similarly, applying (46) to g, we prove (48)ρp,q,g=lim⁡supn→∞⁡logp-1nlogq-1-1/nlog⁡an/n!k/2r∗n=lim⁡supn→∞⁡logp-1nlogq-1-1/nlog⁡βn.
Then result (44) is found from the two relations ρ(p,q,f) and ρ(p,q,g) above and relation (37).
Letρ be the common (p,q)-order of f and g.
The(p,q)-type of ϕ is defined according to Theorem 5 as(49)Tp,q,ϕ=lim⁡supn→∞logp-2nlogq-2-1/nlog⁡anρp,q,ϕ,and we can easily prove that(50)Tp,q,f=lim⁡supn→∞logp-2nlogq-2-1/nlog⁡βnρ,and(51)Tp,q,g=lim⁡supn→∞logp-2nlogq-2-1/nlog⁡βnρ.
Equation (45) now follows in view of (37) and (39).
Hence the proof is completed.Theorem 11.
Letu be an entire GASHE function of (p,q)-order ρ(p,q,u), and βn/βn+1 is a nondecreasing function of n for n>n0. Then(52)ρp,q,u=lim⁡supn→∞logp-1nlogqβn/βn+1.Proof.
For an entire functionϕz=∑n=0∞anzn, according to Theorem 2,(53)ρp,q,ϕ=lim⁡supn→∞logp-1nlogqan/an+1,provided an/an+1 is a nondecreasing function of n for n>n0.
Applying this equation onf we get (54)ρp,q,f=lim⁡supn→∞logp-1nlogqγn/γn+1=lim⁡supn→∞logp-1nlogqan/an+1n+1!/n!2/k=lim⁡supn→∞logp-1n·logqan·Γn+μ+2·Γn+μ+1an+1·Γn+μ+1·Γn+μ+2n+12k-1=lim⁡supn→∞logp-1nlogqβn/βn+1Γn+μ+1/Γn+μ+2n+12/k.
SinceΓ(n+μ+1)/Γ(n+μ+2)(n+1)≈(n+1)/(n+μ+1)≈1 as n→∞ then (55)ρp,q,f=lim⁡supn→∞logp-1nlogqβn/βn+12/k=lim⁡supn→∞logp-1nlogqβn/βn+1.By the same way, we prove(56)ρp,q,g=lim⁡supn→∞logp-1nlogqβn/βn+1.Relation (52) now follows on using (37). Hence the proof is completed.Theorem 12.
Letu be an entire GASHE function of (p,q)-order ρ(p,q,u), lower (p,q)-order λ(p,q,u), and lower (p,q)-type t(p,q,u) and let βn/βn+1 be a nondecreasing function of n for n>n0. Then(57)λp,q,u=lim⁡infn→∞logp-1nlogq-1-1/nlog⁡βn,(58)tp,q,u=lim⁡infn→∞logp-2nlogq-2-1/nlog⁡βnρ.Proof.
For an entire functionϕz=∑0∞anzn, and according to Theorem 3,(59)λp,q,ϕ=lim⁡infn→∞logp-1nlogq-1-1/nlog⁡an.We know that if βn/βn+1 form a nondecreasing function of n for n>n0, then, also, γn/γn+1 and δn/δn+1.
Applying (59) to f, we get (60)λp,q,f=lim⁡infn→∞⁡logp-1nlogq-1-1/nlog⁡an/n!k/2n=lim⁡infn→∞⁡logp-1⁡n∗logq-1-1nloganΓn+μ+1Γn+μ+1n!k2n-1=lim⁡infn→∞⁡logp-1nlogq-1-1/nlog⁡βn+o1=lim⁡infn→∞⁡logp-1nlogq-1-1/nlog⁡βnSimilarly, applying (59) to g, we prove that (61)λp,q,g=lim⁡infn→∞⁡logp-1nlogq-1-1/nlog⁡an/n!k/2r∗n=lim⁡infn→∞⁡logp-1nlogq-1-1/nlog⁡βn.Then result (57) is now followed by (38) and above by the two relations for λ(p,q,f) and λ(p,q,g).
Ifϕz=∑0∞anzn is an entire function of (p,q)-order ρ(p,q,ϕ) and lower (p,q)-type t(p,q,ϕ), and if an/an+1 is a nondecreasing function of n for n>n0, then, according to Theorem 6, we have(62)tp,q,ϕ=lim⁡infn→∞logp-2nlogq-2-1/nlog⁡anρp,q,ϕ.
We can easily prove that(63)tp,q,f=lim⁡infn→∞logp-2nlogq-2-1/nlog⁡βnρp,q,f,and(64)tp,q,g=lim⁡infn→∞logp-2nlogq-2-1/nlog⁡βnρp,q,g.Equation (58) now follows in view of (37) and (40). Hence the proof is completed.Theorem 13.
Letu be an entire GASHE function of lower (p,q)-order λ(p,q,u), and let βn/βn+1 be a nondecreasing function of n for n>n0. Then(65)λp,q,u=lim⁡infn→∞logp-1nlogqβn/βn+1.Proof.
Letϕz=∑0∞anzn be an entire function and an/an+1 form a nondecreasing function of n for n>n0, and, according to Theorem 4, we have(66)λp,q,ϕ=lim⁡infn→∞logp-1nlogqan/an+1.Applying this on f and g, we can easily prove that(67)λp,q,f=lim⁡infn→∞logp-1nlogqβn/βn+1,and(68)λp,q,g=lim⁡infn→∞logp-1nlogqβn/βn+1.
Thus, we obtain relation (65) by using (38) and the two equalities above.Theorem 14.
Letu be an entire GASHE function of (p,q)-order ρ, (0<ρ<∞), (p,q)-type T(p,q,u), and lower (p,q)-type t(p,q,u). Then(69)lim⁡infn→+∞Nρ·logp-2nlogq-1βn/βn+1ρ≤tu,p,q≤Tu,p,q≤lim⁡supn→+∞Nρ·logp-2nlogq-1βn/βn+1ρ.Proof.
Ifϕz=∑n=0∞anzn is an entire function of (p,q)-type T(p,q,g) and lower (p,q)-type t(p,q,g), then in view of Theorem 7 we have(70)lim⁡infn→∞Nρ·logp-2nlogq-1an/an+1ρ-C≤t≤T.(71)T≤lim⁡supn→∞Nρ∗logp-2nlogq-1an/an+1ρ-C.
Applying inequality (71) to f(z)=∑0∞γnzn, we get (72)Tf≤lim⁡supn→∞Nρ·logp-2nlogq-1γn/γn+1ρ-C≤lim⁡supn→∞Nρ·logp-2nlogq-1an/an+1n+1!/n!2/kρ-C≤lim⁡supn→∞Nρ·logp-2nlogq-1,βn/βn+1ρ-C,and then(73)Tu=Tf≤lim⁡supn→∞Nρ·logp-2nlogq-1βn/βn+1ρ-Cand applying (70) to the function g(z)=∑0∞δnzn we get(74)lim⁡infn→+∞Nρ·logp-2nlogq-1δn/δn+1ρ≤tg≤tu,and we can easily prove(75)lim⁡infn→+∞logp-2nlogq-1δn/δn+1ρ=lim⁡infn→+∞logp-2nlogq-1βn/βn+1ρ,and, thus,(76)lim⁡infn→+∞Nρ·logp-2nlogq-1βn/βn+1ρp,q,f≤tu,and thus the proof is completed.



---

*Source: 7293160-2018-10-03.xml*