# On the Growth Order and Growth Type of Entire Functions of Several Complex Matrices

Authors: M. Abul-Ez; H. Abd-Elmageed; M. Hidan; M. Abdalla
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we establish an explicit relation between the growth of the maximum modulus and the Taylor coefficients of entire functions in several complex matrix variables (FSCMVs) in hyperspherical regions. The obtained formulas enable us to compute the growth order and the growth type of some higher dimensional generalizations of the exponential, trigonometric, and some special FSCMVs which are analytic in some extended hyperspherical domains. Furthermore, a result concerning linear substitution of the mode of increase of FSCMVs is given.


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## Body

## 1. Introduction

The study of the asymptotic mode of increase behavior of entire functions in one and several complex variables is one of the classical central topics in complex analysis. Basic tools to study the mode of increase behavior of holomorphic functions are entities such as growth orders, the growth type, the maximum term, and the central index. Several developments made in this direction started with the early works of Lindelöf [1], Pringsheim [2], Valiron [3], Shah [4, 5], and Gol’dberg [6]. Their results turned out to be very useful in the study of partial differential equations and elsewhere. Generalization to higher dimensions has been given by many other authors (see, e.g., [7–10]) where they have contributed to the study of the order and type of entire functions of several complex variables. As Clifford analysis offers another possibility of generalizing complex function theory to higher dimensions, many authors introduced a study on the mode of increase of entire monogenic functions (see [11–16]). In recent years, the theory of matrix functions has grown significantly, with new applications appearing and the literature expanding at the last rate (see [12, 17]). From this point of view, many authors have dealt with different problems concerning functions in several complex matrix variables (see [18–20]). However, a number of central questions concerning the asymptotic growth behavior of functions in several complex matrix variables are still open. Recently, Kishka et al. [18] obtained the order and type of entire functions of two complex matrices in complete Reinhardt domains. The present paper is devoted to investigate the mode of increase of entire functions in several complex matrix variables in hyperspherical regions independent of the scalar entire functions of several complex variables associated with them (e.g., [16, 17]). The obtained results are applicable by illustrative examples. We end our study with an interesting result on linear substitution of entire functions in several complex matrix variables in hyperspherical regions.Following Nassif [21], we give some notations and associated properties in the framework of several complex variables.Assumem=m1,m2,…,mk and n=n1,n2,…,nk belonging to ℕ0k are k-dimensional multi-indices, and(1)m=m1+m2+⋯+mk,n=n1+n2+⋯+nk,zm=z1m1⋅z2m2⋅⋯⋅zkmk,tm=t1m1⋅t2m2⋅⋯⋅tkmk,0=0,0,…,0,where z1,z2,…,zk∈ℂ and t1,t2,…,tk are nonnegative numbers. For the k-dimensional space, we write z=z1,z2,…,zk and t=t1,t2,…,tk, and for r>0, we set(2)Sr=z∈ℂk:z<r,S¯r=z∈ℂk:z≤r,where Sr is an open spherical region of radius r and S¯r is its closure.Consider the functionfz, which is regular in S¯r; then, (see [19, 21])(3)fz=∑m=0∞amzm.The maximum modulus offz is denoted by(4)Mf;S¯r=supz∈S¯rfz.Note that (2) leads to(5)zs≤rts:t=1;s=1,2,…,k⊂S¯r.Cauchy’s inequality for functions of several complex variables was introduced by Nassif in the form (see [21])(6)am≤σmMf;S¯rrm,where(7)σm=inft=11tm=mm/2∏s=1kmsms/2and 1≤σm≤km on the assumption that msms/2=1, whenever ms=0.The numberσm in (7) is considered to be a generalization to the number σh,k in the two-complex variable case (cf. [22]), where(8)σh,k=h+kh+k/2hh/2kk/2,h,k≥1,0,hork=0.Now, theradius of convergence of the power series (3) is defined in the open sphere Sr by(9)Rf=1lim supm⟶∞am/σm1/m.Then, the functionfz is an entire function if Rf=∞. The growth order of the function fz is given in [21] in the form(10)ϱ=lim supr⟶∞ln lnMf;S¯rlnr=lim supm⟶∞mlnmlnσm/am.If0<ϱ<∞, we can prove using the same way as in the single complex variable case (see [23–25]) that the growth type θ of fz can be given in the form(11)θ=lim supr⟶∞lnMf;S¯rrϱ=1eϱlim supm⟶∞mamσmϱ/m.


## 2. Functions of Several Complex Matrix Variables (FSCMVs)

We shall cite some preliminaries and notations which are essential to establish our main results (see [20]).


### 2.1. Preliminaries and Notations

LetℳNℂ be the space of N×N matrices whose entries are complex numbers. Let A∈ℳNℂ,A=auv,u,v=1,2,…,N. The multiplication law of matrices takes the form(12)A2uv=∑s=1NAusAsv,and in general,(13)Aμuv=∑j1,j2,…,jμ−1Auj1Aj1j2…Ajμ−2jμ−1Ajμ−1v,μ∈ℕ,where the summation includes all symbols jv independently, from 1 to N. Following [20], suppose that A,B∈ℳNℂ,A=auv,B=buvu,v=1,2,…,N. We shall use the following notation:(14)Auv=Auv,which means that a matrix A whose each of its elements have been taken to be moduli of the elements.If a matrixB has positive elements which are greater than the elements of the matrix A, we have A<B. In other words, this inequality is equivalent to the following system of N×N inequalities:(15)Auv<Buv,u,v=1,2,…,N.Also referring to [20], the notation d shall mean that a matrix in which all its elements are equal to the number d and determine its positive integral powers as follows:(16)d2uv=dd+dd+⋯+dd=Nd2,u,v=1,2,…,N.Thus,d2=Nd2, and generally for positive integral powers we have(17)dμ=Nμ−1dμ,μ∈ℕ.


### 2.2. Convergence Property of FSCMVs

In the light of Section2.1, we discuss convergence property of a power series of several complex matrix variables in hyperspherical regions by the convergence of a power series of several complex variables without any restrictions on the coefficients.LetX=xs;ij;s=1,2,…,k be commutative matrices in ℳNℂ, and then the function ℱX of several complex matrices can be written as a power series in the form(18)ℱX=∑n=0∞an1,n2,…,nk,X1n1,X2n2,…,Xknk,=∑n=0∞anXn=∑n=0∞anZ.SinceZ=Zij1≤i,j≤N and Z∈ℳNℂ, Z may enjoy the same notation given in Section 2.1.Thus, we write(19)Z=X1n1,X2n2,…,Xknk,Zij=∑s1,s2,…,sk−1X1n1is1,X2n2s1s2,…,Xknksk−1j=∑s1,s2,…,sk−1∑j11,j21,…,jn1−11x1;ij11,x1;j11j21,…,x1;jn1−11s1,∑j12,j22,…,jn2−12x2;s1j12,x2;j12j22,…,x2;jn2−12s2,⋮∑j1k,j2k,...,jnk−1kxk;sk−1j1kxk;j1kj2k…xk;jnk−1kj.Hence,ℱX=fij1≤i,j≤N, where(20)fij=∑n=0anZij,i,j=1,2,…,N.Let us investigate the convergence of series (20) in a domain which is a subset of the space ℳNℂ determined by the following inequalities:(21)X1<rt1,X2<rt2,…,Xk<rtk.For this purpose, let(22)fij=∑n=0anZij,i,j=1,2,…,N.The convergence of this series guarantees the convergence of series (20), and in this case, series (20) will be absolutely convergent.To show the sufficient condition for the absolute convergence of series (20), suppose that the scalar function Fz=∑n=0anzn of several complex variables associated with the matrix function in (18) is an analytic function in the region S¯NR, where NR is the radius of convergence of this series, N is the common order of our matrices, and R is a positive number. Since(23)Fz=∑nanzn,MF;S¯NR=supz∈S¯NRFz,using similar procedure in deriving (6), we can deduce the following inequality for the coefficients of series (23), taking into account the common order of matrices N. Thus,(24)an≤σnMF;S¯NRNRn,where(25)σn=inft=11tn=nn/2∏s=1knsns/2.We obviously have(26)X1n1<Nn1−1rt1n1,X2n2<Nn2−1rt2n2,…,Xknk<Nnk−1rtknk.Thus, using (24) and (26) in (22), it follows that(27)fij=∑n=0anZij≤∑nan∑s1,s2,…,sk−1X1n1is1,X2n2s1s2,…,Xknksk−1j≤∑n=0anNn1−1+n2−1+⋯+nk−1+k−1rntn=MN∑n=0rRn=MN∑μ=0∞rRμ=M/N1−rR,R>r,that is, the power series in (18) will be absolutely convergent. We thus have the following theorem.Theorem 1.
If the radius of convergence of the series in (23) is equal to NR, then series (18) will be absolutely convergent for all matrices situated in the neighborhood of domain (21).


## 2.1. Preliminaries and Notations

LetℳNℂ be the space of N×N matrices whose entries are complex numbers. Let A∈ℳNℂ,A=auv,u,v=1,2,…,N. The multiplication law of matrices takes the form(12)A2uv=∑s=1NAusAsv,and in general,(13)Aμuv=∑j1,j2,…,jμ−1Auj1Aj1j2…Ajμ−2jμ−1Ajμ−1v,μ∈ℕ,where the summation includes all symbols jv independently, from 1 to N. Following [20], suppose that A,B∈ℳNℂ,A=auv,B=buvu,v=1,2,…,N. We shall use the following notation:(14)Auv=Auv,which means that a matrix A whose each of its elements have been taken to be moduli of the elements.If a matrixB has positive elements which are greater than the elements of the matrix A, we have A<B. In other words, this inequality is equivalent to the following system of N×N inequalities:(15)Auv<Buv,u,v=1,2,…,N.Also referring to [20], the notation d shall mean that a matrix in which all its elements are equal to the number d and determine its positive integral powers as follows:(16)d2uv=dd+dd+⋯+dd=Nd2,u,v=1,2,…,N.Thus,d2=Nd2, and generally for positive integral powers we have(17)dμ=Nμ−1dμ,μ∈ℕ.


## 2.2. Convergence Property of FSCMVs

In the light of Section2.1, we discuss convergence property of a power series of several complex matrix variables in hyperspherical regions by the convergence of a power series of several complex variables without any restrictions on the coefficients.LetX=xs;ij;s=1,2,…,k be commutative matrices in ℳNℂ, and then the function ℱX of several complex matrices can be written as a power series in the form(18)ℱX=∑n=0∞an1,n2,…,nk,X1n1,X2n2,…,Xknk,=∑n=0∞anXn=∑n=0∞anZ.SinceZ=Zij1≤i,j≤N and Z∈ℳNℂ, Z may enjoy the same notation given in Section 2.1.Thus, we write(19)Z=X1n1,X2n2,…,Xknk,Zij=∑s1,s2,…,sk−1X1n1is1,X2n2s1s2,…,Xknksk−1j=∑s1,s2,…,sk−1∑j11,j21,…,jn1−11x1;ij11,x1;j11j21,…,x1;jn1−11s1,∑j12,j22,…,jn2−12x2;s1j12,x2;j12j22,…,x2;jn2−12s2,⋮∑j1k,j2k,...,jnk−1kxk;sk−1j1kxk;j1kj2k…xk;jnk−1kj.Hence,ℱX=fij1≤i,j≤N, where(20)fij=∑n=0anZij,i,j=1,2,…,N.Let us investigate the convergence of series (20) in a domain which is a subset of the space ℳNℂ determined by the following inequalities:(21)X1<rt1,X2<rt2,…,Xk<rtk.For this purpose, let(22)fij=∑n=0anZij,i,j=1,2,…,N.The convergence of this series guarantees the convergence of series (20), and in this case, series (20) will be absolutely convergent.To show the sufficient condition for the absolute convergence of series (20), suppose that the scalar function Fz=∑n=0anzn of several complex variables associated with the matrix function in (18) is an analytic function in the region S¯NR, where NR is the radius of convergence of this series, N is the common order of our matrices, and R is a positive number. Since(23)Fz=∑nanzn,MF;S¯NR=supz∈S¯NRFz,using similar procedure in deriving (6), we can deduce the following inequality for the coefficients of series (23), taking into account the common order of matrices N. Thus,(24)an≤σnMF;S¯NRNRn,where(25)σn=inft=11tn=nn/2∏s=1knsns/2.We obviously have(26)X1n1<Nn1−1rt1n1,X2n2<Nn2−1rt2n2,…,Xknk<Nnk−1rtknk.Thus, using (24) and (26) in (22), it follows that(27)fij=∑n=0anZij≤∑nan∑s1,s2,…,sk−1X1n1is1,X2n2s1s2,…,Xknksk−1j≤∑n=0anNn1−1+n2−1+⋯+nk−1+k−1rntn=MN∑n=0rRn=MN∑μ=0∞rRμ=M/N1−rR,R>r,that is, the power series in (18) will be absolutely convergent. We thus have the following theorem.Theorem 1.
If the radius of convergence of the series in (23) is equal to NR, then series (18) will be absolutely convergent for all matrices situated in the neighborhood of domain (21).


## 3. Mode of Increase of FSCMVs

In this section, we prove two main results which provide the mode of increase of the entire FSCMVs. These results then enable us to compute the growth order and the growth type of some entire FSCMVs.LetℱX be an entire function of several complex matrix variables of common order N with Taylor expansion:(28)ℱX=∑n=0∞anXn,and the maximum modulus(29)Mℱ;S¯R=maxi,jmaxX1<rt1…Xk<rtkℱX.Therefore, Cauchy’s inequality for the matrix functionℱX can be given in the form(30)an≤NMℱ;S¯rrNnσn.For the description of the asymptotic growth behavior of the maximum modulus of entire matrix functions, we write the following.Definition 1.
LetℱX be an entire function of several complex matrices. Then, the order of growth of the maximum modulus of an entire function of several complex matrices is described by(31)ρℱ=lim supr⟶∞ln lnMℱ;S¯rlnr.
Now, we are going to prove our main result which provides us with a generalization to the context of entire FSCMVs of the famous theorem of Lindelöf and Pringsheim in terms of the relation between the growth order and the Taylor coefficients.Theorem 2.
For an entire function in several complex matrix variables with a Taylor series representation of form (28), let(32)Ω=lim supn⟶∞nlnn−lnan/Kn,Kn=σnNn.
Then,ρℱ=Ω.Proof.
We first show thatρℱ≥Ω. For Ω=0, this inequality is trivial. So, let us assume that 0<Ω≤+∞ in what follows. In view of (32), there exist infinitely many n∈ℕ0k with(33)nlnn≥−blnanKn,where b is a real constant to be chosen such that b=Ω−ε>0 with an ε>0 if Ω<∞. In the case where Ω=∞, one can take for b any arbitrary positive real number. We then have(34)lnanKn≥−nblnn.
The coefficients of a matrix Taylor series (28) satisfy Cauchy’s inequality in the form(35)an≤NKnr−nMℱ;S¯r.
Thus, we may infer that(36)lnMℱ;S¯r≥lnanKnrnN=lnanKn+nlnrN1/n.
From relation (33), we find that(37)lnMℱ;S¯r≥nlnrN1/n−nblnn.
Now, letr=en1/b in (37), and we get for this arbitrarily large r the following inequality:(38)lnMℱ;S¯r≥nlnen1/bN1/n−nblnn=nb1−lnNb/n=rbbe1−lnNbe/rb.
This permits us to conclude that(39)ρℱ=limsupr⟶∞lnlnMℱ;S¯rlnr≥b=Ω−ε,for arbitrary ε>0. For ε⟶0, one gets ρℱ≥Ω.
For the caseΩ≥ρℱ we see the following.
In the case whereΩ=+∞, there is nothing to prove. So, let us assume without loss of generality that 0≤Ω<∞. Since ℱX is an entire matrix function, we have that limn⟶∞an=0. Because of this property and in view of (32), one can find that, for all ε>0, κ∈ℕ such that for all multi-indices n with n≥κ,(40)0≤nlnn−lnan/Kn≤Ω+ε.
For all these multi-indices withn≥κ, it holds that(41)anKn≤n−n/Ω+ε.
Therefore,(42)Mℱ;S¯r≤maxi,jmaxS¯r∑n=0anXn≤1N∑n=0∞Nrnanσn,that is,(43)Mℱ;S¯r≤1N∑n=0κ+∑n=κ+1∞Nrnanσn≤1Nℒ1+∑n=κ+1∞rΩ+εnn/Ω+ε,where ℒ1 is a positive real constant. Choose the number r0>1 such that(44)2r0Ω+ε>μforr>r0,μ∈ℕ,and then fix the positive integer ε such that(45)κ<ε≤2r0Ω+ε<ε+1;r>r0,Mℱ;S¯r≤1Nℒ1+∑n=κ+1εrΩ+εnn/Ω+ε+∑n=ε+1∞rΩ+εnn/Ω+ε.
Thus,(46)∑n=κ+1εrΩ+εnn/Ω+ε<rε∑n=κε1κ+1n/Ω+ε<rε∑n=0∞1κ+1n/Ω+ε<r2r0Ω+ε1−1κ+11/Ω+ε−2=ℒ2r2r0Ω+ε,∑n=ε+1∞rΩ+εnn/Ω+ε≤∑n=ε+1∞rΩ+εε+1n/Ω+ε<∑n=0∞12n/Ω+ε=ℒ3.
Summarizing, we have obtained(47)Mℱ;S¯r≤ℒ4exp2rΩ+εlnr,where ℒ2,ℒ3, and ℒ4 are constants. Making r⟶∞ such that ρℱ≤Ω+ε for ε tends to zero, and then we arrive at the desired estimate ρℱ≤Ω, and the theorem is hereby deduced.
To get a finer classification of the growth behavior within the set of entire matrix functions that have the same growth order, one further introduces the growth type of an entire matrix functions as follows.Definition 2.
For any entire function of several complex matricesℱX of order ρ0<ρ<∞, the growth type τ is given by(48)τ=lim supr⟶∞lnMℱ;S¯rrρ.
The following is the second main result which provides us with a generalization of the famous Pringsheim–Lindelöf result on the relation between the growth type and the Taylor coefficients (see, e.g., [1, 2]) in the framework of matrix function theory.Theorem 3.
IfℱX is an entire matrix function of finite growth order ρ0<ρ<∞ and growth type τ, then(49)τ=Nρeρlim supn⟶∞nanσnρ/n.
To prove this theorem, we need the following lemma.Lemma 1.
LetℱX be a function in several complex matrix variables of common order N which have a Taylor series expansion in (28). Suppose there are numbers α>0andβ>0 and an integer γ=γα,β>0 such that an/σn<eαβ/Nαnn/α for all n>γ. Then, ℱX is an entire matrix function, and given any ε>0, there is a number R=Rε>0 such that(50)Mℱ;S¯r<expβ+εrαfor allr>R.Proof.
Sincean/σn<eαβ/Nαnn/α, for all these multi-indices with n>γ, it holds that(51)Nnanσn1/n<eαβn1/α.
Hence,Nnan/σn1/n⟶0,n⟶∞, and ℱX is an entire matrix function.Furthermore,(52)Nrnanσn1/n<eαβn1/αr<1/2,if multi-indices with n>κ=κr=2αeαβrα. Choosing R1=R1α,β>1, which is so large that κr>γ, and if r>R1, then(53)Nrnanσn<1/2n,provided multi-indices with n>κ.
We now establish an upper bound forMℱ;S¯r:(54)Mℱ;S¯r≤maxi,jmaxS¯r∑n=0∞anXn≤1N∑n=0∞anσnNrn=1N∑n=0κanσnNrn+1N∑n=κ+1∞anσnNrn<∑n=0κanσnNrn+∑n=κ+1∞12n<∑n=0κanσnNrn+1,ifr>R1.
However,(55)∑n=0κanσnNrn=∑n=0γanσnNrn+∑n=γ+1κanσnNrn<rγ∑n=0γanσnNn+κ−γmaxγ+1≤n≤κanσnNrn,maxγ+1≤n≤κanσnNrn≤maxγ+1≤nanσnNrn<maxγ+1≤neαβNαnn/αNrn≤max1≤neαβnn/αrn=expβrα.
The maximum is achieved for all these multi-indices withn=βαrα; hence,(56)maxγ+1≤n≤κanσnNrn<expβrα.
Therefore, ifr>R1,(57)Mℱ;S¯r<rγ∑n=0γanσnNn+κ−γexpβrα+1=rγ∑n=0γanσnNn+2αeαβrα−γexpβrα+1=expβrα2αeαβrα−γ+exp−βrα+rγexp−βrα∑n=0γanσnNn.
Given anyε>0, there is a number R=Rε>R1 such that the expression in brackets is less than expεrα provided that r>R. Hence,(58)Mℱ;S¯r<expβ+ϵrαfor allr>R.Now Lemma 1 can be used to determine Theorem 3 as follows.Proof of Theorem 3.2.
Supposeτ is finite. Then, given any π>τ, there is a number R=Rπ>0 such that Mℱ;S¯r<expπrρ, for r>R. According to Cauchy’s inequality in (30), we have(59)Nnanσn≤NMℱ;S¯rrn<Nexpπrρrn,for allr>R.
The minimum value ofexpπrρ/rn occurs for r=n/πρ1/ρ; thus,(60)Nnanσn<Neρπnn/ρ,for all these multi-indices with n>γ and r=n/πρ1/ρ>Rπ.
Rewriting,(61)Nρ/nπ>Nρeρnanσnρ/n.
Hence,(62)π≥Nρeρlim supn⟶∞nanσnρ/n.
Sinceπ is an arbitrary number exceeding τ,(63)τ≥Nρeρlim sup⊏n⊐⟶∞nanσnρ/n,where the right-hand side is clearly finite. Now let π1 be any number exceeding the right-hand side of (49). Then, there is a number γ=γπ1>0 such that(64)Nnanσn<eρπ1nn/ρ,for alln>γ.
Applying Lemma 1 withβ=π1 and α=ρ, given such that(65)Mℱ;S¯r<expπ1+ϵrρ,for allr>R.
Therefore,τ≤π1 and because of the choice of π1,(66)τ≤Nρeρlim supn⟶∞nanσnρ/n.
Thus, the result is derived. Also, if the right-hand side of (49) is finite so is τ, and if τ is infinite, so is the right-hand side of (49).


### 3.1. Basic Applications

Theorem 2 and Theorem 3 allow us to evaluate the exact value of the growth order and the growth type for some elementary matrix generalizations of the classical exponential, trigonometric, and other special functions. To illustrate this, let us afford here some important examples.Example 1.
Let(67)ℱX≔eΣs=1kbsXs,s=1,2,…,k,be an entire function of several complex matrix variables X=X1,X2,…,Xk all of which are of common order N in S¯r; bs are positive numbers, and then the growth order ρℱ=1 and the growth type τ=kb12+b22+⋯+bk2.Proof.
Definition 1 and Definition 2 tell us that(68)ρ=lim supr⟶∞ln lnMℱ;S¯rlnr,τ=lim supr⟶∞lnMℱ;S¯rrρ.
Thus,(69)ℱX=eΣs=1kbsXs,s=1,2,…,k,Mℱ;S¯r=maxi,jmaxX∈S¯rℱX=maxi,jmaxX∈S¯reΣs=1kbsXs=ekrb12+b22+⋯+bk2.
This leads to(70)ρ=lim supr⟶∞lnkrb12+b22+⋯+bk2lnr=1,τ=lim supr⟶∞krb12+b22+⋯+bk2r=kb12+b22+⋯+bk2.Remark 1.
One could expect to get entire function of several complex matrices of growth orderpp∈ℕ and growth type kb12+b22+⋯+bk2p just by taking(71)ℱX=eΣs=1kbsXsp,s=1,2,…,k.Example 2.
The generalized cosine function of several complex matricesX=X1,X2,…,Xk, all of which are of common order N in S¯r, given by(72)cosX≔∑n=0∞−1n2n!X12n1,X22n2,…,Xk2nk,where 2n=2n1,2n2,…,2nk have growth order ρ=1 and growth type τ=Nk.Proof.
First, we note that the power series in (72) has indeed an infinite radius of convergence in S¯r; thus, the essential(73)1N∑2n=0∞12n!⋅Nr2nσ2n,converges for all 0≤r<∞ in S¯r.
Applying Theorem2 and Theorem 3 lead immediately to(74)ρ=lim supn⟶∞2nln2nlnσ2n/−1n/2n!N2n=lim supn⟶∞2nln2nlnσ2n+ln2n!−2nlnN.
According to Stirling’s formula in the form(75)n!∼nn+1/22πe−n,(see [13] and [14]) we get ρ=1, and if n is sufficiently large we have(76)2nk!k≤2n!≤2n!.
This leads to(77)τ=Nρeρlim supn⟶∞2n−1n/2n!σ2nρ/2n=Nelim supn⟶∞2n1/2n!σ2n1/2n=Nelim supn⟶∞eln2n−1/2nln2n!+lnσ2n=Nelim supn⟶∞eln2n−1/2nkln2n/k!+lnσ2n=Nk.Example 3.
Similarly, one can show that the generalized sine function of several complex matricesX=X1,X2,…,Xk all of which are of common order N in S¯r by(78)sinX=∑n=0∞−1n2n+1!X12n1+1,X22n2+1,…,Xk2nk+1,where 2n+1=2n1+1,2n2+1,…,2nk+1 also have growth order ρ=1 and growth type τ=Nk.Example 4.
We can use the same construction principle to get examples of the rational growth order and growth type. This elementary example is, for instance, the generalized Bessel matrix function of order 0 of several complex matricesX=X1,X2,…,Xk all of which are of common order N in S¯r defined by(79)J0X≔∑n=0∞1n!2X1n1,X2n2,…,Xknk,and then using Theorem 2 and Theorem 3, we have its growth order ρ=1/2 and growth type τ=0.Example 5.
According to Theorem2 and Theorem 3. All matrix power series(80)PX≔∑n=0NanXnhave growth order ρ=0 and growth type τ=∞.


## 3.1. Basic Applications

Theorem 2 and Theorem 3 allow us to evaluate the exact value of the growth order and the growth type for some elementary matrix generalizations of the classical exponential, trigonometric, and other special functions. To illustrate this, let us afford here some important examples.Example 1.
Let(67)ℱX≔eΣs=1kbsXs,s=1,2,…,k,be an entire function of several complex matrix variables X=X1,X2,…,Xk all of which are of common order N in S¯r; bs are positive numbers, and then the growth order ρℱ=1 and the growth type τ=kb12+b22+⋯+bk2.Proof.
Definition 1 and Definition 2 tell us that(68)ρ=lim supr⟶∞ln lnMℱ;S¯rlnr,τ=lim supr⟶∞lnMℱ;S¯rrρ.
Thus,(69)ℱX=eΣs=1kbsXs,s=1,2,…,k,Mℱ;S¯r=maxi,jmaxX∈S¯rℱX=maxi,jmaxX∈S¯reΣs=1kbsXs=ekrb12+b22+⋯+bk2.
This leads to(70)ρ=lim supr⟶∞lnkrb12+b22+⋯+bk2lnr=1,τ=lim supr⟶∞krb12+b22+⋯+bk2r=kb12+b22+⋯+bk2.Remark 1.
One could expect to get entire function of several complex matrices of growth orderpp∈ℕ and growth type kb12+b22+⋯+bk2p just by taking(71)ℱX=eΣs=1kbsXsp,s=1,2,…,k.Example 2.
The generalized cosine function of several complex matricesX=X1,X2,…,Xk, all of which are of common order N in S¯r, given by(72)cosX≔∑n=0∞−1n2n!X12n1,X22n2,…,Xk2nk,where 2n=2n1,2n2,…,2nk have growth order ρ=1 and growth type τ=Nk.Proof.
First, we note that the power series in (72) has indeed an infinite radius of convergence in S¯r; thus, the essential(73)1N∑2n=0∞12n!⋅Nr2nσ2n,converges for all 0≤r<∞ in S¯r.
Applying Theorem2 and Theorem 3 lead immediately to(74)ρ=lim supn⟶∞2nln2nlnσ2n/−1n/2n!N2n=lim supn⟶∞2nln2nlnσ2n+ln2n!−2nlnN.
According to Stirling’s formula in the form(75)n!∼nn+1/22πe−n,(see [13] and [14]) we get ρ=1, and if n is sufficiently large we have(76)2nk!k≤2n!≤2n!.
This leads to(77)τ=Nρeρlim supn⟶∞2n−1n/2n!σ2nρ/2n=Nelim supn⟶∞2n1/2n!σ2n1/2n=Nelim supn⟶∞eln2n−1/2nln2n!+lnσ2n=Nelim supn⟶∞eln2n−1/2nkln2n/k!+lnσ2n=Nk.Example 3.
Similarly, one can show that the generalized sine function of several complex matricesX=X1,X2,…,Xk all of which are of common order N in S¯r by(78)sinX=∑n=0∞−1n2n+1!X12n1+1,X22n2+1,…,Xk2nk+1,where 2n+1=2n1+1,2n2+1,…,2nk+1 also have growth order ρ=1 and growth type τ=Nk.Example 4.
We can use the same construction principle to get examples of the rational growth order and growth type. This elementary example is, for instance, the generalized Bessel matrix function of order 0 of several complex matricesX=X1,X2,…,Xk all of which are of common order N in S¯r defined by(79)J0X≔∑n=0∞1n!2X1n1,X2n2,…,Xknk,and then using Theorem 2 and Theorem 3, we have its growth order ρ=1/2 and growth type τ=0.Example 5.
According to Theorem2 and Theorem 3. All matrix power series(80)PX≔∑n=0NanXnhave growth order ρ=0 and growth type τ=∞.


## 4. Linear Substitution for FSCMs

Letℱ⋆X=ℱX+A be an entire matrix function of growth order ρ⋆ and growth type τ⋆, where A=as;ij,s=1,2,…,k,i,j=1,2,…,N are any constant numbers. It is required in this section to establish ρ⋆ and τ⋆ as follows.IfX=xs;ij,s=1,2,…,k, are several complex matrices situated in the neighbourhood of the origin given by (21), then(81)∑s=1kxs;ij+as;ij≤supt=1rk+2rt1+t2+⋯+tk+⋯+kθk=r+kθk=Rk;θ=supi,ja1;ij,a2;ij,…,ak;ij.The matricesX + A are situated in the neighbourhood of the origin given by(82)X+A<Rt,t=1,R=r+kθ.Thus,(83)Mℱ⋆;S¯r≤Mℱ;S¯r.Now, suppose thatε>0 is any number, then there is a positive number r1 such that, applying (48) and (83), one gets(84)Mℱ⋆;S¯r≤Mℱ;S¯r<eεRρ,forR>r1.That is,(85)ρ⋆=lim supr⟶∞ln lnMℱ⋆;S¯rlnr≤lim supr⟶∞lnϵr+kθρlnr=ρ.On the other hand, ifℱ⋆X−A=ℱX, then ρ≤ρ⋆, and thus ρ=ρ⋆.Considering the growth typeτ⋆, we see from (84) and (48) that(86)τ⋆=lim supr⟶∞lnMℱ⋆;S¯rrρ⋆≤lim supr⟶∞ϵr+kθρrρ=ϵ.Since ϵ can be chosen arbitrarily as near toτ as we please, we infer that τ⋆≤τ. In the usual way, if ℱ⋆X−A=ℱX, we infer that τ⋆≥τ, and hence τ⋆=τ. Thus, the following theorem is determined.Theorem 4.
The entire matrix functionℱ⋆X is of the same growth order and growth type as the matrix function ℱX.
Now, we consider the matrix functionℱ⋆bX+A,b≠0, and ℱX are an entire matrix function of growth order ρ and growth type τ. Then, we obtained the following result by modicum calculations based on Theorem 4 and relations (31) and (48).Theorem 5.
LetℱX be an entire matrix function of positive order ρ and type τ, and let b≠0 be the real constant. Then, the matrix function ℱ⋆bX+A is an entire matrix function of the growth order ρ and the growth type τbρ [26].



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