# Analytical Properties of the Generalized Heat Matrix Polynomials Associated with Fractional Calculus

Authors: Mohamed Abdalla; Salah Mahmoud Boulaaras
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

In this paper, we introduce a matrix version of the generalized heat polynomials. Some analytic properties of the generalized heat matrix polynomials are obtained including generating matrix functions, finite sums, and Laplace integral transforms. In addition, further properties are investigated using fractional calculus operators.


---

## Body

## 1. Overture

In the past few decades, matrix versions of the orthogonal polynomials have attracted a lot of research interest due to their close relations and various applications in many areas of mathematics, statistics, physics, and engendering, for example, see [1–11].The recent advances of fractional order calculus (FOC) are dominated by its multidisciplinary applications. Moreover, special functions of fractional order calculus have many applications in various areas of mathematical analysis, probability theory, control systems, and engineering (see, for example, [12–15]).Moreover, the development of fractional calculus associated with special matrix functions and polynomials has been investigated by many researchers, for example, the recent works [16–22].Among these classical polynomials are the heat polynomials (also designated as Temperature polynomials) that are polynomial solutions of the heat equation and also are particularly useful in solving the Cauchy problem (see [23–26]). Special functions, such as the confluent hypergeometric function, integral error functions, and Laguerre polynomials, have a close link with the generalized heat polynomials introduced [27–29]. Further, the generalized heat polynomials are mainly used to construct an approximate solution of a given problem as a linear combination of the polynomials. Such solution satisfies the governing equation and other equations (cf., e.g., [30–35]).In our investigation here, we define a generalized heat matrix polynomialℍℙmT;ξ,v. We then establish certain generating matrix functions, finite sum formulas, Laplace transforms, and fractional calculus operators for these polynomials in Sections 3, 4, 5, and 6, respectively. Further, some interesting special cases and concluding remarks of our main results are pointed out in Section 7.


## 2. Preliminaries

In this section, we give some basic definitions and terminologies; for more details, we can be referred to [36, 37].Here and through the work, letℂd×d be the vector space of all the square matrices of order d∈ℕ, (ℕ is the set of all positive integers) whose entries are in the set of complex numbers ℂ. For a E∈ℂd×d, let σE be the set of all eigenvalues of E which is called the spectrum of E. We have
(1)μE≔maxRξ:ξ∈σEandμ~E≔minRz:z∈σE,which imply μ~E=−μ−E. Here, μE is called the spectral abscissa of E, and the matrix E is said to be positive stable if μ~E>0. Further, let I and 0 denote the identity and zero matrices corresponding to a square matrix of any order, respectively.IfE is a positive stable matrix in ℂd×d, then the gamma matrix function ΓE is well defined as follows (cf., e.g., [11, 38, 39]):
(2)ΓE=∫0∞e−zzE−Idz,zE−I≔expE−Ilnz.Moreover, ifE is a matrix in ℂd×d which gratifies
(3)E+nIisinvertibleforeveryintegern∈ℕ0=ℕ∪0,then ΓE is invertible, its inverse coincides with Γ−1E. Under condition (3), we can write the following Pochhammer matrix symbol
(4)En=EE+I⋯E+n−1I=ΓE+nIΓ−1En∈ℕ.Letr,k∈ℕ0. Also let Sr and Qk be arrays of r commutative matrices S1,S2,⋯,Sr and k commutative matrices Q1,Q2,⋯,Qk in ℂd×d, respectively, such that Qk+nI are invertible for 1≤d≤k and all n∈ℕ0. Then, the generalized hypergeometric matrix function rFkSr;Qk;zz∈ℂ can be defined by (see, e.g., [11, 39])
(5)rFkSr;Qk;z=∑s=0∞∏j=1rSjs∏i=1kQis−1zss!.In particular, the hypergeometric matrix function2F1S,P;C;z≡FS,P;C;z is defined by
(6)FR,P;C;z=∑s=0∞RsPsCs−1zss!,for matrices R,P,C in ℂd×d such that C+nI is invertible for all n∈ℕ0. Also, note that for r=1,k=0 in (9), we have the Binomial type matrix function 1F0R;−;z as follows:
(7)1F0R;−;z=1−z−R=I+Rz+RR+Iz22!+⋯+Rnznn!+⋯,z<1.LetE be a positive stable invertible matrix in ℂd×d. Then, the nth Laguerre matrix polynomial is defined in the form (see, e.g., [11, 40])
(8)LnEz=∑m=0n−1mzmm!n−m!S+InE+Im−1=E+Inn!F11−nI;E+I;z,n∈N0.The Laplace transform offξ is defined by [7].
(9)Lfξ;λ=∫0∞e−λξfξdξ,Rλ>0,provided that the improper integral exists.Lemma 1.
(see [7]). Let S be a positive stable invertible matrix in ℂd×d. Then, we have
(10)LξS;λ=λ−S+IΓS+I,Reλ>0.


## 3. Generalized Heat Matrix Polynomial and Generating Matrix Functions

A generalized heat matrix polynomial is defined in (11) below; then, a family of generating matrix functions are proposed, see Theorem 4 and Theorem 8 of this section.Definition 2.
LetT be a positive stable matrix in the complex space Cd×d satisfying the spectral condition (3). Then, we define a generalized heat matrix polynomial of degree m∈ℕ0 in the following explicit form:
(11)ℍℙmT;ξ,v=∑s=0m22smsΓT+m+12IΓ−1T+m−s+12Iξ2m−2svs=m!4vmLmT−12I−ξ24v;v>0,where LmTξ is the Laguerre matrix polynomial in (8).Remark 3.
Note that(12)ℍℙmT;ξ,v=4vmT+12ImF11−mI,T+12I;−ξ24v,and that for the scalar case d=1, taking T=a and a>0, the mth polynomial ℍℙma;ξ,v coincides with the classical scalar generalized heat polynomial, see [24, 26, 33]. Further, the ordinary heat polynomial defined in [23], when T=0; ℍℙm0;ξ,v=υ2mξ,v.Theorem 4.
Letξ∈ℂ, v>0, m∈ℕ0, ξ2t<1−4vt and T and R be positive stable matrices in Cd×d such that T+nI is invertible for all n∈ℕ0.A generating matrix function of ℍℙmT;ξ,v is
(13)∑m=0∞T+12Im−1RmℍℙmT;ξ,vtmm!=1−4vt−RF11R,T+12I;ξ2t1−4vt.Proof.
For convenience, suppose that the left-hand side of (13) is denoted by J. According to the series expression of (11) and (7) to J, we find that
(14)J=∑m=0∞T+12Im−1RmℍℙmT;ξ,vtmm!=∑m=0∞∑s=0mRmT+1/2Im−s−1s!m−k!ξ2m−2s4vstm=∑m=0∞RmT+12Im−1ξ2tmm!.∑s=0∞R+mIsξvtss!=1−4vt−R∑m=0∞RmT+12Im−1ξ2t/1−4vtmm!.
Upon using the relation (6), the last equality evidently leads us to the required result.Corollary 5.
ForℍℙmT;ξ,v, the following generating matrix function holds true
(15)∑m=0∞T+12Im−1ℍℙmT;ξ,vtmm!=exp4vtF01−,T+12I;ξ2t.Remark 6.
The Bessel matrix functionJRz, for a positive stable matrix R∈Cd×d, is expressible in terms of hypergeometric matrix function as follows (see, e.g., [11, 41])
(16)JRw=w2RΓ−1R+IF01−;R+I,−w24.
Thus, by applying the relation (16) to (15) in Corollary 5, we can deduce the following Corollary.Corollary 7.
ForℍℙmT;ξ,v, the following holds true
(17)∑m=0∞T+12Im−1ℍℙmT;ξ,vtmm!=ΓT+12Iiξt−T−12Iexp4vtJT−12I2iξt.Theorem 8.
Letξ∈ℂ, v>0, m,l∈ℕ0 and T be a positive stable matrix in Cd×d such that T+nI is invertible for all n∈ℕ0. The following relation holds true
(18)∑m=0∞ℍℙm+lT;ξ,vtmm!=1−4vt−T+l+12Iexpξ2t1−4vtℍℙm+lT;ξ1−4vt,v.Proof.
Follows by induction or by the successive application of Theorem4 when R=T+1/2I. The details are omitted.Corollary 9.
Forl=0 in Theorem 8, the following holds true
(19)∑m=0∞ℍℙmT;ξ,vtmm!=1−4vt−T+12Iexpξ2t1−4vt.Remark 10.
The special cases of (18) and (19) when d=1 are seen to yield the classical generating functions of the generalized Heat polynomials (see [24, 33]).


## 4. Finite Sums

Here, various finite sums ofℍℙmT;ξ,v can be obtained in the following results.Theorem 11.
Letξ,z∈ℂ, v>0, m∈ℕ0, and T be a positive stable matrix in Cd×d such that T+nI is invertible for all n∈ℕ0. Then, we have
(20)ℍℙmT;ξz,v=∑s=0mnkT+12ImT+12Is−14v1−z2m−sz2sℍℙsT;ξz,v.Proof.
From (15) and the following fact
(21)e4vt0F1;T+12I;ξ2z2t=e4vt1−z2+4vz2tF01;T+12I;ξ2z2t.We thus find that
(22)∑m=0∞∑s=0∞ℍℙmT;ξ,vT+12Is−14v1−z2mz2tsm!s!=∑m=0∞∑s=0m4v1−z2m−sz2tsm−s!s!T+12Is−1ℍℙmT;ξ,v.
Comparing the coefficient oftm on both side, we thus get the required a finite sum formula (20).Theorem 12.
Letξ∈ℂ, v>0, m∈ℕ0, also let T and R be positive stable matrices in Cd×d such that T+nI and R+nI are invertible for all n∈ℕ0. Then, we have
(23)ℍℙmT;ξ,v=T+12ImRm−1∑s=0mnsT−R+12IsT+12Is−1×ℍℙsT;iξ,vℍℙm−s2R−T−I;ξ,v.Proof.
By using the series (11) and Theorem 4 with applying to Kummer’s matrix formula (see [5]), we observe that
(24)∑m=0∞T+12Im−1RmℍℙmT;ξ,vtmm!=1−4vt−Rexpξ2t1−4vt1F1T−R+12I,T+12I;−ξ2t1−4vt=1−4vt−2R−T−I+12Iexpξ2t1−4vt1−4vt−T−R+12I1F1T−R+12I,T+12I;−ξ2t1−4vt=∑m=0∞ℍℙm2R−T−I;ξ,vm!tm∑s=0∞T−R+12IsT+12Is−1ℍℙmT;iξ,vm!ts.Equating the coefficient of tm on both sides, we thus arrive at the desired result (23).


## 5. Laplace Transforms

Here, Laplace integral transforms of the generalized heat matrix polynomials are derived as follows.Theorem 13.
Letξ∈ℂ, v>0, Rλ>0, m∈ℕ0. Also, let T and A be a positive stable matrices in Cd×d such that T+nI is invertible for all n∈ℕ0. The following Laplace transform formula hold
(25)LuAℍℙmT;ξu,v:λ=4vmT+12Imλ−A+IΓA+I2F1−mI,A+I;T+12I;−ξ24vλ.Proof.
Making a particular use of (9) with (11), (6) and applying to Lemma 1, yields our desired result (25) in Theorem 13. The details are omitted.A similar procedure yields the following Laplace integral transforms. So we prefer to omit the proofs.Theorem 14.
Letξ∈ℂ, v>0, Rλ>0, m∈ℕ0. Also, let T and A be a positive stable matrices in Cd×d such that T+nI is invertible for all n∈ℕ0. The following Laplace transform formula hold
(26)LuAℍℙmT;ξu,v:λ=4vmT+12Imλ−A+IΓA+IF31−mI,12A+I,12A+2I;T+12I;−ξ2vλ2.Theorem 15.
Letξ∈ℂ, v>0, Rλ>0, m∈ℕ0. Also, let T and A be a positive stable matrices in Cd×d such that T+nI is invertible for all n∈ℕ0. The following Laplace transform formula hold
(27)LuAℍℙmT;u,v:λ=4vmT+12Imλ−A+IΓA+IF31−mI,12A+I,12A+2I;T+12I;−1vλ2.
The above Theorems lead to the following special cases.Corollary 16.
ForT is a positive stable matrix in Cd×d, and Rλ>0, then we have the following Laplace transforms
(28)LuT−12IℍℙmT;ξu,v:λ=4vmλ−A+12ΓA+m+12I1+ξ24vλm,Lu2T−IℍℙmT;ξu,v:λ=4vmT+12Imλ−2TΓ2TF20−mI,T;−;−ξ2vλ2,Lu2T−IℍℙmT;u,v:λ=4vmT+12Imλ−2TΓ2TF31−mI,T;−;−1vλ2.


## 6. Fractional Calculus Approach

Here, we consider the Riemann. Liouville fractional integral and derivative operatorsIγ and Dzγ of order γ∈C,Rγ>0, respectively (see, for details, [19])
(29)Iγfτz=1Γγ∫0zfτz−τγ−1dτ,where fτ is a function of τ and some square matrices so that this integral converges.
(30)Dzγfz=1Γ−γ∫0zz−τ−γ−1fτdτ,Rγ<0,dndznDzγ−nfz,n−1≤Rγ<nn∈N,where
(31)Dzγzβ=Γβ+IΓβ−γ−1Izβ−γRβ>−1.It is noted in passing that (29) and (30) are applied in recent works, for example, (see [16–22]).Theorem 17.
For the generalized heat matrix polynomials of degreem∈ℕ0, the following fractional integral operator holds true:
(32)IγξT−12IℍℙmT;ξ,v=ΓT+m+12IΓ−1T+γ+m+12I×ξT+γ−12IℍℙmT+γI;ξ,v.Proof.
From (29) and (11), we have
(33)IγξT−12IℍℙmT;ξ,v=1Γγ∫0ξξ−τγ−1τT−12IℍℙmT;τ,vdτ.
Settingτ=ξη and after a simplification, we get
(34)IγξT−12IℍℙmT;ξ,v=ξT+γ−1/2IΓγ∫01ηT−12I1−ηγ−1ℍℙmT;ξη,vdη=ξT+γ−12I4vmT+12Im∑s=0m−mIsT+12Is−1ΓT+s+12I×Γ−1T+s+γ+12I−ξ2/4vss!,
whose last summation, in view of (11), is easily seen to arrive at the expression in (32). This completes the proof of Theorem 17.Theorem 18.
For the generalized heat matrix polynomialsℍℙmT;ξ,v, we have the following formula
(35)∑m=0∞T+12Im−1βmγmℍℙmT;ξ,vtmm!F11γ−β;γ+m;4vt=exp4vtF11βI;T+12I;ξ2t,Rβ>0,Rγ>0.Proof.
According to (15), we find that
(36)exp−4vt∑m=0∞T+12Im−1ℍℙmT;ξ,vtmm!=F01−;T+12I;ξ2t.
Assume that the left-hand side of (35) be denoted by J. Multiply the left-hand side (35) by tβ−1 and applying the relation (31), we get
(37)J=ΓβΓγtγ−1∑m=0∞∑s=0∞T+12Im−1ℍℙmT;ξ,v4vs−1sβm+stm+sγm+ss!m!=ΓβΓγtγ−1∑m=0∞T+12Im−1ℍℙmT;ξ,vβmtmγmm!F11β+m;γ+12;−4vt.
Invoking Kummer’s matrix transformation [5] leads to
(38)J=ΓβΓγtγ−1exp−4vt∑m=0∞T+12Im−1ℍℙmT;ξ,vβmtmγmm!F11γ−β;γ+m;4vt.
It is easy to multiply right-hand side of (35) by tβ−1 and applying the fractional differentiation operator Dtβ−γ from (31), whereupon this completes the establishment of the Theorem 18.


## 7. Concluding Remarks

In [23], Rosenbloom and Widder investigated expansions of solutions ux,t of the heat equation uxx=ut in series of polynomial solutions υnx,t. Further, Haimo and Markett [33, 34] discussed the generalized heat equation
(39)Δxux,t=∂∂tux,t,where Δxhx=h′′x+2ν/xh′x,ν is a fixed positive number and they introduced the generalized heat polynomial solution of (39) as
(40)ℙn,νx,t=∑j=0n22jnjΓν+1/2+nΓν+1/2+n−jx2n−2jtj.Note thatℙn,0x,t=υ2nx,t is the ordinary heat polynomials of even order. Also, ℙn,0x,−1=H2nx/2 is the Hermite polynomials of even order.Recently, many studies and extensions of the well-known special matrix polynomials have been in a focus of increasing attention leading to new and interesting problems. In this perspective, we defined the generalized heat matrix polynomials. Also, we have given some of their main properties, namely, the generating matrix functions, finite sum formulas, Laplace integral transforms, and fractional calculus operators. Further, This study is assumed to be a generalization of the scalar cases [27, 33, 34] to the matrix setting. In addition, this approach allows to derive several new integral and differential representations that can be used in theoretical, applicable aspects like the boundary value problems and the numerical algorithms. Additional research and application on this topic is now under preparation and will be presented in forthcoming works.



---

*Source: 4065606-2021-06-09.xml*