On A Matrix Hypergeometric Differential Equation

SEBHA UNIVERSITY JOURNAL OF PURE & APPLIED SCIENCES VOL.20 NO. 1 2021 ‫مجلة جامعة سبها للعلوم البحتة والتطبيقية‬ Sebha University Journal of Pure & Applied Sciences Journal homepage: www.sebhau.edu.ly/journal/index.php/jopas On A Matrix Hypergeometric Differential Equation Salah Hamd1, Faisal Saleh Abdalla2, Ahmed Shletiet3 of Mathematics, College of Education, Omar Al-Mukhtar University, Al-Bayda,Libya. of general science, College of mechanical engineering technology – Benghazi, Benghazi,Libya 3Department of Mathematics, faculty of science, University of Ajdabiya, Ajdabiya, Libya. 1Department 2Department Keywords: Special functions. A matrix Hypergeometric differential Equation. Matrix. Gamma function. Jordan Canonical Form. ABSTRACT In this paper we consider a matrix Hypergeometric differential equation, which are special matrix functions and solution of a specific second order linear differential equation. The aim of this work is to extend a well known theorem on Hypergeometric function in the complex plane to a matrix version, and we show that the asymptotic expansions of Hypergeometric function in the complex plane ” that are given in the literature are special members of our main result. Background and motivation are discussed. ‫حل املعادلة التفاضلية لدالة فوق هندسية في الصورة املصفوفية‬ ‫ و أحمد شليتيت‬2‫و فيصل صالح عبدهللا‬1‫صالح حمد‬ 3 ‫ ليبيا‬،‫ جامعة عمر املختار‬،‫ كلية التربية‬،‫قسم الرياضيات‬1 ‫ ليبيا‬،‫ جامعة بتغازى‬،‫ كلية تقنية الهندسة امليكاتيكية‬،‫قسم العلوم العامة‬2 ‫ ليبيا‬،‫ جامعة اجدابيا‬،‫ كلية العلوم‬،‫قسم الرياضيات‬3 :‫الكلمات املفتاحية‬ ‫امللخص‬ .‫دوال خاصة‬ .‫دالة جاما‬ ‫في هذا البحث تمت دراسة املعادلة التفاضلية لدالة فوق هندسية في الصورة املصفوفية والتي تمثل دالة‬ ‫ والهدف من هذه الدراسة هو تعميم نظرية‬.‫مصفوفية خاصة لحل معادلة تفاضلية خطية من الرتبة الثانية‬ ‫مشهورة على الدوال فوق الهندسية لتكون في الصيغة املصفوفية بدال من الصورة التي كانت عليها في املستوى‬ ‫ باألضافة إلى ذلك تم أثبات تقريب املقرب للدوال فوق الهندسية في املستوى املركب والتي تم عرضها في‬،‫املركب‬ ‫ خالل هذا البحث تم مناقشة االساسيات والدوافع‬.‫بحوث سابقة كحالة خاصة من النتيجة التي توصلنا إليها‬ .‫التي استندت عليها هذه الدراسة‬ .‫صيغة جوردان القانونية‬ .‫مصفوفة‬ .‫معادلة تفاضلية فوق هندسية‬ Introduction Generalization and extension of scalar special functions to matrix special functions have been developed in the past two decades The Gamma matrix function, whose eigenvalues are all in the right open half-plane is presented and investigated by L. J´odar, J. Cort´es [1] for matrices in 𝐶 𝑟𝑥𝑟 . Hermite matrix polynomials are introduced and discussed by L. J´odar et al [2] and some of their properties are provided in E. Defez, L. J´odar [3]. Other classical orthogonal polynomials as Laguerre and Chebyshev have been extended to orthogonal matrix polynomials, and some results have been studied in L. J´odar, J. Sastre [4] and E. Defez, L. J´odar [5]. Relations between the Beta, Gamma and the Hypergeometric matrix function are given in L. J´odar, J. G. Cort´es [6] and R. S. Batahan [7]. These special functions of matrices have developed an important tool in both theory and applications. The main our goal is that , some cases of the asymptotic expansions of 2F1(a, b; c; z) have been provided in the literature, they are all limited by a narrow domain of validity in the complex plane of the variable. Overcoming this restriction, we provide new asymptotic expansion for the matrix hypergeometric function .The order of presentation in this article is as follows. In section 2 we provide basic necessary notation, definitions and auxiliary theorems that need to be cited in the sequel. In section 3 we provide our main results. Preliminaries In this part we elaborate on some necessary language that is adopted from L. J´odar, J. Sastre [4] and N. J. Higham [8]. Denote by *Corresponding author: E-mail addresses: salahh92@gmail.com, (F. S. Abdalla2) faisal@ceb.edu.ly, (A. Shletiet) Ahmed.shletiet@uoa.edu.ly Article History : Received 27 March 2021- Received in revised form 25 May 2021 - Accepted 01 june 2021 On A Matrix Hypergeometric Differential Equation Hamd et al. λ1, λ2 · · · , λn the distinct eigenvalues of a matrix P ∈ C r×r . The spectrum σ(P) of P ∈ Cr×r , denotes the set of all the eigenvalues of P. We put γ(P). and 𝝔(P) the real numbers 𝜸(𝑷) = 𝒎𝒂𝒙{𝑹𝒆(𝝀): 𝝀 ∈ 𝝈(𝑷)}, (𝟐. 𝟏) 𝝔(𝑷) = 𝐦𝐢𝐧{𝑹𝒆(𝝀): 𝝀 ∈ 𝝈(𝑷)} holomorphic function f(λ) at a point was defined as a regular analytic function in a neighborhood of the point, see e.g. W. Wasow [9]. It is called holomorphic in a set if it is holomorphic at every point of the set. A matrix is called holomorphic if every entry of it is a holomorphic function. If 𝒇(𝝀) 𝒂𝒏𝒅 𝒈(𝝀) are homomorphic function of the complex variable λ, which are defined in an open set Ω of the complex plane, and P is matrix in Cr×r with σ(P) ⊂ Ω, than from the properties of the matrix functional calculus, see N. Dunford, J. Schwartz [10], it follows that 𝒇(𝑷)𝒈(𝑷) = 𝒈(𝑷)𝒇(𝑷) (2.2) A set of complex numbers is called positive stable if all the elements of the set have positive real part and a square matrix P is called positive stable if σ(P) is positive stable. If P is a positive stable matrix in Cr×r , than Γ(P) is well defined, see L. J´odar, J. G. Cort´es [1] ∞ 𝜞(𝑷) = ∫ 𝒆−𝒕 𝒕𝒑−𝟏 𝒅𝒕 (𝟐. 𝟑) 𝟎 If 𝒇(𝑷) is well defined and T is an invertible matrix in 𝒇〖(TP𝑻−𝟏 )〗= 𝑻 𝒇(𝑷)𝑻−𝟏 (2.4) It is a standard result that for any matrix P ∈Cr×r there exist a nonsingular matrix T ∈Cr×r such that 𝑻−𝟏 𝑷 𝑻 = 𝑱 = 𝒅𝒊𝒂𝒈(𝑱𝟏 , 𝑱𝟐 , … … , 𝑱𝑵 ) (𝟐. 𝟓) Where 𝝀𝒌 𝟏 𝟎 … 𝟎 ⋮ 𝟎 𝝀𝒌 𝟏 ⋱ Jk=Jk(λk)= ⋮ (2.6) ⋱ ⋱ ⋱ 𝟎 ∈𝑪𝒎𝒌 ×𝒎𝒌 ⋱ 𝝀𝒌 𝟏 ⋮ [ 𝟎 … … 𝟎 𝝀𝒌 ] Cr×r , then Lemma2.1 (matrix function via Jordan canonical form). Let f be defined on σ(P), P ∈C r×r and let p have the Jordan canonical form (2.5) subject to (2.6).Then 𝐟(𝐏) = 𝐓 𝐟(𝐉)𝐓 −𝟏 = 𝐓 𝐝𝐢𝐚𝐠(𝐟(𝐉𝟏 ), 𝐟(𝐉𝟐 ),· … (𝟐. 𝟕) · , 𝐟(𝐉𝐬 ))𝐓 −𝟏 Where (𝝀𝒌 ) 𝒇(𝝀𝒌 ) 𝒇(𝟏) (𝝀𝒌 ) … (𝒎 −𝟏)! 𝒌 𝒇(𝝀𝒌 ) ⋱ ⋮ 𝒇(𝑱𝒌 )= 𝟎 ∈𝑪𝒎𝒌 ×𝒎𝒌 (2.8) ⋱ ⋱ 𝒇(𝟏) (𝝀𝒌 ) ⋮ … 𝟎 𝒇(𝝀 ) ] [ 𝟎 𝒌 Proof: The proof of this lemma is already proved in [8] A hypergeometric function is the sum of a hypergeometric series, which is defined as follows see e.g. F. W. J. Olver [11] . The hypergeometric function 𝒑𝑭𝒒 (a1, a2, · · · ap; b1, b2, · · · bq; z) is defined by means of a hypergeometric series as 𝒑𝑭𝒒 (𝒂1, 𝒂2,· · · 𝒂𝒑 ; 𝒃1, 𝒃2,· · · 𝒃q; 𝒛) ∞ =∑ 𝑛=0 (𝑎1 )𝑛(𝑎2 )𝑛 · · · (𝑎𝑃 )𝑛 𝑧 𝑛 . (𝑏1 )𝑛(𝑏2 )𝑛 · · · (𝑏𝑞 )𝑛 𝑛! Recall that the shifted factorial (a)n is defined b (𝒂)𝒏 = 𝒂(𝒂 + 𝟏)(𝒂 + 𝟐) · · · (𝒂 + 𝒏 − 𝟏) , 𝒏 ∈ 𝑵 𝒂𝒏𝒅 (𝒂)𝟎 = 𝟏. Gauss’s hypergeometric equation is a second order differential equation with three regular singular points {0, 1,∞}, that is 𝒛(𝟏 − 𝒛)𝒇 ′′ + [𝒄 − (𝟏 + 𝒂 + 𝒃)𝒛]𝒇′ − 𝒂𝒃𝒇 = 𝟎. JOPAS Vol.20 No. 1 2021 (𝒄)𝒏 𝒏! Theorem 2.2 for Re 𝒄 > 𝑹𝒆 𝒃 > 𝟎 we have 2F1 (𝒂, 𝒃; 𝒄; 𝒛) = 𝜞(𝒄) 𝜞(𝒃)𝜞(𝒄 − 𝒃) 𝟏 ∫𝟎 𝒕 𝒃−𝟏 (𝟏 − 𝒕) 𝒄−𝒃−𝟏 (𝟏 − 𝒛𝒕)−𝒂 𝒅𝒕, (2.9) for all 𝒛 ∈ 𝑪 and | 𝒛 | < 𝟏. The hypergeometric function with matrix arguments F(A, B; C; z), see L. J´odar and J. G. Sastre [6], is a solution of the differential equation 𝒛(𝟏 − 𝒛)𝑾 (𝟐) − 𝒛𝑨𝑾(𝟏) + 𝑾(𝟏) (𝑪 + 𝒛(𝒏 − 𝟏)𝑰) + 𝒏𝑨𝑾 = 𝟎 which is defined by 𝑭(𝑨, 𝑩; 𝑪; 𝒛) = 𝟏 (∫ (1 − 𝒕𝒛)−𝑨 𝒕 𝑩−𝑰 (1 − 𝒕)𝑪−𝑩−𝑰 𝒅𝒕) 𝜞 −𝟏 (𝑩) 𝜞 −𝟏 𝟎 (𝑪 − 𝑩)𝜞(𝑪) (𝟐. 𝟏𝟎) Proposition 2.3 Let f(t) be a complex valued function of a real variable t such that (𝑖) 𝑓(𝑡)𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑜𝑛 (0, ∞), ∞ (𝑖𝑖) 𝑓(𝑡) ∼ ∑ 𝑎𝑛𝑡 𝜉𝑛−1 𝑎𝑠 𝑡 ⟶ 0 𝑤𝑖𝑡ℎ 0 < 𝜉 0 < 𝜉1 < 𝜉 2 𝑛=0 The symbols 𝓞, o and ∼, due to Bachmann and Landau (1927), which are also used by e.g. F. W. J. Olver [11] and A. Erd´elyi [12]. Concerning the definition and elementary properties of asymptotic series we refer to W. Wa-sow [9] and A. Erd´elyi [12]. 𝒇(𝒎𝒌 −𝟏) and has a solutions 𝒏 ∞ (𝒂)𝒏 (𝒃)𝒏 𝒛 2F1 (𝒂, 𝒃; 𝒄; 𝒛) = ∑𝒏=𝟎 < …… 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑙𝑜𝑠𝑠 𝑜𝑓 𝑔𝑒𝑛𝑒𝑟𝑎𝑙𝑖𝑡𝑦 𝑎𝑠𝑠𝑢𝑚𝑒 𝑎0 ≠ 0, (𝑖𝑖𝑖) 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑓𝑖𝑥𝑒𝑑 𝑐 > 0 𝑓(𝑡) = 𝑂{𝑒 𝑐𝑡 } 𝑎𝑠 𝑡 ⟶ ∞. Then we have ∞ −𝜆 𝑡 𝑑𝑣 𝑑𝑛 𝛤(𝜉𝑛 + 𝑣) 𝑘 𝑎𝑠 𝑅𝑒 𝜆𝑘 ⟶ 𝑓(𝑡)𝑑𝑡) ∼ ∑∞ 𝜉𝑛+𝑣 𝑣 (∫0 𝑒 𝑛=0 𝑑𝜆𝑘 𝜆𝑘 ∞ (2.11) where 𝑑𝑛 = (−1)𝑣 𝑎𝑛 and 𝜋 𝜋 | 𝑎𝑟𝑔(𝜆𝑘 )| ≤ − 𝛿 < 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝛿 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 0 < 𝛿 2 2 𝜋 . < 2 Proof: The proof of this proposition is already proved in [13] . Lemma 2.4 Suppose 𝑄 is a positive stable matrix in Cr×r and suppose also that 𝑓(𝑡)𝐼 ∈Cr×r , where f(t) is a function of a real variable t such that (𝑖) 𝑓(𝑡) 𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑜𝑛 (0, ∞). ∞ (𝑖𝑖) 𝑓(𝑡) ∼ ∑ 𝑎𝑛 𝑡 𝜉𝑛−1 𝑎𝑠 𝑡 ⟶ 0 With 𝑛=0 (2.12) 0 < 𝜉0 < 𝜉1 < 𝜉2 < · · · · · · (𝑖𝑖𝑖) 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑓𝑖𝑥𝑒𝑑 𝑐 > 0, 𝑓(𝑡) = 𝑂{𝑒 𝑐𝑡 } 𝑎𝑠 𝑡⟶ ∞ (iv) 𝑄 = 𝑃 has the Jordan canonical form subject to (2.5) and (2.6) with 𝜋 𝜋 | 𝑎𝑟𝑔(𝜆𝑘 ) | ≤ − 𝛿 < for some δ such that 0 < 𝜋 2 2 . 𝛿 < 2 Then we have ∞ 𝑻−𝟏 [∫𝟎 𝒆−𝑸𝒕 𝒇(𝒕) dt] T ∼ 𝜱𝑸 as Re λk → ∞ (2.13) Where 𝜱𝑸 = diag(𝜱𝟏 , 𝜱𝟐 , · · · , 𝜱𝑺 ) is a square block diagonal matrix in Cr×r with blocks Φk ∈ ℂ𝒎𝒌 ×𝒎𝒌 . 𝜱𝒌 = 𝒅𝒏 𝜞(𝝃𝒏 +𝒎𝒌 −𝟏) ∞ ∑ ) 𝒅𝒏 𝜞(𝝃𝒏 𝒅𝒏 𝜞(𝝃𝒏 + 𝟏) 𝒌 −𝟏 ∑∞ ∑∞ … 𝒏=𝟎 𝝀 𝝃𝒏+𝒎 (𝒎𝒌 −𝟏)! 𝝃𝒏+𝟏 𝒏=𝟎 𝝀 𝝃𝒏 𝒏=𝟎 𝒌 𝝀𝒌 𝒌 ⋮ 𝒅𝒏 𝜞(𝝃𝒏 ) ∑∞ ⋱ ∑∞ 𝒅𝒏𝜞(𝝃𝒏 + 𝟏) 𝟎 𝒏=𝟎 𝝀 𝝃𝒏 𝝃𝒏+𝟏 𝒏=𝟎 𝒌 𝝀𝒌 ⋱ ⋱ ⋮ 𝒅𝒏 𝜞(𝝃𝒏 ) ∑∞ … 𝟎 𝒏=𝟎 𝝀 𝝃𝒏 𝟎 [ ] (2.14) or we can write 𝜱𝒌 𝒂𝒔 𝒌 183 On A Matrix Hypergeometric Differential Equation where ∑∞ 𝒏=𝟎 𝜱𝒌 = 𝑫𝒌 + 𝑬𝒌 = 𝑫𝒌 [ 𝑰𝒎𝒌 + 𝑫−𝟏 𝒌 𝑬𝒌 ] 𝑫𝒌 = 𝒅𝒏 𝜞(𝝃𝒏 ) 𝟎 𝝃𝒏 𝝀𝒌 And ⋮ 𝟎 𝟎 ∑∞ 𝒏=𝟎 [ ⋮ ⋮ 𝟎 [ ∑∞ 𝒏=𝟎 𝑬𝒌 = 𝒅𝒏 𝜞(𝝃𝒏 + 𝟏) 𝝃𝒏+𝟏 𝝀𝒌 ⋱ 𝟎 … 𝒅𝒏 𝜞(𝝃𝒏 ) ⋱ … ⋱ 𝝃𝒏 𝝀𝒌 … ⋱ ⋱ … ⋱ 𝟎 ∑∞ 𝒏=𝟎 𝟎 ⋮ 𝟎 𝒅𝒏 𝜞(𝝃𝒏 ) ∑∞ 𝒏=𝟎 𝝀 𝝃𝒏 𝒌 ∈ ℂ𝒎𝒌 ×𝒎𝒌 ] 𝝃𝒏+𝒓𝒌 −𝟏 ∑∞ 𝒏=𝟎 ⋮ (𝒓𝒌 −𝟏)! 𝒅𝒏 𝜞(𝝃𝒏 + 𝟏) 𝝃𝒏+𝟏 𝝀𝒌 ∈ ℂ𝒎𝒌 ×𝒎𝒌 … ] 𝟎 Proof: The proof of this lemma is already proved in [13] . 3 On a matrix Hypergeometric Differential Equation. In this section we apply the machinery of the previous sections to show that in a certain sense 𝐹(𝑎𝐼, 𝑏𝐼, 𝐶, 𝑧) ∼ 𝐼 𝑎𝑠 𝑅𝑒 𝜆 ⟶ ∞ 𝑎𝑛𝑑 𝐹(𝑎𝐼, 𝑏𝐼, 𝜃𝑄, 𝑧) ∼ 𝐼 𝑎𝑠 𝜃 ⟶ ∞ , with 𝑄 a fixed matrix. Theorem 3.1 Suppose C - bI ∈ ℂ r×r has a Jordan canonical form subject to (2.5) and (2.6) then for any fixed matrices aI, bI in ℂ r×r, b > 0 we have 𝑇 −1 𝐹(𝑎𝐼, 𝑏𝐼; 𝐶; 𝑧)𝑇 𝐼 𝑎𝑠 𝑅𝑒 𝜆𝑘 ⟶ ∞ (3.1) uniformly for | z |≤ δ < 1 with δ a fixed number. Proof Consider the following integral as a function of a matrix C 1 𝜑(C) = ∫0 (1 − 𝑧𝑡)−𝑎𝐼 t (b−1)I (1 − t) C−(b+1)I dt. (3.2) Setting 1 − t = e −u , dt = e −u du when t = 1 ⇒ u → ∞, also when t = 0 ⇒ u = 0 , so we have ∞ φ(C) = ∫0 (1 + z(e−u − 1))−aI (1- e−u ) (b−1)I e−u(C- (b+1)I) e−u dt . ∞ = ∫0 (1 + z(e−u − 1))−aI (1- e−u ) (b−1)I e−u(C-bI) dt . Note that 𝑢2 −𝑢 𝑢3 (1 + 𝑧(𝑒 −𝑢 − 1))−𝑎𝐼 =(1 + 𝑧( + − + 1! 2! 3! Therefore (1 + 𝑧(𝑒 −𝑢 − 1))−𝑎𝐼 ∼ I as u→ 0 Also (1 + 𝑒 −𝑢 )−(𝑏−1)𝐼 =(1 − 1 + −𝑢 −𝑢 𝑢2 1! + 𝑢3 𝑢2 2! − 𝑢4 𝑢3 3! + 𝑢4 4! 𝑢4 4! … ))−𝑎𝐼 … )(𝑏−1)𝐼 𝑢(𝑏−1) (1 − + − + … … )(𝑏−1) 𝐼 2! 3! 4! 1! When u⟶ 0 we have (1 + 𝑒 −𝑢 )(𝑏−1)𝐼 ∼ ub-1 I thus (1 + 𝑧(𝑒 −𝑢 − 1))−𝑎𝐼 (1- 𝑒 −𝑢 ) (b−1)I ∼ub-1 I dt as u⟶ 0. Now lemma 2.4 implies that 1 T −1 [∫0 (1 − 𝑧𝑡)−𝑎𝐼 t (b−1)I (1 − t) C−(b−1)I ] T 1 ∫0 (1 𝑧𝑡)−𝑢𝐼𝑘 (𝑏−1)𝐼𝑘 𝑡)𝐽𝑘 −𝐼 1 = ∫0 t (b−1)I (1 − t)Jk −I dt ∼ ΦC−bI as Re λk ⟶ ∞ k = 1,2 ….. It is readily observed that F(aI, bI; C; z) 1 𝒅𝒏 𝜞(𝝃𝒏 +𝒓𝒌 −𝟏) 𝝀𝒌 Hamd et al. and obtain 1 T−1[β(Bi ,C−bI)]T=T−1[∫0 t (b−1)I (1 − t) C−(b+1)I dt] T (1 − dt ∼ ΦC−bI as = − 𝑡 Re λk ⟶ ∞ (3.3) Note that by letting a = 0 in equation (3.3), we obtain the following representation for the Beta matrix. Namely 1 β(C −bI, bI) = ∫0 𝑡 (𝑏−1)𝐼 (1 − 𝑡)𝐶−(𝑏+1)𝐼 dt = Γ−1 (C)Γ(C − bI)Γ(bI) (3.4) = ( ∫0 (1 − 𝑡𝑧) −𝑎𝐼 t (b−1)I (1 − t) C−(b+1)I dt ) Γ −1 (bI)Γ−1 (C − bI)Γ(C) (3.5) in the sense that T −1 F(aI, bI; C; z)T ∼ I . Example Let C = c ∈ (0, ∞) in the equation (3.5), then for any fixed a and b > 0 we have 1 𝜑(C) = ∫0 𝑡 𝑏−1 (1 − t) c−b−1 (1 − zt) −a dt. Setting 1 − t = e −u , dt = e −u du when t = 1 ⇒ u ⟶ ∞ , also when t = 0 ⇒ u = 0 so we have 1 𝜑(C)=∫0 (1 + 𝑧(𝑒 −𝑢 − 1))−𝑎 (1-1 − 𝑒 −𝑢 )b-1 e-u(c-b) du . Note that (1 + 𝑧(𝑒 −𝑢 − 1))−𝑎 (1-1 − 𝑒 −𝑢 )b-1 ~ ub-1 as u→ 0 Now Watson’s lemma implies that 1 𝛤(𝑏) ∫0 (1 + 𝑧(𝑒 −𝑢 − 1))−𝑎 (1-1 − 𝑒 −𝑢 )b-1 e-u(c-b) du ~ 𝑏 as c→ ∞ or 1 ∫0 t b−1 (1 − t) c−b−1 (1 − zt) −a dt . ~ (𝑏−𝑐) Γ(b) (b−c)b as c→ ∞ (3.6) By assuming a=0 in (3.6) we obtain 1 β(b, c − b) = ∫0 t b−1 (1 − t) c−b−1 dt ~ Γ(b) (3.7) (b−c)b By (3.6) and (3.7) we derive asymptotic expansion for 2F1(a,b;c; z) when c Approaches infinity, (3.8) 2F1(a,b;c; z) ~ 1 as c→ ∞ Corollary 3.2 Suppose 𝐶 − 𝑏𝐼 = 𝜃𝑄 where θ ∈ (0, ∞) and 𝑄 is a constant matrix and has a Jordan canonical form subject to (2.5) and (2.6) then 1 T −1 [∫0 (1 − zt)−aI t (b−1)I (1 − t)c−(b+1)I dt] T ∼ Ψ θQ as θ → ∞ (3.9) Where 𝛹𝜃𝑄 = 𝑑𝑖𝑎𝑔(𝛹1, 𝛹2,· · · , 𝛹𝑠) is a square block diagonal matrix in ℂ r×r, with blocks 𝛹𝑘 ∈ ℂ𝑚𝑘×𝑚𝑘 , Ψk= ∑∞ 𝑛=0 𝑎𝑛 𝛤(𝜉𝑛 ) 𝜃 𝜉𝑛 ∑∞ 𝑛=0 𝑑𝑛 𝜆𝑘 𝛤(𝜉𝑛 + 1) (𝜃𝜆𝑘 )𝜉𝑛 + 1 𝑎 𝛤(𝜉 ) ∞ ∑𝑛=0 𝑛 𝜉𝑛 𝑛 𝜃 ⋯ ∑∞ 𝑛=0 𝑑𝑛 𝜆𝑘 𝛤(𝜉𝑛 +𝑚𝑘 −1) (𝜃𝜆𝑘 )𝜉𝑛+𝑚𝑘 −1 (𝑚𝑘 −1)! ⋮ 𝑑𝑛 𝜆𝑘 𝛤(𝜉𝑛 + 1) ⋱ 0 ∑∞ 𝑛=0 (𝜃𝜆 )𝜉𝑛 + 1 𝑘 ⋱ ⋱ ⋮ 𝑎𝑛 𝛤(𝜉𝑛 ) ∞ ∑𝑛=0 𝜉𝑛 0 … 0 [ ] 𝜃 Moreover, T −1 F(aI, bI; θC; z)T ∼ I as θ ⟶ ∞ (3.10) Uniformly for | z | ≤ δ < 1 with 𝛿 a fixed number. Proof: By Watson’s lemma we have 1 𝑎𝑛 𝛤(𝜉𝑛 ) ∫0 (1 − 𝑧𝑡) −𝑎 𝑡 (𝑏−1) (1 − 𝑡)(𝜃𝜆𝑘)−1 dt ∼ ∑∞ 𝑛=0 𝜉𝑛 θ→∞. and by proposition 2.3 as 𝜃 → ∞ we get 1 dv ( ∫0 1 − zt) −a t dθv v dn λk Γ(ξn + v) ∑∞ n=0 (θλ )ξn+v . k t (b−1) as (𝜃𝜆𝑘 ) (1 − t) (θλk )−1 dt ) ∼ Thus by the lemma 2.1 we have JOPAS Vol.20 No. 1 2021 184 On A Matrix Hypergeometric Differential Equation 1 ∫0 (1 zt)−aIk t (b−1)Ik t)θJk −1 − (1 − dt ∼ Ψk and the results (3.9) and (3.10) follow. Conclusion We show that and Hamd et al. as θ ⟶ ∞ , 𝐹(𝑎𝐼, 𝑏𝐼, 𝐶, 𝑧) ∼ 𝐼 𝑎𝑠 𝑅𝑒 𝜆 ⟶ ∞ 𝐹(𝑎𝐼, 𝑏𝐼, 𝜃𝑄, 𝑧) ∼ 𝐼 𝑎𝑠 𝜃 → ∞ with 𝑄 a fixed matrix. References: [1]- J´odar L. and Cort´e J. G. , Some Properties Of Gamma And Beta Matrix Function, Appl . Math. Lett, 11, (1998), 89-93 [2]- J´odar L . and Company R. Hermite Matrix Polynomials And Second Order Matrix Differential Equation , Approximation Theory and its Application 12(2) , 20-30, (1996). [3]- Defez E. and J´odar, L. Some Applications of the Hermite Matrix Polynomials Series Expansions, Journal Of Computational Applied Mathematics 99, 105-117, (1998). [4]- J`odar L. and Sastre J. The growth Of Laguerre Matrix Polynomials On Bounded Intervals, appl. math. lett, 13, (2000),21-26. [5]- Defez E. , and J´odar L . Chebyshev Matrix Polynomials And Second Order Matrix Differential Equation. Utilitas Mathematica , (2002). 61. 107-123. [6]- J´odar L. and Cort´es J. G. On the Hypergeometric Matrix Function, J. Comp. Appl . Math.99 , (1998),205-217. [7]- Batahan R. S. Generalized Form Of Hermite Matrix Polynomials Via the Hypergeometric Matrix Function, Advances In Linear Algebra And Matrix Theory, (2014) ,4,134-141. JOPAS Vol.20 No. 1 2021 185