Details

Autor(en) / Beteiligte

• Mennou, S
• Salhi, S
• Kacha, A

Titel

Matrix continued fraction representation of the Gauss hypergeometric function

Ist Teil von

• New trends in mathematical sciences, 2021-01, Vol.9 (1), p.42-51

Ort / Verlag

Istanbul: Yildiz Technical University, Faculty of Chemistry and Metallurgy

Erscheinungsjahr

2021

Quelle

EZB Electronic Journals Library

Beschreibungen/Notizen

• Recently, the extension of continued fractions theory from real numbers to the matrix case has seen several developments and interesting applications (see [1,6,7]). Since calculations involving matrix valued functions with matrix arguments are feasible with large computers, it will be interesting attempt to develop such matrix theory. The theory of the algorithmic discovery of identities remains an active research topic. 2 Preliminary and notations Throughout this paper, we denote by JZm the set of mxm real (or complex) matrices endowed with the subordinate matrix infinity norm defined by, This norm satisfies the inequality Let A e J{m, A is said to be positive semidefinite (resp. positive definite) if A is symmetric and where (.,.) denotes the standard scalar product of Rm. If the function f(x) can be expanded in a power series in the circle \ x - xq \<r by then this expansion remains valid when the scalar argument x is replaced by a matrix A whose characteristic values lie within the circle of convergence. 3 Main results 3.1 The real case The Gauss hypergeometric function 2F\ is defined in [10] as follows for a,ft,xel,ce R/Z such that | x \ < 1, by where, for some parameter /z, the Pochhammer symbol (p)j is defined as (p)o = 1; (p)j = /J-(p + l)...(/i. + j - 1), 7 = 1,2,... The representation continued fraction of the Gauss hypergeometric function 2Fi(a,b;c;A) is Proof. Since A is a positive definite matrix, then there exists an invertible matrix X such that A =XDX1 where D = diag(Xi,X,2,...,Xm) and A,- > 0 for all 1 < i < m. As the function F = 2Fi(a,b;c;) is analytic in a domain /=] - !, 1[ then Let us define the sequences (P) and (Q) by We see that Pand Q are diagonal matrices.