Relations Between the Matrix Algebraic Factorized Type Solutions at Different Singular Points for Generalized Hypergeometric Functions of Typep+1Fp

Abstract

The generalized hypergeometric function \[ u(z)={}_{p+1}\!F_p(a_1,\dots ,a_{p+1};b_1,\dots ,b_p| z) \] is defined as \(u(z)= \sum_{k=0}^\infty \frac{(a_1)_k\dots (a_{p+1})_k}{(b_1)_k\dots (b_p)_k} \frac{z^k}{k!}\), where \((a)_k=a(a+1)\dots (a+k-1)\). Based on the identity \((z\frac{d}{dz}+\alpha )z^k=(\alpha +k)z^k\), the authors prove that \(u\) satisfies a linear differential equation \(P(z\frac{d}{dz})u(z)=\frac{1}{z}Q(z\frac{d}{dz})u(z)\), where \(P\) and \(Q\) are polynomials of degree \(p+1\). This equation is converted into a first order differential equation \({\mathbf u}'(z)=(\frac{1}{z}{\mathbf A}_1+\frac{1}{1-z}{\mathbf A}_2){\mathbf u}(z)\) of \((p+1)\) unknowns \(u_k(z)=(z\frac{d}{dz})^ku(z)\), where \(0\leq k\leq p\). The solution of the last equation can be written in the form \({\mathbf u}(z)={\mathbf U}(z){\mathbf c}\), where \({\mathbf c}\) is an arbitrary vector and \({\mathbf U}(z)\) is the `evolution matrix'. The authors present a factorization scheme to express the solution in an infinite product of exponential matrices, at both regular and singular points. The factorization at a given point can be expressed as postmultiplied by a constant matrix of the factorization at another given point. The determination of the constant matrix relating such two factorizations is the main purpose of the paper.