Generalized-Hypergeometric Solutions of the General Fuchsian Linear ODE Having Five Regular Singularities

Abstract

We show that a Fuchsian differential equation having five regular singular points admits solutions in terms of a single generalized hypergeometric function for infinitely many particular choices of equation parameters. Each solution assumes four restrictions imposed on the parameters: two of the singularities should have non-zero integer characteristic exponents and the accessory parameters should obey polynomial equations.

Double-Step Shape Invariance of Radial Jacobi-Reference Potential and Breakdown of Conventional Rules of Supersymmetric Quantum Mechanics

“…Since the TP degree cannot exceed 2, the rational DT (RDT) of the JRef CSLE using each of the basic solutions (1) as its TF generally leads to the Fuchsian RCSLEs with five regular singular points (including infinity). Their common remarkable feature (compared with the general case examined in [19,20]) is that the eigenfunctions of the RCSLEs constructed in such a way are expressible in terms of finite sequences of polynomials referred to by us as 'Jacobi-seed' (JS) Heine polynomials, though it is worth mentioning that the coefficient functions of the first derivative in the corresponding Fuchsian equations generally depend on the polynomial degree (in contrast with the standard definition of Heine polynomials [21][22][23]).…”

Transformation formulae for terminating balanced $_4F_3$-series and implications

“…The hypergeoemtric series plays an important role in mathematics and physics (cf. [14,15]). There exist numerous summation and transformation formulae of classical hypergeometric series (see [3,Chapter 8] and [6][7][8][9][10]12,13,16,17,20]).…”

Solving the First Order Differential Equations using Newton's Interpolation and Lagrange Polynomial

“…Many mathematical models in science and engineering fields ( [3], [4], [5], [6], [7], [10], [13]) can be formulated in the form of linear and nonlinear ordinary differential equations ( [9], [12]) which need an analytical method ( [2], [15], [17]) to solve the exact equations. However in some problems, we cannot obtain the exact solutions by the analytical method in [7] for example y ′ = x 2 + y 2 .…”