Hypergeometric solutions for variants of the $q$-hypergeometric equation

Abstract

We introduce a configuration of a q-difference equation and characterize the variants of the q-hypergeometric equation, which were defined by Hatano-Matsunawa-Sato-Takemura, by configurations. We show integral solutions and series solutions for the variants of the q-hypergeometric equation.

Heun’s differential equation, integral transformation and q-deformation

“…(1) 7 through the q-middle convolution is not fully understood. The paper [3] by Fujii and Nobukawa might be related to this problem.…”

Connection Problem for an Extension of q-Hypergeometric Systems

“…When M = 1, the equation E 1 y = 0 is equivalent to the variant of q-hypergeometric equation of degree three [11]. Lemma 3.5 and Proposition 3.6 for M = 1 were obtained in [8], and integral solutions for the variant were given. See also [5] for the integral solutions.…”

“…This integral is a q-analog of In [8], a q-difference equation associated with the Jackson integral of Jordan-Pochhammer type:…”

“…They [8] also gave integral solutions and series solutions, which are q-analogs of (1.4) and (1.5) respectively, for the variant. For integral solutions, see also [5].…”

“…This system is an extension of the variant of q-hypergeometric equation of degree three [11]. It was found by Fujii and the author [8] that this variant can be regarded as a q-analog of the Riemann-Papperitz system, so we call the system in Theorem 3.11 q-RP M . The Riemann-Papperitz system is a second order Fuchsian differential equation with three singularities {t 1 , t 2 , t 3 } on the Riemann sphere P 1 = C ∪ {∞}, and this has solutions of the following forms:…”