Result: The \(q\)-hypergeometric equation of Gauss and a description of its series and integral solutions

Title:
The \(q\)-hypergeometric equation of Gauss and a description of its series and integral solutions
Authors:
Gel'fand, I. M., Graev, M. I., Retakh, V. S.
Publisher Information:
American Mathematical Society (AMS), Providence, RI
Subject Terms:
Generalized basic hypergeometric series, \(q\)-hypergeometric series, Other basic hypergeometric functions and integrals in several variables, \(q\)-hypergeometric equations, Discrete version of topics in analysis, Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics, Additive difference equations
Document Type:
Academic journal Article
File Description:
application/xml
Access URL:
https://zbmath.org/726900
Accession Number:
edsair.c2b0b933574d..d5c5305ce4544350d11b84a4d33255b1
Database:
OpenAIRE

Further Information

It is well-known that the Gauss hypergeometric function \(F(a, b; c; x)\) satisfies the Gauss hypergeometric equation \(x(1- x)d^2_x F+(c- (a+b +1)x) d_x F- abF_0\). \(q\)-Analogues of this equation have been considered by a number of authors. In this paper a new (more natural) \(q\)-analogue of this equation is given. The authors study its solutions in the form of series of hypergeometric form and in the form of triple integrals (GG-functions). This new \(q\)-analogue of the Gauss hypergeoemtric function depends on 8 parameters \(r_i\) and \(s_i\), \(i=1, 2, 3, 4\), in addition to the parameters \(a\), \(b\), \(c\). The previously known \(q\)-analogues of the Gauss equations and their solutions in the form of Heine \(q\)-hypergeometric series correspond to special values of the parameters \(r_i\) and \(s_i\). The structure of solutions of the \(q\)-hypergeometric function, defined in this paper, in the form of convergent series of Laurent type depends in an essential way on the parameters \(r_i\) and \(s_i\). If there do not exist solutions in the form of Laurent series, then it is possible to construct triple integral solutions.