Difference analogs and \((r,s)\)-analogs of \(GG\)-functions
A new approach to hypergeometric functions was recently developed. Namely, these functions were defined as solutions of GG-systems which are collections of linear relations for a function, its first partial derivative and translations with respect to its parameters. A GG-system is defined by a linear subspace \(L\subset {\mathbb C}^N\) and consists of equations \((\partial _i f)(\alpha ,a)=f(\alpha -e_i ,a)\), \(i=1,\dots ,N\), \(f(\alpha +l,a)=f(\alpha ,a)\), \(l\in L\), \(\sum _{i=1}^N \omega _i (\theta _i f)(\alpha ,a)=\langle \omega ,\alpha\rangle f(\alpha ,a)\), \(\omega =(\omega _1,\dots ,\omega _N)\in ({\mathbb C}^N)'\), \(\omega \bot L\). Here \(\partial _i =\partial /\partial a_i\), \(\theta _i =a_i \partial /\partial a_i\), and \(\{ e_i\}\) is the standard basis in \({\mathbb C}^N\). Holomorphic solutions of this system are called GG-functions associated with the subspace \(L\). In the present paper, the above GG-system of equations is considered for which \(\partial _i\) and \(\theta _i\) are operators of more general form. The author studies detailly partial cases of such systems, namely, \((r,s)\)-analogues and difference analogs of GG-systems. For these systems, solutions (analogous to solutions for the case \(\partial _i =\partial /\partial a_i\) and \(\theta _i =a_i \partial /\partial a_i\)) are constructed. Simple examples of such solutions are considered in detail.
scientific article; zbMATH DE number 1205905