\(GG\)-functions of one variable (Q1595586)

From the text (translated from the Russian): We consider the hypergeometric Gaussian equation \[ \Bigl(\beta_1+x \frac{d}{dx}\Bigr) \Bigl( \beta_2+x \frac{d}{dx}\Bigr) F=\frac{d}{dx} \Bigl(\beta_3+x \frac{d}{dx} \Bigr)F. \tag{1} \] We denote by \(F(\beta_1,\beta_2,\beta_3;x)\) its solution, regular at the point \(x=0\) and satisfying the condition \[ F(\beta_1,\beta_2,\beta_3;0)= {\Gamma(\beta_1)\Gamma(\beta_2)\over \Gamma (\beta_3+1)}. \tag{2} \] The function \(F\) can also be defined in another, equivalent, way -- as being a solution, regular at \(x=0\) and satisfying condition (2), of the system of equations \[ \begin{aligned}\frac{dF(\beta_1,\beta_2,\beta_3;x)}{dx} & =F(\beta_1+1,\beta_2+1,\beta_3+1;x),\\ F(\beta _1+1,\beta_2,\beta_3;x) &-xF(\beta_1+1,\beta_2+1,\beta_3+1;x) =\beta_1F(\beta_1,\beta_2,\beta_3;x),\\ F(\beta_1,\beta_2+1, \beta_3;x) &-xF(\beta_1+1,\beta_2+1,\beta_3+1;x)=\beta_2F(\beta _1,\beta_2,\beta_3;x),\\ F(\beta_1,\beta_2,\beta_3-1;x) &-xF (\beta_1+1,\beta_2+1,\beta_3+1;x)=\beta_3F(\beta_1,\beta_2, \beta_3;x). \tag{3}\end{aligned} \] There is a simple connection between the solutions of (3) and (1): every solution of (3) satisfies equation (1) and every solution of (1) can be represented as a linear combination of two solutions of system (3) with coefficients depending only on \(\beta\). We generalize this situation, when the function \(F(\beta_1,\beta_2,\beta_3;x)\) is defined on the basis of the system of equations (3) rather than on the differential equation (1). Let there be given an arbitrary nonzero vector \(\ell=(l_l,\dots,l _n)\in \mathbb{C}^n\). The function \(F(\beta,x)\) with \(x\in\mathbb{C}\) and parameter \(\beta=(\beta_l,\dots, \beta_n)\in\mathbb{C}^n\) is said to be a GG-function associated with the vector \(l\) if it satisfies the system of equations (4) \(df (\beta,x)/dx =f(\beta+l,x)\), (5) \(f(\beta-e_i,x)-l_ixf(\beta+l,x) =\beta _if(\beta,x)\), \(i=1,\dots,n\), where \(e_1,\dots,e_n\) is a standard basis in \(\mathbb{C}^n\); system (4)--(5) is said to be a GG-system associated with \(l\). We suppose that this is a more general and natural system of equations for hypergeometric functions of one variable. The solutions of such a system depend on the additional parameters \(l_l,\dots,l_n\). For rational \(l_l,\dots, l_n\), hypergeometric functions arise, including generalized hypergeometric functions \(_pF_q\). The interrelation between GG-functions and \(_pF_q\) is very interesting and allows one to reorganize the theory. An unexpected result in this approach is that when the rational number \(l_l,\cdots,l_n\) tend to a real limit then a new class of functions is obtained at the limit.