Hypergeometric functions interpolating Appell-Lauricella's \(F_D\) and \(F_A\) (Q2002613)

The Appell-Lauricella hypergeometric system \(E_{D}(a,\mathbf{b}, c)\) of differential equations of type \(D\) in \(n\) real variables admits solutions represented by integrals \(I_{R} \mathbf{(x})= \int_{R} \omega_{n}, \) where \(\omega_{n}= \prod t_{i}^{b_{i}-1}(1- |\mathbf{t}|)^{c-1-|\mathbf{b}|}(1- \langle \mathbf{x},\mathbf{t} \rangle)^{-a} d\mathbf{t}\) and \(R\) is a region cut out from the real \(n\)-dimensional space by the \(n+2\) hyperplanes \(H_{0}= \{\mathbf{t}: \sum t_{i}=1\},\) \(H_{i}= \{ \mathbf{t}: t_{i}=0 \}, i=1,2, \dots, n,\) and \(H_{\mathbf{x}}= \{ \mathbf{t} : \langle \mathbf{x},\mathbf{t} \rangle=1 \}\). As usual, \(\mathbf{b}= (b_{1}, b_{2}, \dots, b_{n})\) denotes a vector in the \(n\)-dimensional Euclidean space, \(|\mathbf{b}|= \sum_{k=1}^{n} b_{k},\) and \(d\mathbf{t}= \prod_{k=1}^{n}d t_{k}\). If the region \(R\) is the simplex \(\{ \mathbf{t}: 1- \sum t_{i}> 0, t_{i} >0, i=1,2, \dots, n\},\) then the integral is given in terms of the Appell- Lauricella function \(F_{D}(a,\mathbf{b}, c; \mathbf{x})= \sum_{|\mathbf{m}|=0}^ {\infty} \frac{(a)_{|\mathbf{m}|} \prod (b_{i})_{m_{i}}}{(c)_{|\mathbf{m}|}}\frac{\prod x_{i}^{m_{i}}} {m_{i}!}\) up to a constant factor. Again, \(\mathbf{m}= (m_{1}, m_{2}, \dots, m_{n})\) denotes an \(n\)-dimensional vector with nonnegative integers as entries. In the paper under review, the authors deal with integrals \(\int_{C_{k}} \omega_{n}, \) over regions \(C_{k}= \{ \mathbf{t}: 1- |\mathbf{t}|> 0, t_{i}<0, i=1,2, \dots, k, t_{i} >0, i=k+1, k+2, \dots, n\}\). Up to a constant factor they can be given in terms of series \(F_{k}^{n}(a, \mathbf{b}, c_{1}, c_{2}; \mathbf{x})\) introduced in \textit{H. Exton} [Multiple hypergeometric functions and applications. Mathematics and Its Applications. Chichester: Ellis Horwood Limited, Publisher; New York etc.: Halsted Press, a division of John Wiley \& Sons, Inc. (1976; Zbl 0337.33001)]. The rank and the singular locus of the system of differential equations satisfied by this series is analyzed. Further generalizations of such series based on ordered partitions of the integer number \(n\) are presented as well as fundamental properties are discussed. Notice that if the length of the partition is \(1\), then you have the Appell-Lauricella function, while if the length of the partition is \(n,\) then you have the hypergeometric function \(F_{A}\). In some sense, the above generalizations interpolate \(F_{D}\) and \(F_{A}\).