Hypergeometric functions interpolating Appell-Lauricella's \(F_D\) and \(F_A\)
type: Item
instance of: scholarly article
title: Hypergeometric functions interpolating Appell-Lauricella's \(F_D\) and \(F_A\)
zbMATH Open document ID: 1426.33043
author: Masaaki Yoshida
published in: Hokkaido Mathematical Journal
publication date: 2019-07-12T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.hokmj/1562810511
review text: 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 b…
Mathematics Subject Classification ID: 33C65
zbMATH DE Number: 7080097
zbMATH Keywords: Appell-Lauricella hypergeometric functions
zbMATH Keywords: hyperplanes
zbMATH Keywords: chambers
zbMATH Keywords: singular locus
zbMATH Keywords: rank
reviewed by: Francisco Marcellán
MaRDI profile type: Publication
OpenAlex ID: W2957864481
DOI: 10.14492/HOKMJ/1562810511
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description: scientific article; zbMATH DE number 7080097