Certain integral representations of Euler type for the Exton function \(X_2\). (English) Zbl 1234.33010
Summary: H. Exton [J. Indian Acad. Math. 4, No. 2, 113–119 (1982; Zbl 0509.33002)], introduced 20 distinct triple hypergeometric functions whose names are \(X_i(i=1,\dots,20)\) to investigate their twenty Laplace integral representations whose kernels include the confluent hypergeometric functions \(_0F_1\), \(_1F_1\), a Humbert function \(\Psi_2\), a Humbert function \(\Phi_2\). The object of this paper is to present 16 (presumably new) integral representations of Euler type for the Exton hypergeometric function \(X_2\) among his twenty \(X_i\) \((i=1, \dots,20)\), whose kernels include the Exton function \(X_2\) itself, the Appell function \(F_4\), and the Lauricella function \(F_C\).
MSC:
| 33C20 | Generalized hypergeometric series, \({}_pF_q\) |
| 33C65 | Appell, Horn and Lauricella functions |
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
| 33C60 | Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions) |
| 33C70 | Other hypergeometric functions and integrals in several variables |
