The incomplete Lauricella functions of several variables and associated properties and formulas.

*(English)*Zbl 1400.33024Summary: Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [H. M. Srivastava et al., Integral Transforms Spec. Funct. 23, No. 9, 659–683 (2012; Zbl 1254.33004)] and the second Appell function [A. Çetinkaya, Appl. Math. Comput. 219, No. 15, 8332–8337 (2013; Zbl 1318.33001)], we introduce here the incomplete Lauricella functions \(\gamma^{(n)}_A\) and \(\Gamma^{(n)}_A\) of \(n\) variables. We then systematically investigate several properties of each of these incomplete Lauricella functions including, for example, their various integral representations, finite summation formulas, transformation and derivative formulas, and so on. We provide relevant connections of some of the special cases of the main results presented here with known identities. Several potential areas of application of the incomplete hypergeometric functions in one and more variables are also pointed out.

##### MSC:

33C65 | Appell, Horn and Lauricella functions |

33B15 | Gamma, beta and polygamma functions |

33B20 | Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) |

33C20 | Generalized hypergeometric series, \({}_pF_q\) |

##### Keywords:

gamma functions; incomplete gamma functions; Pochhammer symbol; incomplete Pochhammer symbols; incomplete generalized hypergeometric functions; Lauricella functions; Appell functions; Laguerre polynomials; Bessel and modified Bessel functions; incomplete second Appell functions; incomplete Lauricella functions of several variables; Marcum \(q\)- and \(Q\)-functions
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\textit{J. Choi} et al., Kyungpook Math. J. 58, No. 1, 19--35 (2018; Zbl 1400.33024)

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