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The incomplete Lauricella functions of several variables and associated properties and formulas. (English) Zbl 1400.33024
Summary: 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, $${}_pF_q$$
DLMF; Equator
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