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The incomplete Lauricella and fourth Appell functions. (English) Zbl 1315.33015
Summary: Recently, H. M. Srivastava et al. [Integral Transforms Spec. Funct. 23, No. 9, 659–683 (2012; Zbl 1254.33004)] introduced the incomplete Pochhammer symbol and studied some fundamental properties and characteristics of a family of potentially useful incomplete hypergeometric functions. Here we introduce the incomplete Lauricella function $$\gamma^{(n)}_C$$ and $$\Gamma^{(n)}_C$$ of $$n$$ variables, and investigate certain properties of the incomplete Lauricella functions, for example, their various integral representations, differential formula and recurrence relation, in a rather systematic manner. Some interesting special cases of our main results are also considered.

##### 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) 33C05 Classical hypergeometric functions, $${}_2F_1$$ 33C15 Confluent hypergeometric functions, Whittaker functions, $${}_1F_1$$ 33C20 Generalized hypergeometric series, $${}_pF_q$$
DLMF; Equator
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