Some generating relations for extended Appell’s and Lauricella’s hypergeometric functions. (English) Zbl 1308.33009

Using the extended beta function, see M. A. Chaudhry et al. [Appl. Math. Comput. 159, No. 2, 589–602 (2004; Zbl 1067.33001)], the author of the paper has generalized Appell’s hypergeometric functions and Lauricella’s hypergeometric function. These may be considered as very effective extensions of the results of M. Ali Özarslan and E. Özergin [Math. Comput. Modelling 52, No. 9–10, 1825–1833 (2010; Zbl 1205.33021)]. Integral formulae for these generalized hypergeometric functions are established, see Theorems 2.1–2.3 of the paper. Each of them has a confluent hypergeometric function in the result, owing to the generalized beta function definition, see E. Özergin et al. [J. Comput. Appl. Math. 235, No. 16, 4601–4610 (2011; Zbl 1218.33002)]. The formulae, see Theorems 3.1–3.5 of this paper, related to Mellin transforms of these three generalized hypergeometric functions, are given in Section 3. Theorems 4.1–4.6 give the differentiation formulae of these generalized functions. In Section 5, using the recurrence formulae of Kummer’s function, some recurrence relations for these new functions are proved.


33C65 Appell, Horn and Lauricella functions
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
33B15 Gamma, beta and polygamma functions
44A20 Integral transforms of special functions
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
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