Abstract
The aim of this manuscript is to establish several finite summation formulas (FSFs) for the generalized Kampé de Fériet series (GKDFS). Moreover, the particular result for confluent forms of Lauricella series in n variables and four generalized Lauricella functions are obtained from the finite summation formulas for the GKDFS.
Highlights
Special functions are essential tools in several equations arising in natural science. e hypergeometric series and its generalizations are appeared in many mathematical problems and their applications. e theory of hypergeometric functions in many variables by the fact that the solutions of partial differential equations appearing in several applied problems of mathematical physics has been presented in terms of such hypergeometric functions [1,2,3,4]
Some particular cases yielding to finite summation formulas (FSFs) for four generalized Lauricella functions and confluent forms of Lauricella series in n variables are given
Characterizing the parameters in (24), we established the following summation identity for F(Dn) and Φ(Dn):
Summary
Special functions are essential tools in several equations arising in natural science. e hypergeometric series and its generalizations are appeared in many mathematical problems and their applications. e theory of hypergeometric functions in many variables by the fact that the solutions of partial differential equations appearing in several applied problems of mathematical physics has been presented in terms of such hypergeometric functions [1,2,3,4]. In 2016, Wang and Chen [10] derived FSFs of double hypergeometric functions involving some summation theorems. E rth derivative on τ1 of GKDFS is obtained as follows: Drτ. Fpl::υμ11++11;υ;μ22;.;....;.υ;μnn ⎡⎢⎢⎢⎢⎢⎢⎢⎣ ap: a(i 1) + r, b(μ11 ),i; b(μ22 ); . Combine τ1ai+r−1 with the variable τ1 in the GKDFS and put the derivative operator r-times on τ1 to get the following:. We have the following FSF of GKDFS: . Due to GKDFS and the Leibnitz formula for differentiation of a product of two functions, one gets. We have the following FSFs of GKDFS:. E following FSFs of GKDFS are satisfied:. E following FSFs of GKDFS are verified: established . Equating the above two identities yields (28). e other equality (29) can be established
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