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Linear relations for Lauricella $F_D$ functions and symmetric polynomial

In this paper we develop an algorithm for obtaining some new linear relations among the Lauricella $F_D$ functions. Relations we obtain, generalize those hinted in the work of B. C. Carlson. The coefficients of these relations are contained in the ring of polynomials in the variables $x_1,\dots,x_N$ or in some exceptional cases in the field of rational functions ${\mathbb R}(x_1,\dots,x_N,p)$. The method is based on expressing suitably chosen Euler type indefinite integrals associated with these functions recursively as linear combination of some other Euler type integrals and elementary functions and then integrating over the interval $[0,1].$ We describe the complete algorithm for obtaining these relations. We believe that such relations might be useful in computations.