On the finite sum representations and transcendence properties of the Lauricella functions FDFD

We first generalize the results in Tan and Zhou (2005) [2] that a Lauricella function FD(a,b1,…,bn;c;x1,…,xn)FD(a,b1,…,bn;c;x1,…,xn) of nn variables can be written as a finite sum of rational functions and logarithm functions of one variable, for a,b1,…,bn,ca,b1,…,bn,c positive integers with c≥a+1c≥a+1, and for distinct x1,…,xnx1,…,xn, to all x1,…,xnx1,…,xn not necessarily distinct. Then we use the finite sum representation to prove that the values of FD(a,b1,…,bn;c;x1,…,xn)FD(a,b1,…,bn;c;x1,…,xn), for positive integers a,b1,…,bn,ca,b1,…,bn,c with c>ac>a, and real algebraic numbers x1,…,xnx1,…,xn with 0