The Lauricella series are generalizations of the Gaussian hypergeometric function to several variables. Since the later function satisfies several interesting transformation formulas it is plausible to expect that similar transformation formulas will hold true for the Lauricella generalization. Now by generalizing the result form here we discovered the following identity.
Let $ \left( \xi_\eta \right)_{\eta=1}^7 $ be parameters. Now define another set of parameters as follows:
\begin{equation} \chi_\eta := -2 \frac{\prod\limits_{\lambda=5}^7 (\xi_\eta - \xi_\lambda)}{\prod\limits_{\lambda=1,\lambda\neq \eta}^4 (\xi_\eta - \xi_\lambda)} \end{equation} for $\eta=1,\cdots,4$. Note that $\sum\limits_{\eta=1}^4 \chi_\eta = - 2 \quad (a)$.
In addition to that we define another parameters viz:
\begin{equation} {\mathfrak x}_\eta := \frac{\xi_\eta-\xi_2}{\xi_1-\xi_2} \cdot {\mathfrak x}_1 \end{equation} for $\eta = 3,4$. Let us also impose the condition that ${\mathfrak x}_\eta < 0 $ for all $\eta=1,3,4$ because otherwise the integral representation of the Lauricella series can have a singularity.
Then the following identity holds true:
\begin{eqnarray} &&\frac{\chi_1+1}{2} \cdot {\mathfrak x}_1 \cdot F_D^{(3)}\left(2,\chi_1+2,\chi_3+1,\chi_4+1,3;{\mathfrak x}_1,{\mathfrak x}_3,{\mathfrak x}_4 \right) + \\ &&\frac{\chi_3+1}{2} \cdot {\mathfrak x}_3 \cdot F_D^{(3)}\left(2,\chi_1+1,\chi_3+2,\chi_4+1,3;{\mathfrak x}_1,{\mathfrak x}_3,{\mathfrak x}_4 \right) + \\ &&\frac{\chi_4+1}{2} \cdot {\mathfrak x}_4 \cdot F_D^{(3)}\left(2,\chi_1+1,\chi_3+1,\chi_4+2,3;{\mathfrak x}_1,{\mathfrak x}_3,{\mathfrak x}_4 \right) + \\ &&F_D^{(3)}\left(1,\chi_1+1,\chi_3+1,\chi_4+1,2;{\mathfrak x}_1,{\mathfrak x}_3,{\mathfrak x}_4 \right) = \\ && \prod\limits_{\eta=1}^4 \left( \xi_1-\xi_2 + (\xi_2-\xi_\eta) \cdot {\mathfrak x}_1 \right)^{-\chi_\eta-1+2 \delta_{\eta,2}} \quad (i) \end{eqnarray}
As always we provide a snippet of code that verifies $(i)$ numerically. We have:
Clear[xi]; Clear[chi]; Clear[xx]; numIs = 100; myll = {};
LauricellaFD[a_, b1_, b2_, b3_, c_, x1_, x2_, x3_] :=
Gamma[c]/(Gamma[a] Gamma[c - a])
NIntegrate[
tt^(a - 1) (1 - tt)^(
c - a - 1) (1 - x1 tt)^-b1 (1 - x2 tt)^-b2 (1 - x3 tt)^-b3, {tt,
0, 1}, WorkingPrecision -> 20];
For[ii = 1, ii <= numIs, ii++,
{xi[1], xi[2], xi[3], xi[4], xi[5], xi[6], xi[7]} =
RandomReal[{-1, 0}, 7, WorkingPrecision -> 50];
xx[1] = -RandomReal[{0, 1}, WorkingPrecision -> 50];
{chi[1], chi[2], chi[3], chi[4]} =
Table[ -2 Product[
xi[p] - xi[q], {q, 5,
7}]/(Product[xi[p] - xi[q], {q, 1, p - 1}] Product[
xi[p] - xi[q], {q, p + 1, 4}]), {p, 1, 4}];
{xx[3],
xx[4]} = { (xi[3] - xi[2])/(xi[1] - xi[2]), (xi[4] - xi[2])/(
xi[1] - xi[2])} xx[1];
If[! (xx[1] < 0 && xx[3] < 0 && xx[4] < 0), Continue[]];
(*Insert the definitions into the identity above and simplify.*)
val = (
(chi[1] + 1) /
2 xx[1] LauricellaFD[2, chi[1] + 2, chi[3] + 1, chi[4] + 1, 3,
xx[1], xx[3], xx[4]] + (chi[3] + 1)/
2 xx[3] LauricellaFD[2, chi[1] + 1, chi[3] + 2, chi[4] + 1, 3,
xx[1], xx[3], xx[4]] + (chi[4] + 1) /
2 xx[4] LauricellaFD[2, chi[1] + 1, chi[3] + 1, chi[4] + 2, 3,
xx[1], xx[3], xx[4]] +
LauricellaFD[1, chi[1] + 1, chi[3] + 1, chi[4] + 1, 2, xx[1],
xx[3], xx[4]] -
Product[((xi[1] - xi[2]) + (xi[2] - xi[eta]) xx[1])^(-chi[eta] -
1 + If[eta == 2, 2, 0]), {eta, 1, 4}]);
myll = Join[myll, {val}];
];
myll
Having said all this my question is how would be go about proving $(i)$ . The idea is to treat both sides of that identity as an infinite series in ${\mathfrak x}_1 $ and then prove that all the coefficients of both sides match. We already see that the coefficient at $({\mathfrak x}_1)^0$ in both sides is equal to unity, which easily follows from the identity $(a)$ above. What about the remaining coefficients?
As I said above we treat both sides as series expansions in ${\mathfrak x}_1$ and compare the coefficients of those series. Let us start with the left hand side. We have:
\begin{eqnarray} &&\left[ @{\mathfrak x}_1^n \right](lhs)= \\ && \left. \frac{\chi_1+1}{2} \cdot \frac{2}{n+1} \cdot \frac{(\chi_1+2)^{(i_1)}}{i_1!} \cdot \frac{(\chi_3+1)^{(i_2)}}{i_2!} \cdot \frac{(\chi_4+1)^{(i_3)}}{i_3!} \cdot \left( \frac{\xi_3-\xi_2}{\xi_1-\xi_2}\right)^{i_2} \cdot \left( \frac{\xi_4-\xi_2}{\xi_1-\xi_2}\right)^{i_3} \right|_{i_1+i_2+i_3=n-1} + \\ && \left. \frac{\chi_3+1}{2} \cdot \frac{2}{n+1} \cdot \frac{(\chi_1+1)^{(i_1)}}{i_1!} \cdot \frac{(\chi_3+2)^{(i_2)}}{i_2!} \cdot \frac{(\chi_4+1)^{(i_3)}}{i_3!} \cdot \left( \frac{\xi_3-\xi_2}{\xi_1-\xi_2}\right)^{i_2+1} \cdot \left( \frac{\xi_4-\xi_2}{\xi_1-\xi_2}\right)^{i_3}\right|_{i_1+i_2+i_3=n-1} + \\ && \left. \frac{\chi_4+1}{2} \cdot \frac{2}{n+1} \cdot \frac{(\chi_1+1)^{(i_1)}}{i_1!} \cdot \frac{(\chi_3+1)^{(i_2)}}{i_2!} \cdot \frac{(\chi_4+2)^{(i_3)}}{i_3!} \cdot \left( \frac{\xi_3-\xi_2}{\xi_1-\xi_2}\right)^{i_2} \cdot \left( \frac{\xi_4-\xi_2}{\xi_1-\xi_2}\right)^{i_3+1}\right|_{i_1+i_2+i_3=n-1} + \\ && \left. \frac{1}{n+1} \cdot \frac{(\chi_1+1)^{(i_1)}}{i_1!} \cdot \frac{(\chi_3+1)^{(i_2)}}{i_2!} \cdot \frac{(\chi_4+1)^{(i_3)}}{i_3!} \cdot \left( \frac{\xi_3-\xi_2}{\xi_1-\xi_2}\right)^{i_2} \cdot \left( \frac{\xi_4-\xi_2}{\xi_1-\xi_2}\right)^{i_3}\right|_{i_1+i_2+i_3=n} \quad = (i)\\ &&------------\\ && \left. \frac{1}{n+1} \cdot \frac{(\chi_1+1)^{(i_1+1)}}{i_1!} \cdot \frac{(\chi_3+1)^{(i_2)}}{i_2!} \cdot \frac{(\chi_4+1)^{(i_3)}}{i_3!} \cdot \left( \frac{\xi_3-\xi_2}{\xi_1-\xi_2}\right)^{i_2} \cdot \left( \frac{\xi_4-\xi_2}{\xi_1-\xi_2}\right)^{i_3} \right|_{i_1+i_2+i_3=n-1} + \\ && \left. \frac{1}{n+1} \cdot \frac{(\chi_1+1)^{(i_1)}}{i_1!} \cdot \frac{(\chi_3+1)^{(i_2+1)}}{i_2!} \cdot \frac{(\chi_4+1)^{(i_3)}}{i_3!} \cdot \left( \frac{\xi_3-\xi_2}{\xi_1-\xi_2}\right)^{i_2+1} \cdot \left( \frac{\xi_4-\xi_2}{\xi_1-\xi_2}\right)^{i_3}\right|_{i_1+i_2+i_3=n-1} + \\ && \left. \frac{1}{n+1} \cdot \frac{(\chi_1+1)^{(i_1)}}{i_1!} \cdot \frac{(\chi_3+1)^{(i_2)}}{i_2!} \cdot \frac{(\chi_4+1)^{(i_3+1)}}{i_3!} \cdot \left( \frac{\xi_3-\xi_2}{\xi_1-\xi_2}\right)^{i_2} \cdot \left( \frac{\xi_4-\xi_2}{\xi_1-\xi_2}\right)^{i_3+1}\right|_{i_1+i_2+i_3=n-1} + \\ && \left. \frac{1}{n+1} \cdot \frac{(\chi_1+1)^{(i_1)}}{i_1!} \cdot \frac{(\chi_3+1)^{(i_2)}}{i_2!} \cdot \frac{(\chi_4+1)^{(i_3)}}{i_3!} \cdot \left( \frac{\xi_3-\xi_2}{\xi_1-\xi_2}\right)^{i_2} \cdot \left( \frac{\xi_4-\xi_2}{\xi_1-\xi_2}\right)^{i_3}\right|_{i_1+i_2+i_3=n} \quad = (ii)\\ &&------------\\ && \left. \frac{1}{n+1} \cdot \frac{(\chi_1+1)^{(i_1)}}{(i_1-1)!} \cdot \frac{(\chi_3+1)^{(i_2)}}{i_2!} \cdot \frac{(\chi_4+1)^{(i_3)}}{i_3!} \cdot \left( \frac{\xi_3-\xi_2}{\xi_1-\xi_2}\right)^{i_2} \cdot \left( \frac{\xi_4-\xi_2}{\xi_1-\xi_2}\right)^{i_3} \right|_{i_1+i_2+i_3=n} + \\ && \left. \frac{1}{n+1} \cdot \frac{(\chi_1+1)^{(i_1)}}{i_1!} \cdot \frac{(\chi_3+1)^{(i_2)}}{(i_2-1)!} \cdot \frac{(\chi_4+1)^{(i_3)}}{i_3!} \cdot \left( \frac{\xi_3-\xi_2}{\xi_1-\xi_2}\right)^{i_2} \cdot \left( \frac{\xi_4-\xi_2}{\xi_1-\xi_2}\right)^{i_3}\right|_{i_1+i_2+i_3=n} + \\ && \left. \frac{1}{n+1} \cdot \frac{(\chi_1+1)^{(i_1)}}{i_1!} \cdot \frac{(\chi_3+1)^{(i_2)}}{i_2!} \cdot \frac{(\chi_4+1)^{(i_3)}}{(i_3-1)!} \cdot \left( \frac{\xi_3-\xi_2}{\xi_1-\xi_2}\right)^{i_2} \cdot \left( \frac{\xi_4-\xi_2}{\xi_1-\xi_2}\right)^{i_3}\right|_{i_1+i_2+i_3=n} + \\ && \left. \frac{1}{n+1} \cdot \frac{(\chi_1+1)^{(i_1)}}{i_1!} \cdot \frac{(\chi_3+1)^{(i_2)}}{i_2!} \cdot \frac{(\chi_4+1)^{(i_3)}}{i_3!} \cdot \left( \frac{\xi_3-\xi_2}{\xi_1-\xi_2}\right)^{i_2} \cdot \left( \frac{\xi_4-\xi_2}{\xi_1-\xi_2}\right)^{i_3}\right|_{i_1+i_2+i_3=n} \quad = (iii)\\ &&------------\\ && \left. \frac{(i_1+i_2+i_3)}{n+1} \cdot \frac{(\chi_1+1)^{(i_1)}}{i_1!} \cdot \frac{(\chi_3+1)^{(i_2)}}{i_2!} \cdot \frac{(\chi_4+1)^{(i_3)}}{i_3!} \cdot \left( \frac{\xi_3-\xi_2}{\xi_1-\xi_2}\right)^{i_2} \cdot \left( \frac{\xi_4-\xi_2}{\xi_1-\xi_2}\right)^{i_3} \right|_{i_1+i_2+i_3=n} + \\ && \left. \frac{1}{n+1} \cdot \frac{(\chi_1+1)^{(i_1)}}{i_1!} \cdot \frac{(\chi_3+1)^{(i_2)}}{i_2!} \cdot \frac{(\chi_4+1)^{(i_3)}}{i_3!} \cdot \left( \frac{\xi_3-\xi_2}{\xi_1-\xi_2}\right)^{i_2} \cdot \left( \frac{\xi_4-\xi_2}{\xi_1-\xi_2}\right)^{i_3}\right|_{i_1+i_2+i_3=n} \quad = (iv) \\ &&------------\\ &&\left. \frac{(\chi_1+1)^{(i_1)}}{i_1!} \cdot \frac{(\chi_3+1)^{(i_2)}}{i_2!} \cdot \frac{(\chi_4+1)^{(i_3)}}{i_3!} \cdot \left( \frac{\xi_3-\xi_2}{\xi_1-\xi_2}\right)^{i_2} \cdot \left( \frac{\xi_4-\xi_2}{\xi_1-\xi_2}\right)^{i_3}\right|_{i_1+i_2+i_3=n} \quad = (v) \\ &&------------\\ &&\left[ @{\mathfrak x}_1^n \right](rhs) (vi) \end{eqnarray}
The line $(i)$ is self-explanatory-- here we just used the definition of the Lauricella series. In $(ii)$ we absorbed the pre-factors into the appropriate Pochhammer symbols. In $(iii)$ we substituted $i_1+1 \rightarrow i_1$, $i_2+1 \rightarrow i_2$ and $i_3+1 \rightarrow i_3$ in the first, second and third lines respectively. In $(iv)$ we multiplied the numerator and the denominator by $i_1$, by $i_2$ and by $i_3$ in the first, second and the third lines respectively and then we took out the common factor. In $(v)$ we simplified the result. In $(vi)$ we used the fact that $\sum\limits_{\eta=1}^4 \chi_\eta = -2 $ and the multi-nomial expansion series. This completes the proof.