A note on two integrals involving product of two generalized hypergeometric functions (Q6619571)

Consider the function \(f(x)= x^{c-1} (1-x) ^{2c-1} (1- \frac{x}{3})^{2c-1} (1- \frac{x}{4})^{c-1}.\) For \(\mathrm{Re}(c)>0\) and \(\mathrm{Re} (2c-a-b)>-1,\) the authors obtain, in terms of gamma functions, the expression of the integrals\N\[\N\int_{0}^{1_2}F_{1} (A, B; C; g(x)) \; \cdot\; {}_{2}F_{2}( D, E; G,H; h(x)) f(x) dx,\N\]\Nwhen \(A=a, B= b, C=\frac{a+b+1}{2}, D=\frac{2c-a+1}{2}, E= \frac{2c-b+1}{2}, G=c, H=\frac{2c-a-b+1}{2}\). Here \(h(x)=9x (1-x)^2 (1- \frac{x}{3})^2 (1- \frac{x}{4})\) and either \(g(x)=\frac{9}{4} x(1- \frac{x}{3})^{2},\) or \(g(x)=(1- \frac{x}{4})(1- x) ^{2}.\)\N\NThe proof is based on a result in [\textit{M. A. Rakha} et al., J. Inequal. Spec. Funct. 3, No. 1, 10--27 (2012; Zbl 1312.33013)], concerning the expression of \[\int_{0}^{1 _2}F_{1} (A, B; C; g(x)) f(x) dx\] for the two choices of \(g\) as above.\N\NAs an application, for some particular values of \(a,b\) the corresponding integrals are deduced.