An identity on $\small{}_pF_q\left(\left.\begin{array}{c} a_1+1,a_2+1,\dots ,a_p+1\\ b_1+1,b_2+1,\dots ,b_q+1\end{array}\right| z\right)$

We first use the differentiation formula for the generalized hypergeometric function \begin{equation} \frac{a_1a_2\dots a_{p}}{b_1b_2\dots b_q}{}_pF_q\left(\left.\begin{array}{c} c_1,c_2,\dots ,c_p\\ d_1,d_2,\dots ,d_q \end{array}\right| z\right)=\frac{d}{dz}{}_pF_q\left(\left.\begin{array}{c} a_1,a_2,\dots ,a_p\\ b_1,b_2,\dots ,b_q \end{array}\right| z\right) \end{equation} Then, the LHS of the proposed identity can be written as \begin{equation} _pF_q\left(\left.\begin{array}{c} a_1,a_2,\dots ,a_p\\ b_1,b_2,\dots ,b_q \end{array}\right| z\right)+z\,\frac{a_1a_2\dots a_{p-1}}{b_1b_2\dots b_q}{}_pF_q\left(\left.\begin{array}{c} c_1,c_2,\dots ,c_p\\ d_1,d_2,\dots ,d_q \end{array}\right| z\right)=\left( 1+\frac{z}{a_p}\frac{d}{dz} \right){} _pF_q\left(\left.\begin{array}{c} a_1,a_2,\dots ,a_p\\ b_1,b_2,\dots ,b_q \end{array}\right| z\right)\tag{1}\label{eq1} \end{equation} To differentiate the hypergeometric function, we use the Euler's integral transform \begin{align} & _pF_q\left(\left.\begin{array}{c} a_1,a_2,\dots ,a_p\\ b_1,b_2,\dots ,b_q \end{array}\right| z\right)\\ &=\frac{\Gamma(b_q)}{\Gamma(a_p)\Gamma(b_q-b_p)} \int_0^1t^{a_p-1}\left( 1-t \right)^{b_q-a_p-1}{}_{p-1}F_{q-1}\left(\left.\begin{array}{c} a_1,a_2,\dots ,a_{p-1}\\ b_1,b_2,\dots ,b_{q-1} \end{array}\right| t\right)\,dt \end{align} Here $b_q=a_p+1$, then \begin{align} _pF_q\left(\left.\begin{array}{c} a_1,a_2,\dots ,a_p\\ b_1,b_2,\dots ,b_q \end{array}\right| z\right)&= a_p \int_0^1t^{a_p-1}{}_{p-1}F_{q-1}\left(\left.\begin{array}{c} a_1,a_2,\dots ,a_{p-1}\\ b_1,b_2,\dots ,b_{q-1} \end{array}\right| zt\right)\,dt\\ &=\frac{a_p}{z^{a_p}} \int_0^zu^{a_p-1}{}_{p-1}F_{q-1}\left(\left.\begin{array}{c} a_1,a_2,\dots ,a_{p-1}\\ b_1,b_2,\dots ,b_{q-1} \end{array}\right| u\right)\,du \end{align} Then \begin{align} \frac{d}{dz}&\,{} _pF_q\left(\left.\begin{array}{c} a_1,a_2,\dots ,a_p\\ b_1,b_2,\dots ,b_q \end{array}\right| z\right)\\ &=\frac{a_p}{z}\,{}_{p-1}F_{q-1}\left(\left.\begin{array}{c} a_1,a_2,\dots ,a_{p-1}\\ b_1,b_2,\dots ,b_{q-1} \end{array}\right| z\right)-\frac{a_p}{z} \,{}_pF_q\left(\left.\begin{array}{c} a_1,a_2,\dots ,a_p\\ b_1,b_2,\dots ,b_q \end{array}\right| z\right) \end{align} Plugging this expression in eq. \eqref{eq1} we find theRHS of the proposed identity.