What kind of hypergeometric function is it?

It is Kampé de Fériet function, introduced in Joseph Kampé de Fériet, "La fonction hypergéométrique.", Mémorial des sciences mathématiques, Paris, Gauthier-Villars.

Its definition is given on Notations page:

_{(source: wolfram.com)}

and, in an alternative form, in Wikipedia:

$${}^{p+q}f_{r+s}\left( \begin{matrix} a_1,\cdots,a_p\colon b_1,b_1{}';\cdots;b_q,b_q{}'; \\ c_1,\cdots,c_r\colon d_1,d_1{}';\cdots;d_s,d_s{}'; \end{matrix} x,y\right)=\\ \sum_{m=0}^\infty\sum_{n=0}^\infty\frac{(a_1)_{m+n}\cdots(a_p)_{m+n}}{(c_1)_{m+n}\cdots(c_r)_{m+n}}\frac{(b_1)_m(b_1{}')_n\cdots(b_q)_m(b_q{}')_n}{(d_1)_m(d_1{}')_n\cdots(d_s)_m(d_s{}')_n}\cdot\frac{x^my^n}{m!n!}.$$

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In this case the Kampé de Fériet function can be represented as an infinite sum of hypergeometric functions:

$$\begin{align*} &\int_0^\infty t^{\alpha-1}J_\lambda(a\,t)\,J_\mu(b\,t)\,J_\nu(c\,t)\, dt=\\&\small\pi^{-1}\,2^{\alpha-1}a^\lambda\,b^\mu\,c^{-\alpha-\lambda-\mu}\sin\left(\frac{\pi}{2}(\alpha+\lambda+\mu-\nu)\right)\times\\&\small\sum_{m=0}^\infty\frac{\Gamma\left(m+\frac{\alpha+\lambda+\mu-\nu}{2}\right)\Gamma\left(m+\frac{\alpha+\lambda+\mu+\nu}{2}\right)\,_2F_1\left(m+\frac{\alpha +\lambda +\mu -\nu}{2},m+\frac{\alpha +\lambda +\mu +\nu}{2};\mu+1;\frac{b^2}{c^2}\right)}{(m!)^2\,\Gamma(m+\lambda+1)}\left(\frac{a}{c}\right)^{2m} \end{align*}$$

Since your question's been answered, let me tell you about the handy listing of notations used by the Wolfram Functions site. In particular, if I scroll down to the "F" section of this page, you'll see an explanation that you are indeed looking at Kampé de Fériet's function.