# What kind of hypergeometric function is it?

I found a formula for an integral of a product of three Bessel functions at The Wolfram Functions Site:

I cannot understand what kind of hypergeometric function it is. The Mathematica code given for it is:

HypergeometricPFQ[{{(α + λ + μ + ν)/2, (α + λ + μ - ν)/2}, {}, {}},
{{}, {λ + 1}, {μ + 1}}, a^2/c^2, b^2/c^2]


When I try to evaluate it in Mathematica 9, the last argument is highlighted in red and I get an error message:

HypergeometricPFQ::argrx: HypergeometricPFQ called with 4 arguments; 3 arguments are expected. >>

• It's a Kampé de Fériet function. Unfortunately, Mathematica does not yet have support for this multivariate hypergeometric function, so I don't quite understand how they generated that Mathematica syntax. Commented May 22, 2013 at 16:16
• What version of Mathematica you have? Commented May 22, 2013 at 16:20
• In that case, it doesn't seem we'll be seeing more multivariate hypergeometric functions anytime soon. All they have is one of Appell's quartet and arguably Meijer's function. Commented May 22, 2013 at 17:07
• @Spawn1701D Several versions, including 9. Commented May 22, 2013 at 17:32
• I knew the Unicode in this case. An easier method would be to type out the $\LaTeX$, wait for MathJax to render the Greeks, and then copy whatever MathJax is displaying. Commented May 22, 2013 at 17:52

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:

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

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.