# How to explain a formula from a generalized hypergeometric function

I am asking about the formula 07.27.03.0003.01 found here https://functions.wolfram.com/PDF/Hypergeometric3F2.pdf, specifically the second formula on page 2 of the PDF file of the link above.

What is the meaning of the vertical line on the second line of this formula, and what about the quotients on the left and the right of such vertical line? Are they a product?

\begin{aligned}_3F_2(a,b,c;d,e;1)&=\sqrt{\Gamma(1-a)}~\sqrt{\Gamma(1-b)}~\sqrt{\Gamma(1-c)}~\Gamma(d)\Gamma(e)\sqrt{\Gamma(-a-b-c+d+e)}\\&\left\langle\frac{d-a-b-1}2~\frac{e-a-c-1}2~\frac{b+d-a-1}2~\frac{a-c-e+1}2~\left|~\frac{d-a-b-1}2~\frac{e-a-c-1}2~\frac{d+e-b-c}2~-1~\frac{b+d-c-e}2\right\rangle\right/\\&\left(\sqrt{d+e-b-c-1}~\sqrt{\Gamma(d-a)}~\sqrt{\Gamma(d-b)}~\sqrt{\Gamma(d-c)}~\sqrt{\Gamma(e-a)}~\sqrt{\Gamma(e-b)}~\sqrt{\Gamma(e-c)}\right)/;\Re(d+e-a-b-c)>0\end{aligned}

Where $$\langle j_1\ j_2\ m_1\ m_2 | j_1\ j_2\ j\ m \rangle$$ means Clebsch-Gordan Coefficient ClebschGordan[{j1, m1}, {j2, m2}, {j, m}]

It looks like ThreeJSymbol in OutputForm. The inputForm used what is called ClebschGordan.

expr=HypergeometricPFQ[{a,b,c},{d,e},1]==Sqrt[Gamma[1-a]] Sqrt[Gamma[1-b]] Sqrt[Gamma[1-c]] Gamma[d] Gamma[e] Sqrt[Gamma[d+e-a-b-c]] (ClebschGordan[{(d-a-b-1)/2,(b+d-a-1)/2},{(e-a-c-1)/2,(a+1-c-e)/2},{(d+e-b-c)/2-1,(b+d-c-e)/2}]/(Sqrt[-1-b-c+d+e] Sqrt[Gamma[-a+d]] Sqrt[Gamma[-b+d]] Sqrt[Gamma[-c+d]] Sqrt[Gamma[-a+e]] Sqrt[Gamma[-b+e]] Sqrt[Gamma[-c+e]]))/;Re[d+e-a-b-c]>0


 TraditionalForm[expr]


In otherwords this part in pdf

is meant to be this part

It could be the Latex used to generate the PDF is faulty. I tried lulatex in TL 2023, and I do not get the vertical line.

and what about the quotients on the left and the right of such vertical line? are they a product?

You can see they are not products from the above.

• For your first line about Input/OutputForm, do you mean the two are the same function, but just visually different? If so, the two functions are slightly different functions with this identity Commented May 18 at 17:15
• @ТymaGaidash٠ In the input, from the above linked to web page, you see Mathematica had ClebschGordan. In the Output it had ThreeJSymbol. I do not know anything about these as I never used them so I can not really answer about the math itself. But the point is, the Latex generated using TeXForm does not look like what is in the PDF file. i.e. there is no vertical line. So my guess is that the vertical line came from using old Latex conversion from old version or something like this. I am using V 14 and do not see vertical line in the Latex. You can try yourself and see also. Commented May 18 at 20:14