On Humbert matrix functions and their properties. (English) Zbl 1301.33004
Summary: In this paper, we consider a Humbert matrix function in the following form:
\[
J_{A,B}(z)=\left(\frac{z}{3}\right)^{A+B}\Gamma^{-1}(A+I)\Gamma^{-1}(B+I)\,_0F_2\left(-,-;A+I,B+I; -\frac{z^3}{27}\right),\quad |z|<\infty,
\]
where
\[
\begin{aligned} _0F_2\left(-,-;A+I,B+I;-\frac{z^3}{27}\right)= & \Gamma(A+I)\Gamma(B+I)\\ & \times\sum\limits_{k=0}^\infty\frac{(-1)^k\Gamma^{-1}(A+(k+1)I)\Gamma^{-1}(B+(k+1)I)}{k!}\\ & \times\left(\frac{z}{3}\right)^{3k},\end{aligned}
\]
and for this function we present order and type, integral representations and differential recurrence relations.
MSC:
33C05 | Classical hypergeometric functions, \({}_2F_1\) |
33C20 | Generalized hypergeometric series, \({}_pF_q\) |
33C60 | Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions) |
33C65 | Appell, Horn and Lauricella functions |
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