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 |
References:
| [1] | Constantine A.G., Muirhead R.J.: Partial differential equations for hypergeometric functions of two argument matrix. J. Multivar. Anal. 3, 332–338 (1972) · Zbl 0256.33007 · doi:10.1016/0047-259X(72)90020-6 |
| [2] | James A.T.: Special functions of matrix and single argument in statistics. In: Askey, R.A. (ed) Theory and Application of Special Functions, Academic Press, New York (1975) · Zbl 0326.33010 |
| [3] | Jódar L., Sastre J.: On the Laguerre matrix polynomials. Util. Math. 53, 37–48 (1998) · Zbl 0990.33008 |
| [4] | Jódar L., Cortés J.C.: Some properties of Gamma and Beta matrix functions. Appl. Math. Lett. 11, 89–93 (1998) · Zbl 1074.33002 · doi:10.1016/S0893-9659(97)00139-0 |
| [5] | Jódar L., Cortés J.C.: On the hypergeometric matrix function. J. Comp. Appl. Math. 99, 205–217 (1998) · Zbl 0933.33004 · doi:10.1016/S0377-0427(98)00158-7 |
| [6] | Lebedev N.N.: S.Special Functions and Their Applications. Dover Publications Inc., New York (1972) |
| [7] | Metwally M.S., Mohamed M.T., Shehata A.: On Hermite–Hermite matrix polynomials. Math. Bohemica. 133, 421–434 (2008) · Zbl 1199.15079 |
| [8] | Pasricha B.R.: Some integrals involving Humbert function. Proc. Indian Acad. Sci. 18, 11–18 (1943) · Zbl 0063.06125 |
| [9] | Sayyed K.A.M., Metwally M.S., Batahan R.S.: Gegenbauer matrix polynomials and second order matrix differential equations. Divul. Mat. 12, 101–115 (2004) · Zbl 1102.33010 |
| [10] | Varma R.S.: On Humbert functions. Ann. Math. 42, 429–436 (1941) · Zbl 0027.21301 · doi:10.2307/1968908 |
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