Generating functions of a quadruple hypergeometric polynomial with SL symmetry. (English) Zbl 0831.33007
The quadruple polynomial under consideration is a special case of the author’s \(K_{12}\), viz.,
\[
\sum_p\;{{(a)_{p_1+ p_2+ p_3+ p_4} (-m_1 )_{p_1} (-m_2 )_{p_2} (-m_3 )_{p_3} (-m_4 )_{p_4} x_1^{p_1} x_2^{p_2} x_3^{p_3} x_4^{p_4}} \over {(a)_{p_1+ p_2} (a)_{p_3+ p_4} p_1! p_2! p_3! p_4!}} .
\]
With the aid of his multidimensional extension of Bailey’s transform (1949) an octuple series is established for
\[
\sum_m\;{{D(m_1, m_2, m_3, m_4) t_1^{m_1} t_2^{m_2} t_3^{m_3} t_4^{m_4}} \over {m_1! m_2! m_3! m_4!}} K,
\]
where \(K\) is the above-mentioned \(K_{12}\). By proper choice of the quadruple sequence \(D\), the octuple series reduces to a known function, and a generating function for the \(K\) polynomials results. This is done in several ways, and the generating functions so obtained involve, for instance, Appell functions and modified Bessel functions.
Reviewer: P.W.Karlsson (Virum)
MSC:
33C70 | Other hypergeometric functions and integrals in several variables |
33C65 | Appell, Horn and Lauricella functions |
33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |