New class of generating functions associated with generalized hypergeometric polynomials. (Q1410270)

The authors consider a multiple series, which is here stated in a simplified manner as follows. Let \(q\in\mathbb{N}^r\), \(m\in \mathbb{C}^r\), and set \[ S= \sum^\infty_{n=0} P(n)\Biggl(\sum^{q\cdot k\leq n}_{k_1,\dots, k_r= 0} {(-n)_{q\cdot k} B(k)\over D(k)} A(k)\Biggr){t^n\over n!}, \] where \(A(k)\) is arbitrary, \(P(n)\) is a Pochhammer symbol fraction, \(B(k)\) is a product of factors of the form \((b+ hn)_{m\cdot k}\), and \(D(k)\) is a product of factors of the form \((d+ gn)_{(m+ q)\cdot k}\). Suitable conditions to ensure convergence are understood. By series manipulations the authors obtain a general theorem of the form \(S= T= U\), where \(T\) and \(U\) are \(r\)-dimensional series whose terms involve Wright's hypergeometric function \({_p\Psi_q}[t]\) and \({_pF_q}[t]\), respectively. By specialization, they obtain a number of known generating functions involving various hypergeometric functions.