Extending the factorization principle to hypergeometric series of general form (Q1585641)

In previous works, the author has developed a factorization method, called \(\Omega\) -multiplication, which enables him to express generalized multivariable hypergeometric series in terms of hypergeometric functions of simpler form [see \textit{A. W. Niukkanen}, J. Phys. A 17, L731--L736 (1984; Zbl 0556.33004); Russ. Math. Surv. 43, No. 3, 218-220 (1988; Zbl 0667.33004); Math. Notes 50, No. 1, 702-706 (1991; Zbl 0753.33013)]. The method yields a particularly simple reduction under the assumption that all spectral parameters occurring in the generalized hypergeometric series are nonnegative integers. In the paper under review the author extends the operation of \(\Omega\)-multiplication to remove the non-negativity condition on the spectral parameters. The resulting factorization method can be applied to a much larger class of multivariable hypergeometric series. As an illustration, the author derives transformation formulas for the Horn series \(G_3\). The results are compared with those obtained in the literature by means of much more complicated methods. In particular, the transformation formula relating the Horn function \(G_3\) to the Appell function \(F_2\) occurring as relation (10) in \textit{A. Erdélyi} [Proc. R. Soc. Edinb., Sect. A 62, 378-385 (1948; Zbl 0031.02101)] is shown to be erroneous. The corrected formula is provided.