Treffer: generalised operator reduction formula for multiple hypergeometric series nf x1 xn: Generalised operator reduction formula for multiple hypergeometric series \(^ NF(x_ 1,\dots ,x_ N)\)

The reduction formula obtained in a previous paper [ibid. 16, 1813--1825 (1983; Zbl 0527.33001)] has been generalised as follows. If \(^ NF'\) and \(^ NF''\) are generalised hypergeometric series of \(N\) variables (GHS-N), then, the operation on \(^ NF''(x_ 1t_ 1,...,x_ nt_ n),^ NF'(\partial /\partial t_ 1,...,\partial /\partial t_ N)\) gives, at \(t_ 1=t_ 2=...=t_ N=0\), a function \(^ NF(x_ 1,...,x_ N)\), which is, again, a GHS-N. The differentiation procedure can be regarded as an algebraic \(\Omega\)-multiplication which gives rise to a group-theoretical interpretation of the method. A concept of \(\Omega\)-equivalent relations has been introduced which allows systematisation of numerous results obtained in special functions theory. As the functions \(^ NF\) comprise a number of physically interesting series, the operator factorisation method seems to be applicable to many physical problems providing a possibility of reducing any \({}^ NF\) to simpler functions of the same class.