Generalised operator reduction formula for multiple hypergeometric series NF(x1, . . ., xN)

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Abstract
The reduction formula obtained in a previous paper, (Niukkanen, 1983), has been generalised as follows. If NF' and NF" are generalised hypergeometric series of N variables (GHS-N), then, the operation on NF" (x1t1, . . ., xntn), NF'( delta / delta t1, . . ., delta / delta tN) gives, at t1=t2,= . . . =tN=0, a function NF(x1, . . ., xN), 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.

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