Reduction for a Terminating Bivariate Hypergeometric Appell Series F1 (II)

This paper studies a terminating (“modified”) Appell function F1*(α,β,β′,β+β′;x,y), with α∈Z≥1 and β,β′∈Z≤−1, together with the associated terminating Gauss function 2F1*α,ββ+β′;z. The reduction formula for the Appell function F1*(α,β,β′,β+β′;x,y) to 1(1−y)α2F1α,ββ+β′;x−y1−y breaks down when (β,β′) are integers less than or equal to −1 and it needs to be substituted with a revised identity that includes an explicit additional correction term CORR(α,β,β′)(x,y). This correction term is initially computed for specific cases (particularly for α=1,2,3) and subsequently formulated and verified generally through mathematical induction on α. The final expression demonstrates a structured pattern reminiscent of binomial/Pascal coefficients and leads to various corollaries, including simplified boundary scenarios (such as when α=1).