Pfaff reduction for a terminating bivariate hypergeometric polynomial

This paper studies a terminating ("modified") Appell function

$ F_1^*(\alpha, \beta, \beta, 2\beta;x,y) = \sum\limits_{m = 0}^{-\beta}\sum\limits_{n = 0}^{-\beta}\frac{(\alpha)_{m+n}(\beta)_m(\beta)_n}{(2\beta)_{m+n}}\frac{x^my^n}{m!n!}, $

defined for integers $ \alpha\geq 1 $ and $ \beta\leq -1 $, together with the associated terminating Gauss function

$ _2F_1^*(\alpha, \beta; 2\beta;z) = \sum\limits_{k = 0}^{-\beta}\frac{(\alpha)_{k}(\beta)_k}{(2\beta)_{k}}\frac{z^k}{k!}. $

The classical Pfaff-type reduction for the non-terminating Appell function

$ F_1(\alpha, \beta, \beta', \beta+\beta';x,y) = \sum\limits_{m = 0}^\infty\sum\limits_{n = 0}^\infty\frac{(\alpha)_{m+n}(\beta)_m(\beta')_n}{(\beta+\beta')_{m+n}}\frac{x^my^n}{m!n!} = \frac{1}{(1-y)^{\alpha}}\ _2\!F\!_1\left(\begin{array}{c} \alpha,\beta\\ \beta+\beta'\end{array}; \frac{x-y}{1-y}\right), $

is recalled as background. The paper argues that, for the modified terminating case with $ \beta' = \beta\leq -1 $ and $ \gamma = 2\beta $, the direct Pfaff reduction fails and must be replaced by a corrected identity that involves an explicit additional term $ V^{(\alpha, \beta)}(x, y) $. A derivation of an explicit closed form for the correction term is given; it is first computed in low cases (notably $ \alpha = 1, 2, 3 $), and then stated and proven in general by an induction on $ \alpha $. The final formula exhibits a structured binomial/Pascal-type pattern in its coefficients and yields several corollaries, including simplified boundary cases (for example $ \beta = - 1 $) and an open extension problem for unequal negative integers ($ \beta, \ \beta' $) is stated.