Pfaff reduction for a terminating bivariate hypergeometric polynomial

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

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

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

$${}_2{F}_1^{\ast }(\alpha ,\beta ;2\beta ;z)=\sum _{k=0}^{-\beta }\frac{(\alpha {)}_k(\beta {)}_k}{(2\beta {)}_k}\frac{z^k}{k!}.$$

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

$$F_1(\alpha ,\beta ,{\beta }^{\prime },\beta +{\beta }^{\prime };x,y)=\sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{m+n}(\beta {)}_m({\beta }^{\prime }{)}_n}{(\beta +{\beta }^{\prime }{)}_{m+n}}\frac{x^m{y}^n}{m!n!}=\frac{1}{(1-y{)}^{\alpha }}{}_{2}F{}_1\left(\begin{array}{c}\alpha ,\beta \\ \beta +{\beta }^{\prime }\end{array};\frac{x-y}{1-y}\right),$$

is recalled as background. The paper argues that, for the modified terminating case with ${\beta }^{\prime }=\beta \le -1$ 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 ( $\beta ,{\beta }^{\prime }$) is stated.