A case of a Pfaff transformation is given by the following:

$${}_2{F}_1\left(\begin{array}{c}l,m2m\end{array};x\right)=(1-x{)}^{-l}{}_2{F}_1\left(\begin{array}{c}l,m2m\end{array};\frac{x}{x-1}\right).$$

In this paper, when $m$ is a negative integer, we define the Gaussian hypergeometric series as follows:

$${}_2{F}_1^{\ast }\left(\begin{array}{c}l,m2m\end{array};x\right)=\sum _{k=0}^{-m}\frac{(l{)}_k(m{)}_k}{k!(2m{)}_k}{x}^k,$$

which is well-defined, as it is a terminating hypergeometric series since the summation is only for $k=0,..,-m$; additionally, the fact that $2m$ is a negative integer does not make any harm. With this definition, if we take $m=-1$ and $l=1$, then the left-hand side is a terminating hypergeometric series equal to $1+\frac{x}{2}$, while the right-hand side is also a terminating hypergeometric series, but has $1$ as the pole of multiplicity $2$ given by $-\frac{3x-2}{2(x-1{)}^2}$. More generally, with the definition above, we prove that this case of the Pfaff transformation does not hold for any positive integer $l$ and for any negative integer $m$. Additionally, an analysis aims to solve this situation. In fact, we give a new expression $V^{(l,m)}(x)$ depending on $l,m$, and $x$ such that

$$(1-x{)}^{-l}{}_2{F}_1^{\ast }\left(\begin{array}{c}l,m2m\end{array};\frac{x}{x-1}\right)={}_2{F}_1^{\ast }\left(\begin{array}{c}l,m2m\end{array};x\right)+V^{(l,m)}(x),$$

for any positive integer $l$ and for any negative integer $m$. As a very interesting consequence we present a corollary from the boundary conditions, thereby providing the following:

(1) an expansion of $x^{2n+1}$ as a sum of two terminating hypergeometric series (with symmetric values) with the coefficients given in the integer sequence $A046899$ (these coefficients can be found in Pascal's triangle as an inclined column);

(2) an expansion of $x^{2n+1}(x-2)$ as a sum of two terminating hypergeometric series (with symmetric values) with the coefficients given in the integer sequence $A033184$.

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.