Resolution of an isolated case of Pfaff hypergeometric transformation and new application of integer sequences

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 $A033184$.