A case of a Pfaff transformation is given by the following:
$ \ _2F_1\left(\begin{array}{c} l, m \ 2m \end{array};x\right) = (1-x)^{-l}\ _2F_1\left(\begin{array}{c} l, m \ 2m \end{array}; \frac{x}{x-1}\right) . $
In this paper, when $ m $ is a negative integer, we define the Gaussian hypergeometric series as follows:
$ \ _2F_1^*\left(\begin{array}{c} l, m \ 2m \end{array};x\right) = \sum\limits_{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}\ _2F_1^*\left(\begin{array}{c} l, m \ 2m \end{array}; \frac{x}{x-1}\right) = \ _2F_1^*\left(\begin{array}{c} l, m \ 2m \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 $.
Citation: Mohamed Jalel ATTIA. Resolution of an isolated case of Pfaff hypergeometric transformation and new application of integer sequences[J]. AIMS Mathematics, 2025, 10(9): 20140-20156. doi: 10.3934/math.2025900
A case of a Pfaff transformation is given by the following:
$ \ _2F_1\left(\begin{array}{c} l, m \ 2m \end{array};x\right) = (1-x)^{-l}\ _2F_1\left(\begin{array}{c} l, m \ 2m \end{array}; \frac{x}{x-1}\right) . $
In this paper, when $ m $ is a negative integer, we define the Gaussian hypergeometric series as follows:
$ \ _2F_1^*\left(\begin{array}{c} l, m \ 2m \end{array};x\right) = \sum\limits_{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}\ _2F_1^*\left(\begin{array}{c} l, m \ 2m \end{array}; \frac{x}{x-1}\right) = \ _2F_1^*\left(\begin{array}{c} l, m \ 2m \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 $.
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Mohamed Jalel ATTIA. Resolution of an isolated case of Pfaff hypergeometric transformation and new application of integer sequences[J]. AIMS Mathematics, 2025, 10(9): 20140-20156. doi: 10.3934/math.2025900
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