@article{ATTIA2025, 
author = {Mohamed Jalel ATTIA},
title = {Resolution of an isolated case of Pfaff hypergeometric transformation and new application of integer sequences},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {9},
pages = {20140-20156},
keywords = {hypergeometric series, terminating hypergeometric series, Pfaff transformation, Gauss's hypergeometric theorem, binomial sums, integer sequences, differential equation},
url = {https://www.sciopen.com/article/10.3934/math.2025900},
doi = {10.3934/math.2025900},
abstract = {A case of a Pfaff transformation is given by the following:                   2        F    1        (                                        l            ,            m                        2            m                              ;      x        )    =  (  1  −  x      )          −      l                      2        F    1        (                                        l            ,            m                        2            m                              ;              x                  x          −          1                      )    .In this paper, when    m is a negative integer, we define the Gaussian hypergeometric series as follows:                   2        F    1    ∗        (                                        l            ,            m                        2            m                              ;      x        )    =      ∑          k      =      0              −      m                  (      l              )        k            (      m              )        k                    k      !      (      2      m              )        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    2  m 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  +      x    2  , while the right-hand side is also a terminating hypergeometric series, but has    1 as the pole of multiplicity    2 given by    −            3      x      −      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    ∗        (                                        l            ,            m                        2            m                              ;              x                  x          −          1                      )    =                2        F    1    ∗        (                                        l            ,            m                        2            m                              ;      x        )    +      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          2      n      +      1       as a sum of two terminating hypergeometric series (with symmetric values) with the coefficients given in the integer sequence    A  046899 (these coefficients can be found in Pascal's triangle as an inclined column);(2) an expansion of        x          2      n      +      1        (  x  −  2  ) as a sum of two terminating hypergeometric series (with symmetric values) with the coefficients given in the integer sequence    A  033184.}
}