@article{Attia2026, 
author = {Mohamed Jalel Attia},
title = {Pfaff reduction for a terminating bivariate hypergeometric polynomial},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {4},
pages = {11239-11257},
keywords = {hypergeometric series of one and two variables, summation formulae and transformations, terminating hypergeometric series of one and two variables, Pfaff transformation, binomial sums, integer sequences, Pascal's triangle, differential equation, partial differential equation},
url = {https://www.sciopen.com/article/10.3934/math.2026462},
doi = {10.3934/math.2026462},
abstract = {This paper studies a terminating ("modified") Appell function         F    1    ∗    (  α  ,  β  ,  β  ,  2  β  ;  x  ,  y  )  =      ∑          m      =      0              −      β            ∑          n      =      0              −      β                  (      α              )                  m          +          n                    (      β              )        m            (      β              )        n                    (      2      β              )                  m          +          n                                        x        m                    y        n                    m      !      n      !        ,defined for integers    α  ≥  1 and    β  ≤  −  1, together with the associated terminating Gauss function             2        F    1    ∗    (  α  ,  β  ;  2  β  ;  z  )  =      ∑          k      =      0              −      β                  (      α              )                  k                    (      β              )        k                    (      2      β              )                  k                                z      k              k      !        .The classical Pfaff-type reduction for the non-terminating Appell function         F    1    (  α  ,  β  ,      β    ′    ,  β  +      β    ′    ;  x  ,  y  )  =      ∑          m      =      0        ∞        ∑          n      =      0        ∞              (      α              )                  m          +          n                    (      β              )        m            (              β        ′                    )        n                    (      β      +              β        ′                    )                  m          +          n                                        x        m                    y        n                    m      !      n      !        =      1          (      1      −      y              )                  α                                    2      F          1        (                                        α            ,            β                                                β            +                          β              ′                                          ;                        x          −          y                          1          −          y                      )    ,is recalled as background. The paper argues that, for the modified terminating case with        β    ′    =  β  ≤  −  1 and    γ  =  2  β, the direct Pfaff reduction fails and must be replaced by a corrected identity that involves an explicit additional term        V          (      α      ,      β      )        (  x  ,  y  ). A derivation of an explicit closed form for the correction term is given; it is first computed in low cases (notably    α  =  1  ,  2  ,  3), and then stated and proven in general by an induction on    α. The final formula exhibits a structured binomial/Pascal-type pattern in its coefficients and yields several corollaries, including simplified boundary cases (for example    β  =  −  1) and an open extension problem for unequal negative integers (   β  ,        β    ′  ) is stated.}
}