△_(h)-GOULD-HOPPER APPELL POLYNOMIALS

摘要

In this paper,we introduce the △_(h)-Gould-Hopper Appell polynomials An(x,y;h)via h-Gould-Hopper polynomials G_(n)^(h)(x,y).These polynomials reduces to △_(h)-Appell polynomials in the case y=0,△-Appell polynomials in the case y=0 and h=2D-Appell polynomials in the case h→0,2D △-Appell polynomials in the case h=1 and Appell polynomials in the case h→0 and y=0.We obtain some well known main properties and an explicit form,determinant representation,recurrence relation,shift operators,difference equation,integro-difFerence equation and partial difference equation satisfied by them.Determinants satisfied by △_(h)-Gould-Hopper Appell polynomials reduce to determinant of all subclass of the usual polynomials.Recurrence,shift operators and difference equation satisfied by these polynomials reduce to recurrence,shift operators and difference equation of △_(h)-Appell polynomials,△-Appell polynomials;recurrence,shift operators,differential and integro-differential equation of 2D-Appell polynomials,recurrence,shift operators,integrodifference equation of 2D △-Appell polynomials,recurrence,shift operators,differential equation of Appell polynomials in the corresponding cases.In the special cases of the determining functions,we present the explicit forms,determinants,recurrences,difference equations satisfied by the degenerate Gould-Hopper Carlitz Bernoulli polynomials,degenerate Gould-Hopper Carlitz Euler polynomials,degenerate Gould-Hopper Genocchi polynomials,△_(h)-Gould-Hopper Boole polynomials and △_(h)-Gould-Hopper Bernoulli polynomials of the second kind.In particular cases of the degenerate Gould-Hopper Carlitz Bernoulli polynomials,degenerate Gould-Hopper Genocchi polynomials,△_(h)-Gould-Hopper Boole polynomials and △_(h)-Gould-Hopper Bernoulli polynomials of the second kind,corresponding determinants,recurrences,shift operators and difference equations reduce to all subclass of degenerate so-called families except for Genocchi polynomials recurrence,shift operators,and differential equation.Degenerate Gould-Hopper Carlitz Euler polynomials do not satisfy the recurrences and differential equations of 2D-Euler and Euler polynomials.