
$\Delta _{h}$GOULDHOPPER APPELL POLYNOMIALS
Mehmet Ali ÖZARSLAN, Banu YILMAZ YAŞAR
Acta mathematica scientia,Series B. 2021, 41 (4):
11961222.
DOI: 10.1007/s104730210411y
In this paper, we introduce the $\Delta _{h}$GouldHopper Appell polynomials $\mathcal{A}_{n}(x,y;h)$ via $h$GouldHopper polynomials $ G_{n}^{h}(x,y)$. These polynomials reduces to $\Delta _{h}$Appell polynomials in the case $y=0$, $\Delta $Appell polynomials in the case $y=0$ and $ h=1 $, $2D$Appell polynomials in the case $h\rightarrow 0$, $2D$ $ \Delta$Appell polynomials in the case $h=1$ and Appell polynomials in the case $ h\rightarrow 0$ and $y=0$. We obtain some well known main properties and an explicit form, determinant representation, recurrence relation, shift operators, difference equation, integrodifference equation and partial difference equation satisfied by them. Determinants satisfied by $ \Delta _{h}$GouldHopper 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 $\Delta _{h}$Appell polynomials, $\Delta $Appell polynomials; recurrence, shift operators, differential and integrodifferential equation of $2D$Appell polynomials, recurrence, shift operators, integrodifference equation of $2D$ $\Delta$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 GouldHopper Carlitz Bernoulli polynomials, degenerate GouldHopper Carlitz Euler polynomials, degenerate GouldHopper Genocchi polynomials, $\Delta _{h}$GouldHopper Boole polynomials and $\Delta _{h}$GouldHopper Bernoulli polynomials of the second kind. In particular cases of the degenerate GouldHopper Carlitz Bernoulli polynomials, degenerate GouldHopper Genocchi polynomials, $\Delta _{h}$GouldHopper Boole polynomials and $\Delta _{h}$GouldHopper Bernoulli polynomials of the second kind, corresponding determinants, recurrences, shift operators and difference equations reduce to all subclass of degenerate socalled families except for Genocchi polynomials recurrence, shift operators, and differential equation. Degenerate GouldHopper Carlitz Euler polynomials do not satisfy the recurrences and differential equations of $2D$Euler and Euler polynomials.
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