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$\Delta _{h}$-GOULD-HOPPER APPELL POLYNOMIALS
Mehmet Ali ÖZARSLAN, Banu YILMAZ YAŞAR

Acta mathematica scientia,Series B. 2021, 41 (4 ):
1196-1222.
DOI: 10.1007/s10473-021-0411-y
In this paper, we introduce the $\Delta _{h}$-Gould-Hopper Appell polynomials $\mathcal{A}_{n}(x,y;h)$ via $h$-Gould-Hopper 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, integro-difference equation and partial difference equation satisfied by them. Determinants satisfied by $ \Delta _{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 $\Delta _{h}$-Appell polynomials, $\Delta $-Appell polynomials; recurrence, shift operators, differential and integro-differential equation of $2D$-Appell polynomials, recurrence, shift operators, integro-difference 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 Gould-Hopper Carlitz Bernoulli polynomials, degenerate Gould-Hopper Carlitz Euler polynomials, degenerate Gould-Hopper Genocchi polynomials, $\Delta _{h}$-Gould-Hopper Boole polynomials and $\Delta _{h}$-Gould-Hopper Bernoulli polynomials of the second kind. In particular cases of the degenerate Gould-Hopper Carlitz Bernoulli polynomials, degenerate Gould-Hopper Genocchi polynomials, $\Delta _{h}$-Gould-Hopper Boole polynomials and $\Delta _{h}$-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.

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