@article{Bedratyuk_Luno_2020, title={Some properties of generalized hypergeometric Appell polynomials}, volume={12}, url={https://journals.pnu.edu.ua/index.php/cmp/article/view/3892}, DOI={10.15330/cmp.12.1.129-137}, abstractNote={<p>Let $x^{(n)}$ denotes the Pochhammer symbol (rising factorial) defined by the formulas $x^{(0)}=1$ and $x^{(n)}=x(x+1)(x+2)\cdots (x+n-1)$ for $n\geq 1$. In this paper, we present a new real-valued Appell-type polynomial family $A_n^{(k)}(m,x)$, $n, m \in {\mathbb{N }_0$, $k \in {\mathbb{N },$ every member of which is expressed by mean of the generalized hypergeometric function ${}_{p} F_q \begin{bmatrix} \begin{matrix} a_1, a_2, \ldots, a_p \:\\ b_1, b_2, \ldots, b_q \end{matrix} \: \Bigg| \:z \end{bmatrix}= \sum\limits_{k=0}^{\infty} \frac{a_1^{(k)} a_2^{(k)} \ldots a_p^{(k) }{b_1^{(k)} b_2^{(k)} \ldots b_q^{(k) } \frac{z^k}{k!}$ as follows $$ A_n^{(k)}(m,x)= x^n{}_{k+p} F_q \begin{bmatrix} \begin{matrix} {a_1}, {a_2}, {\ldots}, {a_p}, {\displaystyle -\frac{n}{k }, {\displaystyle -\frac{n-1}{k }, {\ldots}, {\displaystyle-\frac{n-k+1}{k }\:\\ {b_1}, {b_2}, {\ldots}, {b_q} \end{matrix} \: \Bigg| \: \displaystyle \frac{m}{x^k} \end{bmatrix} $$ and, at the same time, the polynomials from this family are Appell-type polynomials.</p> <p>The generating exponential function of this type of polynomials is firstly discovered and the proof that they are of Appell-type ones is given. We present the differential operator formal power series representation as well as an explicit formula over the standard basis, and establish a new identity for the generalized hypergeometric function. Besides, we derive the addition, the multiplication and some other formulas for this polynomial family.</p>}, number={1}, journal={Carpathian Mathematical Publications}, author={Bedratyuk, L. and Luno, N.}, year={2020}, month={Jun.}, pages={129-137} }