Some properties of generalized hypergeometric Appell polynomials
Let x^(n) denotes the Pochhammer symbol (rising factorial) defined by the formulas x^(0)=1 and x^(n)=x(x+1)(x+2)… (x+n-1) for n≥ 1. In this paper, we present a new real-valued Appell-type polynomial family A_n^(k)(m,x), n, m N _0, k N , every member of which is expressed by mean of the generalized hypergeometric function _p F_q bmatrix matrix a_1, a_2, …, a_p \: b_1, b_2, …, b_q matrix \: | \:z bmatrix= _k=0^∞ a_1^(k) a_2^(k) … a_p^(k) b_1^(k) b_2^(k) … b_q^(k) z^kk! as follows A_n^(k)(m,x)= x^n_k+p F_q bmatrix matrix a_1, a_2, …, a_p, -nk , -n-1k , …, -n-k+1k \: b_1, b_2, …, b_q matrix \: | \: mx^k bmatrix and, at the same time, the polynomials from this family are Appell-type polynomials. 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. , number=1, journal=Carpathian Mathematical Publications, author=Bedratyuk, L. and Luno, N., year=2020, month=Jun., pages=129–137