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 is an element of N-0, k is an element of N, everymember of which is expressed bymean of the generalized hypergeometric function F-p(q) [a(1), a(2),... a(p) b(1), b(2),..., b(q)vertical bar z] = Sigma(infinity)(k=0) a(1)((k))a(2)((k))...a(p)((k))/b(1)((k))b(2)((k))...b(q)((k)) z(k)/k! as follows A(n)((k)) (m,x) = x(n) F-k+p(q)[a(1), a(2),..., a(p), -n/k, -n-1/k,...,-n-k+1/k b(1), b(2), ... b(q)vertical bar m/x(k)] 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. Key words and phrases: Appell sequence, Appell polynomial, generalized hypergeometric polynomial, generalized hypergeometric function.