Abstract
Recently, the central Fubini polynomials were introduced in connection with central factorial numbers of the second kind. In this paper, we consider two variable higher-order central Fubini polynomials as a ‘central analogue’ of two variable higher-order Fubini polynomials. We investigate some properties, identities, and recurrence relations for these polynomials by making use of generating functions and umbral calculus. In particular, we obtain various expressions for the two variable higher-order central Fubini polynomials and express them in terms of some families of special polynomials and vice versa.
Highlights
For n ∈ N ∪ {0}, the central factorial x[n] is defined as x[0] = 1, x[n] = x n x+ –1 n x+ –2 n x– +1 (n ≥ 1). (1)As is well known, the central factorial numbers of the second kind T(n, k) (n, k ≥ 0) are defined by n xn = T(n, k)x[k] (n ≥ 0). (2) k=0From (2), we can derive the following generating function for T(n, k): t e2 e– t 2 k= ∞
We introduce some properties and present several identities and recurrence relations for these polynomials by making use of generating functions and umbral calculus
K=m where S2(n, k) are the numbers of the second kind given by n xn = S2(n, k)(x)k (n ≥ 0)
Summary
For n ∈ N ∪ {0}, the central factorial x[n] is defined as x[0] = 1, x[n] = x n x+ –1 n x+ –2. The central factorial numbers of the second kind T(n, k) (n, k ≥ 0) are defined by n xn = T(n, k)x[k] (n ≥ 0) (see [7,8,9, 11, 13, 16,17,18]). It is known that the two variable Fubini polynomials Fn(r)(x; y) of order r are defined by. N=0 where r is a positive integer
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