A Family of Generalized Stirling Numbers of the First Kind

A modified approach via differential operator is given to derive a new family of generalized Stirling numbers of the first kind. This approach gives us an extension of the techniques given by El-Desouky [1] and Gould [2]. Some new combinatorial identities and many relations between different types of Stirling numbers are found. Furthermore, some interesting special cases of the generalized Stirling numbers of the first kind are deduced. Also, a connection between these numbers and the generalized harmonic numbers is derived. Finally, some applications in coherent states and matrix representation of some results obtained are given.


Introduction
Gould [2] proved that where a + and a are boson creation and annihilation operators, respectively, and satisfy the commutation rela- Moreover El-Desouky [1] derived many special cases and some applications. For the proofs and more details, see [1].
The generalized falling factorial of x associated with the sequence where ( ) For more details on generalized Stirling numbers via differential operators, see [7]- [10] and [11].
The paper is organized as follows: In Section 2, using the differential operator ( ) ( )( ) O are given. In Section 4, some applications in coherent states and matrix representation of some results obtained are given. Section 5 is devoted to the conclusion, which handles the main results derived throughout this work. Finally, a computer program is written using Maple and executed for calculating the generalized Stirling numbers of the first kind and some special cases, see Appendix.

Main Results
Let ( ) 1 2 , , , n r r r = r  : be a sequence of real numbers and ( ) 1 2 , , , n s s s = s  : be a sequence of nonnegative integers.

Special Cases
The proof follows directly from Equation (15) by setting i r r = and , 1, 2, , .
hence comparing Equations (19) and (23) we obtain Equation (22). Furthermore we handle the following special cases. i) If 1 r = , then we have

Proof:
The proof follows directly from Equation (21) by setting 1 r = .

Proof
The proof follows directly from Equation (22)

Proof
The proof follows from (22) by setting 1 s = . Also, using the recurrence relation (28) we can find the following explicit formula.
That is the same recurrence relation (28)
Setting e x t = , we have Using, see [12],  Furthermore, using our notations, it is easy from Equation (4.4) in [6] and (41)  Next, we find a connection between ( ) , ; , n k r s s and the generalized harmonic numbers ( ) i n O which are defined by, see [13] and [14], hence, setting 1 s = , we get the identity Now we come back to normal ordering. Using the properties of coherent states, see [7], the coherent state matrix element of the boson string yields the generalized polynomial

Matrix Representation
In this subsection we derive a matrix representation of some results obtained. Let r s be n n × lower triangle matrix, where r s is the matrix whose entries are the numbers ( )

Conclusion
In this article we investigated a new family of generalized Stirling numbers of the first kind. Recurrence relations and an explicit formula of these numbers are derived. Moreover some interesting special cases and new combinatorial identities are obtained. A connection between this family and the generalized harmonic numbers is given. Finally, some applications in coherent states and matrix representation of some results are obtained.