Received 4 March 2014; revised 4 April 2014; accepted 11 April 2014

ABSTRACT

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.

Keywords:

Stirling numbers, Comtet numbers, Creation, Annihilation, Differential operator, Maple program

1. Introduction

Gould [2] proved that

(1)

where and are the usual Stirling numbers and the singles Stirling numbers of the first kind, respectively, defined by

(2)

(3)

and.

These numbers satisfy the recurrence relations

(4)

(5)

EL-Desouky [1] defined the generalized Stirling numbers of the first kind called -Stirling numbers of the first kind by

(6)

for or and where is a sequence of real

numbers and is a sequence of nonnegative integers.

Equation (6) is equivalent to

(7)

where and a are boson creation and annihilation operators, respectively, and satisfy the commutation rela-

tion

The numbers satisfy the recurrence relation

(8)

with the notations and.

The numbers have the explicit formula

(9)

where with and

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 of order n, where

are real numbers, is defined by

Comtet [3] [4] and [5] defined the generalized Stirling numbers of the first kind, which are called Comtet numbers, by

(10)

These numbers satisfy the recurrence relation

(11)

El-Desouky and Cakic [6] defined, the generalized Comtet numbers by

(12)

where for and.

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 we define a new family

of generalized Stirling numbers of the first kind, denoted by. A recurrence relation and an explicit formula of these numbers are derived. In Section 3, some interesting special cases are discussed. Moreover some new combinatorial identities and a connection between and the generalized harmonic numbers 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.

2. Main Results

Let be a sequence of real numbers and be a sequence of nonnegative integers.

Definition 2.1

The generalized Stirlng numbers are defined by

(13)

where for and.

Equation (13) is equivalent to

(14)

Theorem 2.1

The numbers satisfy the recurrence relation

(15)

with the notations

Proof

Equating the coefficients of on both sides yields (15).

Theorem 2.2

The numbers have the explicit formula

(16)

Proof

thus, by iteration, we get

(17)

Setting we obtain

(18)

Comparing (13) and (18) yields (16).

3. Special cases

Setting in (13), we have the following definition.

Definition 3.1

For any real number r and nonnegative integer s, let the numbers be defined by

(19)

where and for.

Equation (19) is equivalent to

(20)

Corollary 3.1

The numbers satisfy the recurrence relation

(21)

Proof

The proof follows directly from equation (15) by setting and

Corollary 3.2

The numbers have the explicit formula

(22)

Proof

By substituting and in Equation (17), yields

then setting we have

(23)

hence comparing equations (19) and (23) we obtain equation (22).

Furthermore we handle the following special cases.

i) If, then we have

Definition 3.2

(24)

where and for

Corollary 3.3

The numbers satisfy the recurrence relation

(25)

Proof:

The proof follows directly from Equation (21) by setting.

Corollary 3.4

The numbers have the explicit formula

(26)

Proof

The proof follows directly from Equation (22) by setting.

ii) If, then we have

Definition 3.3

The numbers are defined by

(27)

where and for

Corollary 3.5

The numbers satisfy the triangular recurrence relation

(28)

Proof

The proof follows easily from (22) by setting.

Corollary 3.6

The numbers have the following explicit formula

(29)

Proof

The proof follows from (22) by setting.

Also, using the recurrence relation (28) we can find the following explicit formula.

Theorem 3.1

The numbers have the following explicit expression

(30)

Proof

For,

For, we get

That is the same recurrence relation (28) for the numbers This completes the proof.

iii) If and, then we have

Definition 3.4

The numbers are defined by

(31)

where and for

Equation (31) is equivalent to

(32)

Corollary 3.7

The numbers satisfy the triangular recurrence relation

(33)

Proof

The proof follows by setting in equation (28).

Corollary 3.8

The numbers have the explicit formula

(34)

Proof

The proof follows by setting in equation (29).

Moreover have the following explicit formula.

Corollary 3.9

The numbers have the following explicit expression

(35)

Proof

The proof follows by setting in (30).

From equations (29) and (30) (also from equations (34) and (35)) we have the combinatorial identities

(36)

(37)

From equations (29) and (34) we obtain that

(38)

Remark 3.1

Operating with both sides of equation (13) on the exponential function, we get

Therefore, since a nonzero polynomial can have only a finite set of zeros, we have

(39)

If, we obtain

(40)

Remark 3.2

From relation (39), by replacing with, and relation (18) we conclude that

(41)

This gives us a connection between and the generalized Comtet numbers, see [6].

Setting and in (39), we get

(42)

hence, we have where see [6].

If, then

(43)

Next we discuss the following special cases of (42) and (43):

i) If, then

(44)

hence we have the generalized Comtet numbers, where

see [6] .

ii) If, then we have

(45)

hence we obtain Comtet numbers, where, see [3] and [4] .

For example if and s = 2 in (43) we have

(46)

Using Table 2,

L.H.S. of (46) = s(3,0;2,2) + s(3,1;2,2) + s(3,2;2,2) + s(3,3;2,2) + s(3,4;2,2) + s(3,5;2,2) + s(3,6;2,2) = 14400 + 22080 +12784 + 3552 + 508 + 36 + 1 = 53361.

R.H.S. of (46) =.

This confirms (46) and hence (43).

Another example if n = 2, r = 2 and s = 3 in (43) we have

(47)

Using Table 3,

L.H.S. of (47) = s(2,0;2,3) + s(2,1;2,3) + s(2,2;2,3) + s(2,3;2,3) + s(2,4;2,3) + s(2,5;2,3) + s(2,6;2,3) = 1728 + 3456 + 2736 + 1088 + 228 + 24 + 1 = 9261.

R.H.S. of (46) =.

This confirms (43).

iii) If, then we get

(48)

hence we have which is a special case of Comtet numbers, where

see [3] and [4] and Table 1.

Setting, we have then substituting in (2.1) it becomes

(49)

Using, see [12] ,

then equation (49) yields

(50)

Comparing this equation with Equation (4.1) in [6] , we get

(51)

where and are the generalized Comtet numbers of the first

kind.

Furthermore, using our notations, it is easy from Equation (4.4) in [6] and (41) to show that

(52)

where and are the Stirling numbers of the second kind.

Next, we find a connection between and the generalized harmonic numbers which are defined by, see [13] and [14] ,

From (42), we have

Equating the coefficients of on both sides, we obtain

(53)

From (22) and (53), we have the combinatorial identity

(54)

hence, setting, we get the identity

(55)

4. Some Applications

4.1. Coherent state and normal ordering

Coherent states play an important role in quantum mechanics especially in optics. The normally ordered form of the boson operator in which all the creation operators stand to the left of the annihilation operators . Using

the properties of coherent states we can define and represent the generalized polynomial and generalized

number as follows.

Definition 4.1

The generalized polynomial is defined by

(56)

and the generalized number

(57)

For convenience we apply the convention

(58)

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

(59)

Definition 4.2

We define the polynomial as

(60)

and the numbers

(61)

For convenience we apply the conventions

(62)

Similarly, using the properties of coherent states and (32) we have

(63)

4.2. Matrix Representation

In this subsection we derive a matrix representation of some results obtained.

Let be lower triangle matrix, where is the matrix whose entries are the numbers,

i.e. Furthermore let be an lower triangle matrix defined by

, is a diagonal matrix whose entries of the main diagonal are,

i.e. and

.

Equation (27), may be represented in a matrix form as

(64)

for example if n = 3 then

(65)

its inverse is given by

(66)

Setting r = 1 in (64), we get

(67)

(68)

hence

For n = 3, we have

(69)

5. 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.

References

Appendix

Tables of calculated using Maple, for some values of n, k, r and s:

Notice that the last column in all tables is just the sum of the entries of the corresponding row.