Abstract
We define two forms of -analogue of noncentral Stirling numbers of the second kind and obtain some properties parallel to those of noncentral Stirling numbers. Certain combinatorial interpretation is given for the second form of the -analogue in the context of 0-1 tableaux which, consequently, yields certain additive identity and some convolution-type formulas. Finally, a -analogue of noncentral Bell numbers is defined and its Hankel transform is established.
1. Introduction
The Hankel matrix of order of a sequence is given by . The Hankel determinant of order of is the determinant of the corresponding Hankel matrix of order . That is, . The Hankel transform of the sequence , denoted by , is the sequence of Hankel determinants of . For instance, the Hankel transform of the sequence of Catalan numbers, , is given byand the sequence of the sum of two consecutive Catalan numbers, , with being the th Catalan numbers, has the Hankel transformwhere is the th Fibonacci number [1].
One remarkable property of Hankel transform is established by Layman [1], which states that the Hankel transform of an integer sequence is invariant under binomial and inverse binomial transforms. We recall that a sequence is a binomial transform of , denoted by , ifand is an inverse binomial transform of , denoted by , ifIn fact, Riordan [2, 3] established the so-called binomial inversion formula which is given by Hence, if is an integer sequence, is binomial transform of , and is the inverse binomial transform of , then This property played an important role in proving that the Hankel transform of the sequence of ordinary Bell number [4] and that of -Bell numbers [5] are equal. Recently, in the paper by R. B. Corcino and C. B. Corcino [6], this property has also been used in proving that the Hankel transform of the sequence of generalized Bell numbers is given bywhere is the sum of -Stirling numbers (see [7, 8]).
In that same paper [6], an open problem is disclosed which is to determine the Hankel transform of a -analogue of generalized Bell numbers. This problem motivates the present authors to define -analogues of noncentral Stirling and noncentral Bell numbers and, eventually, establish the Hankel transform of a -analogue of noncentral Bell numbers. Results in this paper may be useful in proving the said open problem.
2. A -Analogue of : First Form
The noncentral Stirling numbers of the first and second kind [9], denoted by and , respectively, are defined by means of the following inverse relations:where , are any real numbers, is a nonnegative integer, and These numbers are a certain generalization of the classical Stirling numbers of the first and second kind, respectively, which satisfy the following recurrence relations: with initial conditions For a more detailed discussion of noncentral Stirling numbers, one may see [9]. Parallel to the ordinary Bell numbers, the noncentral Bell numbers can also be introduced as the sum of noncentral Stirling numbers of the second kind as follows: These numbers are equivalent to -Bell numbers with the following relation:
Several mathematicians developed a way of obtaining a generalization of some special numbers. One generalization is a -analogue of these special numbers. A -analogue is a mathematical expression parameterized by a quantity that generalizes a known expression and reduces to the known expression in the limit, as . For instance, a polynomial is a -analogue of the polynomial if . The -analogues of , , , and are, respectively, given by The polynomials are usually called the -binomial coefficients. These polynomials possess several properties including the -binomial inversion formula and certain generating function The -binomial inversion formula is a -analogue of the binomial inversion formula in (5).
A -analogue of both kinds of Stirling numbers was first defined by Carlitz in [10], the second kind of which, known as -Stirling numbers of the second kind, is defined in terms of the following recurrence relation:in connection with a problem in Abelian groups, such that when , this gives the triangular recurrence relation for the classical Stirling numbers of the second kind : A different way of defining a -analogue of Stirling numbers of the second kind has been adapted in [11] which is given as follows:This type of -analogue gives the Hankel transform of -exponential polynomials and numbers which are a certain -analogue of Bell polynomials and numbers. In the desire to establish the Hankel transform of -analogue of generalized Bell numbers, the present authors are motivated to define a -analogue of parallel to that of (21) as follows.
Definition 1. For nonnegative integers and and real number , a -analogue of is defined bywhere , for or and .
Remark 2. When , the above definition will reduce to the recurrence relation of the noncentral Stirling numbers of the second kind established by Koutras [9] which is given by
Remark 3. It can easily be verified that
By proper application of (22), we can easily obtain two other forms of recurrence relations and certain generating function.
Theorem 4. For nonnegative integers and and real number , the -analogue satisfies the following vertical and horizontal recurrence relations: respectively, where with initial value . Moreover, a horizontal generating function for is given by
Explicit formulas and generating functions of a given sequence of numbers or polynomials are useful tools in giving combinatorial interpretation of the numbers or polynomials. In the subsequent theorems, we establish the exponential and rational generating functions and two explicit formulas for . One of these explicit formulas is in symmetric function form which will be used to give combinatorial interpretation of in the context of some -tableaux.
A -analogue of the difference operator, known as -difference operator, was defined and thoroughly discussed in [12, 13]. More precisely, the difference operator of degree , denoted by , is defined to be a mapping that assigns to every function the function defined by the rulewhere is the shift operator defined by . When , we use the notationAs convention, define (the identity map). The following is the explicit formula for the -difference operator:
The new -analogue of Newton’s Interpolation Formula in [13] states that, for we havewhere , , such that when and , this can be simplified asUsing (28) with , we getwhich can be expressed further asApplying the above Newton Interpolation Formula and the identity in (31) with , we getThus, we obtain the following explicit formula.
Theorem 5. The explicit formula for is given by
Remark 6. The above theorem reduces to when which is the explicit formula of the noncentral Stirling numbers of the second kind.
Remark 7. For brevity, (39) can be expressed as which reduces to (11) as .
Theorem 8. For nonnegative integers and and real number , the -analogue has a generating function
Proof. Using the explicit formula in Theorem 5, we obtain Using (31), we prove the theorem.
Remark 9. When , the above theorem becomeswhich is the exponential generating function of the noncentral Stirling numbers of the second kind.
The following theorem contains the rational generating function for .
Theorem 10. For nonnegative integers and and complex number , the -analogue satisfies the rational generating function
Proof. For , we haveWith and using Definition 1, we obtainHence,
The rational generating function in Theorem 10 can then be expressed as Hence,This sum may be written further as follows.
Theorem 11. For nonnegative integers and and complex number , the explicit formula for in homogeneous symmetric function form is given by
3. A -Analogue of : Second Form
We now define another -analogue of the noncentral Stirling numbers, denoted by , as follows:Hence,All other properties parallel to those of can easily be established by imbedding the factor in the derivations or multiplying this factor directly by both sides of the equation that yields the formula/identity for .
The following definition contains the concept of -tableau which is useful in describing , combinatorially.
Definition 12 (see [14]). An -tableau is a list of columns of Ferrer’s diagram of a partition (by decreasing order of length) such that the lengths are part of the sequence , a strictly increasing sequence of nonnegative integers.
Let be a function from the set of nonnegative integers to a ring . Suppose is an -tableau with columns of lengths . We use to denote the set of such -tableaux. Then, we setNote that might contain a finite number of columns whose lengths are zero since and if .
From this point onward, whenever an -tableau is mentioned, it is always associated with the sequence .
We are now ready to mention the following theorem.
Theorem 13. Let denote a function from to a ring (column weights according to length) which is defined by , where is a complex number and is the length of column of an -tableau in . Then,
Proof. This can easily be proved using Definition 12 and (53).
3.1. Combinatorics of -Tableaux
Suppose that, for some numbers and , we have . Then, (53) yields That is, for any , where . Note that the weight of each column of can be considered as a finite sum with additive constant ; that is, for each , we can writewhere . The following theorem determines how an additive constant affects the recurrence formula for .
From Theorem 13,where If for some and , then, by (58),
Suppose is the set of all -tableaux corresponding to such that, for each , eitherThen, we may write Now, if columns in have weights other than , thenwhere . Note that, for each , there correspond tableaux with columns having weights . It can easily be verified that Thus, , contains a total of tableaux with columns of weights . However, only tableaux with columns in are distinct. Hence, every distinct tableau with columns of weights other than appearstimes in the collection. Thus, where denotes the set of all tableaux having columns of weights . Reindexing the double sum, we get where is the set of all tableaux with columns of weights for each . Clearly . Hence, Applying Theorem 13, we obtain the following theorem.
Theorem 14. The -analogue satisfies the following identity: where for some numbers and .
Supposeis a tableau with columns whose lengths are in the set ,is a tableau with columns whose lengths are in the set .Then, where and . Notice that, by joining the columns of and , we obtain an -tableau with columns whose lengths are in the set . That is, . Then,Note thatThus,By (53), we obtain the following theorem.
Theorem 15. The -analogue satisfies the following convolution-type identity:
The next theorem provides another form of convolution-type identity.
Theorem 16. The -analogue satisfies the following second form of convolution formula:
Proof. Let be a tableau with columns whose lengths are in ,be a tableau with columns whose lengths are in .Then, and . Using the same argument above, we can easily obtain the convolution formula.
4. -Noncentral Bell Number and Its Hankel Transform
In this section, we define a -analogue of the noncentral Bell numbers and obtain some combinatorial properties that will be used to establish its Hankel transform.
Definition 17. A -analogue of the noncentral Bell numbers, denoted by , is defined by where . For brevity, we use the term -noncentral Bell numbers for .
We observe that, multiplying both of (22) by , we obtainNote that when , we getwhich coincides with the recurrence relation of the modified -Stirling numbers of the second kind in [11] and that of the -Stirling numbers of the second kind in [15]. In fact,That is, is a proper -analogue of that generalizes .
Note that our objective is to define a -analogue of such that when , it reduces to the -exponential number in [11, 15], whose Hankel transform is known to be (see [11, 15, 16]). Hence, it is appropriate to define a -analogue of the noncentral Bell numbers through .
Clearly, when , , the noncentral Bell numbers. The following theorem contains certain recurrence relations for .
Theorem 18. The -noncentral Bell numbers satisfy the following relation:
Proof. By making use of Theorem 14, with and , we have Then,which is exactly the desired relation.
The following corollary is a direct consequence of Theorem 18 which can be proved using the inversion formula by Riordan [2, 3].
Corollary 19. The -noncentral Bell numbers satisfy the following relation:
To establish the Hankel transform of , we need the concept of rising -binomial transform by Spivey and Steil [17] as well as its property in relation to Hankel transform.
Definition 20 (Spivey-Steil [17]). The rising -binomial transform of a sequence is the sequence , where is given by
We use to denote the set of rising -binomial transforms of . That is, . Then, we have the following theorem by Spivey and Steil.
Theorem 21 (Spivey-Steil [17]). Given a sequence , let . Then,If ,
Now, we are ready to state the main result of the paper.
Theorem 22. For , the Hankel transform of the sequence of -noncentral Bell numbers is given by
Proof. We begin with the relation that . Hence, From Corollary 19, we see that is the binomial transform of . This means that Hence, by Layman’s Theorem [1],That is,Now, Corollary 19 can also be stated as is the rising -binomial transform of . Using Spivey-Steil Theorem, with , , and , we haveWe observe that when and using (95), we have Also, when , Continuing this argument, we obtain
Remark 23. When , the Hankel transform in (94) reduces to the Hankel transform of -exponential numbers in (85) and that in [15]. By taking , we further get the Hankel transform of the ordinary Bell numbers. The same Hankel transform is obtained when and taking , which is exactly the same result established by Mező [5] for the Hankel transform of -Bell numbers . That is,
Remark 24. We can define another form of -analogue of the noncentral Bell numbers as follows: In the paper by Cigler [18], certain generalization of -Stirling numbers of the second kind, denoted by , was defined by means of the following recurrence relation: which is equivalent to . That is, Moreover, the first form of generalized -exponential polynomials in [18] was defined by whose Hankel transform is given by where . With and , we obtainThen, by (109),
Remark 25. We can also define a -analogue of the noncentral Bell numbers as follows: Another form of generalized -exponential polynomials in [18] was defined by whose Hankel transform is given by With and , we obtain Then, by (114),
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This research has been partially funded by the research grant of Professor Keonhee Lee of Chungnam National University, Daejeon, South Korea.