The Hankel Transform of q-Noncentral Bell Numbers

We define two forms of q-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 q-analogue in the context of 0-1 tableaux which, consequently, yields certain additive identity and some convolution-type formulas. Finally, a q-analogue of noncentral Bell numbers is defined and its Hankel transform is established.


A 𝑞-Analogue of 𝑆 𝑎 (𝑛,𝑘): 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: ( + 1, ) =   (,  − 1) + ( − )   (, ) ,   ( + 1, ) =   (,  − 1) + ( − )   (, ) , (12) with initial conditions   (0, 0) = 1,   (, 0) = ()  ,   (0, ) = 0, ,  ̸ = 0,   (0, 0) = 1,   (, 0) = (−)  ,   (0, ) = 0, ,  ̸ = 0. ( 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  → 1.For instance, a polynomial   () is a -analogue of the polynomial   if lim  → 1   () =   .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 International Journal of Mathematics and Mathematical Sciences 3 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  → 1, 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 by where   [0, 0]  = 1,   [, ]  = 0 for  <  or ,  < 0 and Remark 2. When  → 1, 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 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 Δ  ,ℎ , is defined to be a mapping that assigns to every function  the function Δ  ,ℎ  defined by the rule where  ℎ is the shift operator defined by  ℎ () = ( + ℎ).
When ℎ = 1, we use the notation As convention, define Δ 0 ,ℎ = 1 (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 have where   =  0 + ℎ,  = 1, 2, . .., such that when  0 = 0 and ℎ = 1, this can be simplified as Using ( 28) with  = , we get which can be expressed further as Applying the above Newton Interpolation Formula and the identity in (31) with ℎ = 1, we get Thus, we obtain the following explicit formula.
Theorem 5.The explicit formula for   [, ]  is given by Remark 6.The above theorem reduces to when  → 1 which is the explicit formula of the noncentral Stirling numbers of the second kind.
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  → 1, the above theorem becomes which 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  = 0, we have International Journal of Mathematics and Mathematical Sciences 5 With  > 0 and using Definition 1, we obtain Hence, The rational generating function in Theorem 10 can then be expressed as Hence, This sum may be written further as follows. )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  = (  ) ≥0 , a strictly increasing sequence of nonnegative integers.
Let  be a function from the set of nonnegative integers  to a ring .Suppose Φ is an -tableau with  columns of lengths || ≤ ℎ.We use    (ℎ, ) to denote the set of such tableaux.Then, we set Note that Φ might contain a finite number of columns whose lengths are zero since 0 ∈  = {0, 1, 2, . . ., } and if (0) ̸ = 0. From this point onward, whenever an -tableau is mentioned, it is always associated with the sequence  = {0, 1, 2, . . ., }.
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).
Notice that, by joining the columns of  (78) The next theorem provides another form of convolutiontype identity.

)
Suppose   is the set of all -tableaux corresponding to  such that, for each  ∈   , either  has no column whose weight is − [ 2 ]  , or  has one column whose weight is − [ 2 ]  , or  has two columns whose weights are − [ 2 ]  , or Now, if  columns in  have weights other than −[ 2 ]  , then 2 , . . .,   ∈ { 1 ,  2 , . . .,  − }.Note that, for each , tableaux with  columns in   are distinct.Hence, every distinct tableau  with  columns of weights other than −[ 2 ]  appears