# Aq-Analogue of Rucinski-Voigt Numbers

Authors: Roberto B. Corcino; Charles B. Montero
Journal: ISRN Discrete Mathematics
 (2012)

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## Abstract

Aq-analogue of Rucinski-Voigt numbers is defined by means of a recurrence relation, and some properties including the orthogonality and inverse relations with the q-analogue of the limit of the differences of the generalized factorial are obtained.


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## Body

## 1. Introduction

Rucinski and Voigt [1] defined the numbers Skn(a) satisfying the relation
(1.1)xn=∑k=0n‍Skn(a)pka(x),
where a is the sequence (a,a+r,a+2r,…) and pka(x)=∏i=0k-1‍(x-(a+ir)) and proved that these numbers are asymptotically normal. We call these numbers Rucinski-Voigt numbers. Note that the classical Stirling numbers of the second kind S(n,k) in [2–4] and the r-Stirling numbers of the second kind [nk]^r of  Broder [5] can be expressed in terms of Skn(a) as follows:
(1.2)S(n,k)=Skn(d),[n+rk+r]^r=Skn(e),
where d and e are the sequences (0,1,2,…) and (r,r+1,r+2,…), respectively. With these observations, Skn(a) may be considered as certain generalization of the second kind Stirling-type numbers.Several properties of Rucinski-Voigt numbers can easily be established parallel to those in the classical Stirling numbers of the second kind. To mention a few, we have the triangular recurrence relation(P1)Skn+1(a)=Sk-1n(a)+(kr+a)Skn(a),
the exponential and rational generating function
(P2)∑n≥0‍Skn(a)xnn!=1rkk!eax(erx-1)k,(P3)∑n≥0‍Skn(a)xn=xk∏j=0k(1-(rj+a)x),
and explicit formulas
(P4)Skn(a)=1rkk!∑j=0k‍(-1)k-j(kj)(rj+a)n,(P5)Skn(a)=∑c0+c1+⋯+ck=n-k‍∏j=0k‍(rj+a)cj.The explicit formula in(P4) can be used to interpret rkk!Skn(a) as the number of ways to distribute n distinct balls into the k+1 cells ( one ball at a time ), the first k of which has r distinct compartments and the last cell with a distinct compartments, such that(i)
the capacity of each compartment is unlimited;(ii)
the firstk cells are nonempty.The other explicit formula(P5) can also be used to interpret Skn(a) as the number of ways of assigning n people to k+1 groups of tables where all groups are occupied such that the first group contains a distinct tables and the rest of the group each contains r distinct tables.The Rucinski-Voigt numbers are nothing else but ther-Whitney numbers of the second kind, denoted by Wm,r(n,k), in Mező [6]. That is, Skn(a)=Wr,a(n,k). It is worth-mentioning that the r-Whitney numbers of the second kind are generalization of Whitney numbers of the second kind in Benoumhani's papers [7–9].On the other hand, the limit of the differences of the generalized factorial [10]
(1.3)Fα,γ(n,k)=limβ→0[Δtk(βt+γ∣α)n]t=0k!βk,(βt+γ∣α)n=∏j=0n-1‍(βt+γ-jα)
was also known as a generalization of the Stirling numbers of the first kind. That is, all the first kind Stirling-type numbers may also be expressed in terms of Fα,γ(n,k) by a special choice of the values of α and γ. It was shown in [10] that
(1.4)∑k=0n‍Fα,γ(n,k)tk=pnb(t),
where b is the sequence (-γ,-γ+α,-γ+2α,…). Recently, q-analogue and (p,q)-analogue of Fα,γ(n,k), denoted by ϕα,γ[n,k]q and ϕα,γ[n,k]pq, respectively, were established by Corcino and Hererra in [10] and obtained several properties including the horizontal generating function for ϕα,γ[n,k]q(1.5)∑k=0nϕα,γ[n,k]qtk=〈t+[γ]q∣[α]q〉nq,
where
(1.6)〈t+[γ]q∣[α]q〉nq=∏j=0n-1(t+[γ]q-[jα]q),〈t+[γ]q∣[α]q〉0q=1.The numbersFα,-γ(n,k) are equivalent to the r-Whitney numbers of the first kind, denoted by wm,r(n,k), in [6]. More precisely, Fα,-γ(n,k)=wα,γ(n,k). These numbers are generalization of Whitney numbers of the first kind in Benoumhani's papers [7–9].In this paper, we establish aq-analogue of Skn(a) and obtain some properties including recurrence relations, explicit formulas, generating functions, and the orthogonality and inverse relations.


## 2. Definition and Some Recurrence Relations

It is known that a given polynomialak(q) is a q-analogue of an integer ak if
(2.1)limq→1ak(q)=ak.For example, the polynomials(2.2)[n]q=qn-1q-1,[n]q!=∏i=1n‍[i]q,[nk]q=∏i=1k‍qn-i+1-1qi-1
are the q-analogues of the integers n, n!, and (nk), respectively, since
(2.3)limq→1[n]q=n,limq→1[n]q!=n!,limq→1[nk]q=(nk).The last two polynomials in (2.2) are called the q-factorial and q-binomial coefficients, respectively. With these in mind, it is interesting also that, for a given property of an integer ak, we can find an analogous property for the polynomial ak(q). For example, the binomial coefficients (nk) satisfy the known inversion formula
(2.4)fn=∑k=0n‍(nk)gk⟺gn=∑k=0n‍(-1)n-k(nk)fk
and Vandermondes identity
(2.5)(m+nk)=∑r=0k‍(mr)(nk-r),
while the q-binomial coefficients [nk]q satisfy the q-binomial inversion formula [3]
(2.6)fn=∑k=0n‍[nk]qgk⟺gn=∑k=0n‍(-1)n-kq(n-k2)[nk]qfk,fn=∑k=0n‍[nk]qβgk⟺gn=∑k=0n‍(-1)n-kqβ(n-k2)[nk]qβfk,
and q-Vandermondes identity [11]
(2.7)[m+nk]q=∑r=0k‍qr(m-k+r)[mr]q[nk-r]q.Carlitz [12] defined a q-Stirling number of the second kind in terms of a recurrence relation
(2.8)Sq[n,k]=Sq[n-1,k-1]+[k]qSq[n-1,k]
in connection with a problem in abelian groups, such that when q→1, this gives the triangular recurrence relation for the classical Stirling numbers of the second kind S(n,k)(2.9)S(n,k)=S(n-1,k-1)+kS(n-1,k).This motivates the authors to define aq-analogue of the Skn(a) as follows.Definition 2.1.
For nonnegative integersn and k and complex numbers β and r, a q-analogue σ[n,k]qβ,r of Skn(c) is defined by
(2.10)σ[n,k]qβ,r=σ[n-1,k-1]qβ,r+([kβ]q+[r]q)σ[n-1,k]qβ,r,
wherec is the sequence (r,r+β,r+2β,…), σ[0,0]qβ,r=1, and σ[n,k]qβ,r=0 for n<k or n,k<0.The numbersσ[n,k]qβ,r may be considered as a q-analogue of Skn(c) since, when q→1,
(2.11)[kβ]q+[r]q→kβ+r
and, hence, the recurrence relation in (2.10) will give the recurrence relation in (P1) for Skn(c) where c is the sequence (r,r+β,r+2β,…). This fact will also be verified in Section 3 (Remark 3.4).The above triangular recurrence relation for theq-Stirling numbers of the second kind can easily be deduced from (2.10) by taking β=1 and r=0.Clearly, using the initial conditions ofσ[n,k]qβ,r, we can have
(2.12)σ[n,0]qβ,r=[r]qn,∀n≥0,σ[n,n]qβ,r=1,∀n≥0.By repeated application of (2.10), we obtain the following theorem.Theorem 2.2.
For nonnegative integersn and k and complex numbers β and r, the q-analogue σ[n,k]qβ,r satisfies the following vertical recurrence relation:
(2.13)σ[n+1,k+1]qβ,r=∑j=kn‍([(k+1)β]q+[r]q)n-jσ[j,k]qβ,r
with initial conditions σ[0,0]qβ,r=1 and σ[n,n]qβ,r=1,σ[n,0]qβ,r=[r]qn  for all n≥0.Using the following notation(2.14){[r]q∣[β]q}k=∏j=0n-1‍([r]q+[jβ]q),{[r]q∣[β]q}0=1,
we can now state the horizontal recurrence relation for σ[n,k]qβ,r.Theorem 2.3.
For nonnegative integersn and k and complex numbers β and r, the q-analogue σ[n,k]qβ,r satisfies the following horizontal recurrence relation:
(2.15)σ[n,k]qβ,r=∑j=0n-k‍(-1)j{[r]q∣[β]q}k+j+1{[r]q∣[β]q}k+1σ[n+1,k+j+1]qβ,r,
with initial condition σ[0,0]qβ,r=1 and σ[n,n]qβ,r=1,σ[n,0]qβ,r=[r]qn  for all n≥0.Proof.
To prove (2.15), we simply evaluate its right-hand side using (2.10) and obtain σ[n,k]qβ,r.It will be shown in Section3 that
(2.16)[nk]q=(q-1)n-kσ[n,k]q1,logq2.By takingβ=1 and r=logq2, (2.13) and (2.15) yield
(2.17)[n+1k+1]q=∑j=kn‍(qk+1)n-j[jk]q,[nk]q=∑j=0n-k‍(-1)jqjk+(j+12)[n+1k+j+1]q,
which are exactly the recurrence relations obtained in [13]. When q→1, these further give the Hockey Stick identities.


## 3. Explicit Formulas and Generating Functions

The next theorem is analogous to that relation in (1.1). This is necessary in obtaining one of the explicit formulas for σ[n,k]qβ,r and the orthogonality and inverse relations of ϕα,γ[n,k]q and σ[n,k]qβ,r.Theorem 3.1.
For nonnegative integersn and k and complex numbers β and r, the q-analogue σ[n,k]qβ,r satisfies the following relation: (3.1)∑k=0n‍σ[n,k]qβ,r〈t∣[β]q〉kq=(t+[r]q)n.Proof.
We proceed by induction onn. Clearly, (3.1) is true for n=0. Assume that it is true for n>0. Then using Definition 2.1, (3.2)∑k=0n+1‍σ[n+1,k]qβ,r〈t∣[β]q〉kq=∑k=0n‍σ[n,k]qβ,r〈t∣〈[β]q〉k+1q+∑k=0n‍([kβ]q+[r]q)σ[n,k]qβ,r〈t∣[β]q〉kq=(t+[r]q)∑k=0n‍σ[n,k]qβ,r〈t∣[β]q〉kq=(t+[r]q)(t+[r]q)n=(t+[r]q)n+1.The newq-analogue of Newton's Interpolation Formula in [14] states that, for
(3.3)fq(x)=a0+a1[x-x0]q+⋯+am[x-x0]q[x-x1]q[x-xm-1]q,
we have
(3.4)fq(x)=fq(x0)+Δqh,hfq(x0)[x-x0]q[1]qh![h]q+Δqh,h2fq(x0)[x-x0]q[x-x1]q[2]qh![h]q2+⋯+Δqh,hmfq(x0)[x-x0]q[x-x1]q…[x-xm-1]q[m]qh![h]qm,
where xk=x0+kh, k=1,2,… such that when x0=0, this can be simplified as
(3.5)fq(x)=fq(0)+Δqh,hfq(0)[x]q[1]qh![h]q+Δqh,h2fq(0)[x]q[x-h]q[2]qh![h]q2+⋯+Δqh,hmfq(0)[x]q[x-h]q…[x-(m-1)h]q[m]qh![h]qm.Using (3.1) with t=[x]q, we get
(3.6)∑k=0n‍σ[n,k]qβ,r〈[x]q∣[β]q〉kq=([x]q+[r]q)n,
which can be expressed further as
(3.7)∑k=0n‍σ[n,k]qβ,rqβ(k2)[x]q[x-β]q⋯[x-(k-1)β]q=([x]q+[r]q)n.Applying the above Newton's Interpolation Formula and the identity in [14]
(3.8)Δq,hnf(x)=∑k=0n‍(-1)kq(k2)[nk]qf(x+(n-k)h),
we get
(3.9)σ[n,k]qβ,rqβ(k2)=Δqβ,βkfq(0)[k]qβ![β]qk=1[k]qβ![β]qk∑j=0k‍(-1)k-jqβ(k-j2)[kj]qβ([jβ]q+[r]q)n.With〈[kβ]q∣[β]q〉kq=qβ(k2)[kβ]q[(k-1)β]q⋯[β]q=qβ(k2)[k]qβ![β]qk, we obtain the following explicit formula.Theorem 3.2.
For nonnegative integersn and k and complex numbers β and r, the q-analogue σ[n,k]qβ,r is equal to
(3.10)σ[n,k]qβ,r=1〈[kβ]q∣[β]q〉kq∑j=0k‍(-1)k-jqβ(k-j2)[kj]qβ([jβ]q+[r]q)n.Remark 3.3.
We can also prove Theorem3.2 using the q-binomial inversion formula in (2.6). That is, by taking t=[kβ]q, (3.1) gives
(3.11)([kβ]q+[r]q)n=∑j=0n‍σ[n,j]qβ,r〈[kβ]q∣[β]q〉jq=∑j=0k‍[kj]qβ{σ[n,j]qβ,r〈[kβ]q∣[β]q〉jq[kj]qβ}.
Applying (2.6), we obtain
(3.12)σ[n,j]qβ,r〈[kβ]q∣[β]q〉kq[kk]qβ=∑j=0k‍(-1)k-jqβ(k-j2)[kj]qβ([jβ]q+[r]q)n.
This is precisely the explicit formula in Theorem3.2.Remark 3.4.
Note that〈[kβ]q∣[β]q〉kq→k!βk, [kj]qβ→(kj), and ([jβ]q+[r]q)n→(jβ+r)n as q→1. Thus, using property (P4), σ[n,k]qβ,r→Skn(c) as q→1. This implies that σ[n,k]qβ,r is a proper q-analogue of Skn(c).
Now, using the explicit formula in Theorem3.2, we obtain
(3.13)∑n≥0‍σ[n,j]qβ,rtnn!=1〈[kβ]q∣[β]q〉kq∑n≥0‍{∑j=0k‍(-1)k-jqβ(k-j2)[kj]qβ([jβ]q+[r]q)n}tnn!=∑j=0k(-1)k-jqβ(k-j2)[kj]qβ{∑n≥0([jβ]q+[r]q)ntn/n!}〈[kβ]q∣[β]q〉kq=∑j=0k(-1)k-jqβ(k-j2)[kj]qβ{∑n≥0(∑i=0n(([r]qt)i/i!)(([jβ]qt)n-i/(n-i)!))}〈[kβ]q∣[β]q〉kq.
Applying Cauchy's formula for the product of two power series [3], we get
(3.14)∑n≥0‍σ[n,k]qβ,rtnn!=1〈[kβ]q∣[β]q〉kq∑j=0k‍(-1)k-jqβ(k-j2)[kj]qβ{∑λ≥0‍([r]qt)λλ!∑μ≥0‍([jβ]qt)μμ!}.
Thus,
(3.15)∑n≥0‍σ[n,k]qβ,rtnn!=1〈[kβ]q∣[β]q〉kq∑j=0k‍(-1)k-jqβ(k-j2)[kj]qβe([jβ]q+[r]q)t.Applying the above identity forΔq,hnf to the function f defined by
(3.16)f(x)=e([xβ]q+[r]q)t〈[kβ]q∣[β]q〉kq,
we can further express the above generating function in terms of a q-difference operator. More precisely,
(3.17)∑n≥0‍σ[n,k]qβ,rtnn!={Δqk(e([xβ]q+[r]q)t〈[kβ]q∣[β]q〉kq)}x=0.This is a kind of exponential generating function forσ[n,k]qβ,r which is included in the next theorem. Together with this, a rational generating function for σ[n,k]qβ,r is also stated in the theorem that will be used to derive another explicit formula for σ[n,k]qβ,r in homogeneous symmetric function form.Theorem 3.5.
For nonnegative integersn and k and complex numbers β and r, the q-analogue σ[n,k]qβ,r satisfies the exponential generating function
(3.18)Φk(t)=∑n≥0‍σ[n,k]qβ,rtnn!=[Δqβk(e([xβ]q+[r]q)t〈[kβ]q∣[β]q〉kq)]x=0,
and the rational generating function
(3.19)ψk(t)=∑n≥k‍σ[n,k]qβ,rtn=tk∏j=0k(1-([jβ]q+[r]q)t).Proof.
We are done with the proof of the first generating function. We are left to prove the second one and we are going to prove this by induction onk. For k=0, we have
(3.20)ψ0(t)=∑n≥0‍σ[n,0]qβ,rtn=∑n≥0‍[r]qntn=1(1-[r]qt).
With k>0 and using Definition 2.1, we obtain
(3.21)ψk(t)=∑n≥k‍σ[n,k]qβ,rtn=t∑n≥k‍σ[n-1,k-1]qβ,rtn-1+([kβ]q+[r]q)t∑n≥k‍σ[n-1,k]qβ,rtn-1=tψk-1(t)+([kβ]q+[r]q)tψ(t).
Hence,(3.22)ψk(t)=t1-([kβ]q+[r]q)tψk-1(t),
which gives
(3.23)ψk(t)=tk∏j=0k(1-([jβ]q+[r]q)t).The rational generating function in Theorem3.5 can then be expressed as
(3.24)σ[n,k]qβ,r=∑s1+s2+⋯+sk=n-k‍∏j=0k‍([jβ]q+[r]q)sj.This sum may be written further as follows.Theorem 3.6.
For nonnegative integersn and k and complex numbers β and r, the explicit formula for σ[n,k]qβ,r in homogeneous symmetric function form is given by
(3.25)σ[n,k]qβ,r=∑0≤j1≤j2≤⋯≤jn-k≤k‍∏i=1n-k‍([jiβ]q+[r]q).This explicit formula is necessary in giving combinatorial interpretation ofσ[n,k]qβ,r in the context of 0-1 tableau. Note that when β=1 and r=0, Theorem 3.6 yields
(3.26)σ[n,k]q1,0=∑0≤j1≤j2≤⋯≤jn-k≤k‍∏i=1n-k‍[ji]q=Sq[n,k],
the q-Stirling numbers of the second kind [12]. Moreover, taking β=1 and r=logq2, Theorem 3.6 reduces to
(3.27)σ[n,k]q1,logq2=∑0≤j1≤j2≤⋯≤jn-k≤k‍∏i=1n-k‍([ji]q+[logq2]q)=(q-1)k-n∑0≤j1≤j2≤⋯≤jn-k≤k‍∏i=1n-k‍qji.Using the representation given in [15] for the q-binomial coefficients, we have
(3.28)[nk]q=(q-1)n-kσ[n,k]1q1,logq2.This is the identity that we used in Section2.


## 4. Combinatorial Interpretation ofσ[n,k]qβ,r

Definition 4.1 (see [15]).
A0-1 tableau is a pair φ=(λ,f), where
(4.1)λ=(λ1≥λ2≥⋯≥λk)
is a partition of an integer m and f=(fij)1≤j≤λi is a “filling” of the cells of corresponding Ferrers diagram of the shape λ with 0's and 1's, such that there is exactly one “1” in each column.Using the partitionλ=(5,3,3,2,1), we can construct 60 distinct 0-1 tableaux. Figure 1 below shows one of these tableaux with f14=f15=f23=f31=f42=1, fij=0 elsewhere such that 1≤j≤λi.Figure 1
The 0-1 tableauφ=(λ,f).Definition 4.2 (see [15]).
AnA-tableau is a list ϕ of column c of a Ferrer's diagram of a partition λ (by decreasing order of length) such that the lengths |c| are part of the sequence A=(ai)i≥0, a strictly increasing sequence of nonnegative integers.
Letω be a function from the set of nonnegative integers N to a ring K. Suppose Φ is an A-tableau with r columns of lengths |c|≤h. Then, we set
(4.2)ωA(Φ)=∏c∈Φ‍ω(|c|).Note thatΦ might contain a finite number of columns whose lengths are zero since 0∈A={0,1,2,…,k} and if ω(0)≠0.From this point onward, whenever anA-tableau is mentioned, it is always associated with the sequence A={0,1,2,…,k}.We are now ready to mention the following theorem.Theorem 4.3.
Letω:N→K denote a function from N to a ring K (column weights according to length) which is defined by ω(|c|)=[|c|β]q+[r]q where β and γ are complex numbers, and |c| is the length of column c of an A-tableau in TA(k,n-k). Then
(4.3)σ[n,k]qβ,r=∑ϕ∈TA(k,n-k)‍∏c∈ϕ‍ω(|c|).Proof.
This can easily be proved using Definition4.2 and Theorem 3.6.Now, we demonstrate simple combinatorics of 0-1 tableaux to obtain certain relation forσ[n,k]qβ,r. To start with, we have, from Theorem 4.3,
(4.4)σ[n,k]qβ,r=∑ϕ∈TA(k,n-k)‍ωA(Φ),
where
(4.5)ωA(Φ)=∏c∈ϕ‍([|c|β]q+[r]q),|c|∈{0,1,2,…,k}.Substitutingji=|c|, we obtain
(4.6)ωA(Φ)=∏i=1n-k‍([jiβ]q+[r]q),ji∈{0,1,2,…,k}.Let[r]q=c1+c2 where c1=[r]q-[r2]q and c2=[r2]q for some numbers r1 and r2. Then, with ω*(j)=[jβ]q+c2, we have
(4.7)ωA(Φ)=∑l=0n-k‍c1n-k-l∑q1≤q2≤⋯≤ql,qi∈{j1,j2,…,jn-k}‍∏i=1l‍ω*(qi).Now, we are going to count the number of tableaux withn-k columns such that n-k-r columns are of weight c1 and r columns are of weight ω*(qi), qi∈{0,1,2,…,k}. Note that there are (n-kr) tableaux with r columns whose lengths are taken from the lengths of the columns of Φ. Since there is a one-to-one correspondence between weights ω(ji) and A-tableaux, the number of A-tableaux Φ in TA(k,n-k) is equal to the number of possible multisets {j1,j2,…,jn-k} with ji in {0,1,2,…,k}. That is,
(4.8)|TA(k,n-k)|=(nk).Thus, for allΦ∈TA(k,n-k), we can generate (nk)(n-kr) tableaux with r columns whose weights are ω*(ji), ji∈{0,1,2,…,k}. However, there are only
(4.9)|TA(k,r)|=(r+kr)distinct tableaux withr columns whose lengths are in {0,1,2,…,k}. Hence, every distinct tableau with n-k columns, r of which are of weight other than c1, appears
(4.10)(nk)(n-kr)(r+kr)=(nr+k)
times in the collection. Thus,
(4.11)∑Φ∈TA(k,n-k)‍ωA(Φ)=∑r=0n-k‍(nr+k)c1n-k-r∑ϕ∈B¯r‍∏c∈ϕ‍ω*(|c|),whereB¯r denotes the set of all tableaux ψ having r columns of weights ω*(ji)=[jiβ]q+c2. Reindexing the double sum, we get
(4.12)∑Φ∈TA(k,n-k)‍ωA(Φ)=∑j=kn‍(nj)c1n-j∑ϕ∈B¯j-k‍∏c¯∈ϕ‍ω*(|c|),
where B¯j-k is the set of all tableaux with j-k columns of weights ω*(ji)=[jiβ]q+c2 for each i=1,2,…,j-k. Clearly, B¯j-k=TA(k,j-k). Therefore,
(4.13)∑Φ∈TA(k,n-k)‍ωA(Φ)=∑j=kn‍(nj)c1n-j∑ϕ∈TA(k,j-k)‍ωA(ϕ).Applying Theorem4.3 completes the proof of the following theorem.Theorem 4.4.
For nonnegative integersn and k and complex numbers β and r, the q-analogue σ[n,k]qβ,r satisfies the following identity:
(4.14)σ[n,k]qβ,r=∑j=kn‍(nj)q(n-j)r2[r1]qn-jσ[j,k]qβ,r2,
where r=r1+r2.Takingβ=1, r2=0, and r=r1=logq2, Theorem 4.4 gives
(4.15)(q-1)n-kσ[n,k]q1,logq2=∑j=kn‍(nj)(q-1)j-kσ[j,k]q1,0.Using (2.16) and (3.26), we obtain
(4.16)[nk]q=∑j=kn‍(nj)(q-1)j-kSq[j,k]
the Carlitz identity in [12]. Hence, we can consider the identity in Theorem 4.4 as a generalization of the above Carlitz identity.


## 5. Orthogonality and Inverse Relations

We notice that (1.4) can be written as
(5.1)∑k=0m‍Fr,-a(m,k)tk=pma(t).Using (1.1), it can easily be shown that
(5.2)∑k=nm‍Fr,-a(m,k)Snk(a)=∑k=nm‍Skm(a)Fr,-a(k,n)=δmn,
where δmn is the Kronecker delta defined by δmn=1 if m=n, and δmn=0 if m≠n. Moreover, the following inverse relations hold:
(5.3)fn=∑k=0n‍Skn(a)gk⟺gn=∑k=0n‍Fr,-a(n,k)fk,(5.4)fk=∑n≥k‍Skn(a)gn⟺gk=∑n≥k‍Fr,-a(n,k)fn.Relation (5.2) is exactly the orthogonality relation for r-Whitney numbers that appeared in [6]. Consequently, the generating functions in (P2) and (P3) can be transformed, respectively, using (5.4) into the following identities:
(5.5)∑n≥k‍Fr,-a(n,k)k!xk1rnn!eax(erx-1)n=1,∑n≥k‍Fr,-a(n,k)xn-k∏j=0n(1-(rj+a)x)=1,
which will reduce to the following interesting identities for Fα,γ(n,k) when x=1:
(5.6)∑n≥k‍Fα,γ(n,k)k!(eα-1)nαnn!eγ=1,∑n≥k‍Fα,γ(n,k)(1+γ∣α)n+1=1.Note that the numberFα,γ(n,k) can be expressed in terms of the unified generalization of Stirling numbers by Hsu  and  Shiue   [16] as Fα,γ(n,k)=S(n,k;α,0,γ). Hence, the identity in  coincides with the identity in [17, Theorem 9] by taking x=1+γ.Parallel to (5.2), (5.3), and (5.4), we will establish in this section the orthogonality and inverse relations of ϕα,γ[n,k]q and σ[n,k]qβ,r.To derive the orthogonality relation forϕα,γ[n,k]q and σ[n,k]qβ,r, we need to rewrite first (1.5) and (3.1). By taking γ=logq(2-qr), (1.5) gives
(5.7)∑k=0n‍ϕα,logq(2-qr)[n,k]qtk=〈t-[r]q∣[α]〉nq,
and, by replacing t with t-[r]q, (3.1) yields
(5.8)∑k=0n‍σ[n,k]qβ,r〈t-[r]q∣[β]〉kq=tn.Using (5.8), (5.7) can be expressed as
(5.9)〈t-[r]q∣[β]〉mq=∑k=0m‍ϕβ,logq(2-qr)[m,k]q{∑n=0k‍σ[k,n]qβ,r〈t-[r]q∣[β]q〉nq}=∑n=0m‍{∑k=nm‍ϕβ,logq(2-qr)[m,k]qσ[k,n]qβ,r}〈t-[r]q∣[β]q〉nq.
Thus
(5.10)∑k=nm‍ϕβ,logq(2-qr)[m,k]qσ[k,n]qβ,r=δmn(m≥n).Theorem 5.1.
For nonnegative integersm, n, and k and complex numbers β and r, the following orthogonality relation holds:
(5.11)∑k=nm‍ϕβ,r-[m,k]qσ[k,n]qβ,r=∑k=nm‍σ[m,k]qβ,rϕβ,r-[k,n]q=δmn(m≥n),
where r-=logq(2-qr).Remark 5.2.
It can easily be shown thatr-=logq(2-qr)→-r as q→1. This implies that ϕβ,r-[m,k]q→Fβ,-r(m,k) as q→1. Since σ[k,n]qβ,r→Snk(c) as q→1, (5.11) yields (5.2) easily.Remark 5.3.
LetM1 and M2 be two n×n matrices whose entries are ϕβ,r-[i,j]q and σ[i,j]qβ,r, respectively. That is, M1=(ϕβ,r-[i,j]q)0≤i,j≤n and M2=(σ[i,j]qβ,r)0≤i,j≤n. Then using Theorem 5.1, M1M2=M2M1=In, the identity matrix of order n. This implies that M1 and M2 are orthogonal matrices.
Using the orthogonality relation in Theorem5.1, we can easily prove the following inverse relation.Theorem 5.4.
For nonnegative integersm, n, and k, and complex numbers β and r, the following inverse relation holds:
(5.12)fn=∑k=0n‍σ[n,k]qβ,rgk⟺gn=∑k=0n‍ϕβ,r-[n,k]qfk,
where r-=logq(2-qr).Proof.
Givenfn=∑k=0n‍σ[n,k]qβ,rgk, we have
(5.13)∑k=0n‍ϕβ,r-[n,k]qfk=∑k=0n‍ϕβ,r-[n,k]q{∑j=0k‍σ[k,j]qβ,rgj}=∑j=0k‍{∑k=jn‍ϕβ,r-[n,k]qσ[k,j]qβ,r}gj=∑j=0k‍δnjgj=gn.The converse can be shown similarly.One can easily prove the following inverse relation.Theorem 5.5.
For nonnegative integersm, n, and k and complex numbers β and r, the following inverse relation holds:
(5.14)fk=∑n=0∞‍σ[n,k]qβ,rgn⟺gk=∑n=0∞‍ϕβ,r-[n,k]qfn,
where  r-=logq(2-qr).Remark 5.6.
The exponential and rational generating functions in Theorem3.5 can be transformed into the following identities for the q-analogue of Fα,γ(n,k):
(5.15)∑n≥0‍ϕβ,r-[n,k]qk!tkΔqn(e([xβ]q+[r]q)t〈[nβ]q∣[β]〉nq)x=0=1,∑n≥0‍ϕβ,r-[n,k]qtn-k∏j=0n(1-([jβ]q+[r]q)t)=1,
when q→1,  will exactly give , respectively.



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*Source: 592818-2012-10-18.xml*