Certain Identities Associated with ( p , q ) -Binomial Coefﬁcients and ( p , q ) -Stirling Polynomials of the Second Kind

: The q -Stirling numbers (polynomials) of the second kind have been investigated and applied in a variety of research subjects including, even, the q -analogue of Bernstein polynomials. The ( p , q ) -Stirling numbers (polynomials) of the second kind have been studied, particularly, in relation to combinatorics. In this paper, we aim to introduce new ( p , q ) -Stirling polynomials of the second kind which are shown to be ﬁt for the ( p , q ) -analogue of Bernstein polynomials. We also present some interesting identities involving the ( p , q ) -binomial coefﬁcients. We further discuss certain vanishing identities associated with the q -and ( p , q ) -Stirling polynomials of the second kind.

The (p, q)-integers are presented by The (p, q)-factorial [µ] p,q ! of µ ∈ N 0 is given by The (p, q)-binomial coefficients [ n µ ] p,q are provided by and otherwise is accepted to be zero.The Pascal-type identity for (p, q)-binomial coefficient is given as (see, e.g., ([42], Equation (4.5))) (n, µ ∈ N 0 with µ ≤ n) , which, upon using the following easily derivable identity from ( 16) gives another Pascal-type identity for (p, q)-binomial coefficient (n, µ ∈ N 0 with µ ≤ n) and vice versa.One observes from either (17) or (19) that the (p, q)-binomial coefficients [ n µ ] p,q are polynomials in both p and q of degree µ(n − µ).
When p = q with p, q ∈ C \ {0}, one gets (see, e.g., ([42], Equation (4.4))) and n The q-Stirling polynomials of the second kind have been investigated in diverse research subjects including, even, the q-analogue of Bernstein polynomials, for example, [4].The (p, q)-Stirling numbers (polynomials) of the second kind have been studied, particularly, in relation to combinatorics.In this paper, we aim to introduce new (p, q)-Stirling polynomials of the second kind (57) which are shown to be fit for the (p, q)-analogue of Bernstein polynomials (59).We also provide some interesting identities involving the (p, q)-binomial coefficients.We further discuss certain vanishing identities associated with the q-and (p, q)-Stirling polynomials of the second kind.
It is noticed that the product in ( 23) is connected with the following well known Euler's identity: One finds easily that the Euler's identity (24), which may be validated by induction on µ or another method (see, e.g., ([16], Chapter 7)), reduces to the ordinary binomial expansion when q = 1.
Phillips [16] used Here we set x j = [j] q /[n] q so that x j ∈ [0, 1] (j ∈ N 0 ) and summarize some slightly modified related formulas in the following theorem.
n] q for j ∈ N 0 , n ∈ N, and q ∈ R + with 0 ≤ j ≤ n.Then a relation between the Newton's divided difference h [x 0 , x 1 , . . . ,x τ ] and the q-difference of order τ in (26) is given as follows: h x j , x j+1 , . . ., for τ ∈ N 0 , where h It is noted that the explicit q-difference formula ( 27) remains the same under the conditions of Theorem 1.The proof is omitted.The interested reader may be referred (for example) to ( [16], p. 268) and [73] (see also [74]). .
If we choose a monomial h(x), say, h(x) = x ( ∈ N) in (34) together with ( 2) and ( 27), we obtain where the involved notations and conditions remain the same as above.From (35), obviously S q ( , τ) = 0 when τ > , which is equivalent to the vanishing identity (30).
It is noted that for a function h, the Newton's divided difference h x j , x j+1 , . . ., x j+τ with τ + 1 distinct points and the q-difference ∆ τ q h of order τ, themselves, vanish when h is a monomial whose degree is less than τ.In this regard, we choose h(x) = x with τ > in (27) to obtain the following mild extension of the vanishing identity (30): ( , τ, µ ∈ N 0 with τ > ) .

Certain Identities Involving the (p, q)-Binomial Coefficients
We recall the following (p, q)-analogue of the Euler's identity (24), which can be verified by induction (consult, e.g., ([73], Equation (1.5))) or the method in the proof of Theorem 2: (p, q ∈ C with p = q; n ∈ N) .
We may give the inverse of the (p, q)-binomial expansion (38), which is asserted in the following theorem.
Theorem 2. Let p, q ∈ C be such that p = q and |q/p| < 1.Furthermore, let n ∈ N.
Proof of Theorem 2. We write where Then we have Replacing x by q p x in F n (x) gives We find from the last two identities that which, upon equating the coefficients of x s , yields Or, equivalently, From the last identity we derive c s = (−1) s n + s − 1 s q/p c 0 (s ∈ N 0 ) , which, upon using (20) and 2 , produces This completes the proof.
If we multiply (38) and (39) side by side and equate the coefficients x r of the resulting identity, we get a vanishing identity, which is given in the following corollary.
Corollary 1.Let p, q ∈ C be such that p = q and |q/p| We provide a (p, q)-analogue of the Chu-Vandermonde identity, which is asserted in the following theorem.Theorem 3. Let p, q ∈ C be such that p = q.Furthermore, let , m ∈ N and r ∈ N 0 .Then Proof of Theorem 3. We rewrite (38) to denote Then we consider which, upon using the right member of (42), yields Equating the coefficients of x r in the last identity yields (41).
We may use the relation (20) to convert certain identities involving q-binomial coefficients into those associated with (p, q)-binomial coefficients.We illustrate two identities in the following theorem (cf., ([16], Problem 8.1.8)).
Theorem 4. Let p, q ∈ C \ {0} be such that p = q.Furthermore, put n, s ∈ N 0 with 0 ≤ s ≤ n.Then Proof of Theorem 4. One may use induction on n to verify these identities.The details are omitted.
Remark 2. We use the same reasoning in ( [16], p. 268) together with (52) to get q p for some ξ ∈ (x 0 , x ν ), when h ∈ C ν [0, 1].In particular, when h(x) is a monomial x whose degree is less than ν, we find that Here, in fact, when ν > with h(x) = x , the (p, q)-difference ∆ ν p,q h 0 and the Newton's divided difference h [x 0 , x 1 , . . . ,x ν ], themselves, are seen to be zero.The identities and arguments here are easily found to reduce to yield those in Theorem 1.

Open Question
Find a recursive relation for the (p, q)-Stirling polynomials of the second kind S p,q ( , ν) in (57) like the (47).

Conclusions
In Section 3, some interesting and new identities involving the (p, q)-binomial coefficients including, in particular, the (p, q)-analogue of the Chu-Vandermonde identity (41), are presented.Using the same technique in Section 3 together with (20) and (21), some known identities associated with the q-binomial coefficients (if any) are believed to yield the corresponding identities involving the (p, q)-binomial coefficients.
In Section 4, the new (p, q)-Stirling polynomials of the second kind are introduced and shown to be fit for the (p, q)-analogue of Bernstein polynomials.A recursive relation for these new (p, q)-Stirling polynomials of the second kind remains to be an open question.More properties and applications of these new (p, q)-Stirling polynomials of the second kind are left to the authors and the interested researchers for future study.
Certain vanishing identities associated with the q-and (p, q)-Stirling polynomials of the second kind are also discussed.