A (p, q)-Deformed Recurrence for the Bell Numbers

Lahcen Oussi Institute of Mathematics University of Wrocław Pl. Grunwaldzki 2/4 50-384 Wrocław Poland

                 lahcen.oussi@math.uni.wroc.pl

Abstract We obtain a (p, q)-deformation of the recurrence formula for the Bell numbers, using algebraic techniques. Specializing to the case p = 1 and the case p = q = 1, respectively, we recover the generalized recurrence formula for Bell numbers as obtained by Katriel and Spivey, in related papers.

1 Introduction

The Bell numbers, denoted by B_{n}, are given by the following formula:

B_{n} = ∑_{k=0}^{n} S(n, k),  (1)

where S(n, k) are the Stirling numbers of the second kind, which appear as coefficients in the expansion of

x^{n} = ∑_{k=0}^{n} S(n, k) ∏_{i=0}^{k−1}(x − i).

The Stirling numbers of the second kind S(n, k) count the number of ways to partition a set of size n into k nonempty subsets.

The Bell numbers (1) satisfy the following recursive formula:

B_{n+1} = ∑_{k=0}^{n} \binom{n}{k}B_{k}.  (2)

In 2008, Spivey [11] obtained the following generalization of the recurrence formula (2):

B_{n+m} = ∑_{j=0}^{n} ∑_{k=0}^{m} k^{n−j}S(m, k)\binom{n}{j}B_{j},  (3)

using techniques from combinatorics. This formula is known in the literature as “Spivey’s Bell number formula”. Subsequently, Katriel [6] proved a q-deformed version of Spivey’s Bell number formula using algebraic techniques. The resulting formula can be written as follows:

B_{n+m}(q) = ∑_{k=0}^{m} ∑_{j=0}^{n} \binom{n}{j}S_{q}(m, k)[k]_{q}^{n−j}q^{jk}B_{j}(q),  (4)

where

S_{q}(m, k) = q^{k−1}S_{q}(m − 1, k − 1) + [k]_{q}S_{q}(m − 1, k),  (5)

are the q-Stirling numbers of the second kind with the initial value S_{q}(0, 0) = 1, and [k]_{q} = \frac{1−q^{k}}{1−q}. Here,

B_{n}(q) = ∑_{k=0}^{n} S_{q}(n, k)  (6)

are the q-Bell numbers.

An alternative derivation of (4) was obtained by Mangontarum [7], using techniques based on the analysis of the creation, annihilation and number operators in the q-Boson-Fock space [1]. It is worthwhile to mention that, Eq. (1) has also been extended in several ways by several authors, including various generalizations of Stirling and Bell numbers. We refer the reader to [3, 5, 9, 12] for the details.

In the present paper, we follow the techniques by Katriel [6], and obtain a (p, q)-deformed version of Spivey’s Bell number formula (3), using algebraic methods.

2 A (p, q)-deformation of Spivey’s Bell number formula

For 0 < q < p ≤ 1, define the following operators:

1. The operator X of multiplication by the variable x:

Xf(x) = xf(x).

2. The (p, q)-derivative operator:

D_{p,q}f(x) = \frac{f(px) − f(qx)}{x(p − q)}.

3. The Fibonacci operator [8]:

N_{p}f(x) = f(px).

These operators satisfy the (p, q)-commutation relation:

D_{p,q}X − qXD_{p,q} = N_{p}.  (7)

We can check easily that

N_{p}X = pXN_{p},  (8)

and

D_{p,q}N_{p} = pN_{p}D_{p,q}.  (9)

Proposition 1. For any positive integer n, we have

D_{p,q}X^{n} = q^{n}X^{n}D_{p,q} + [n]_{p,q}X^{n−1}N_{p},  (10)

where [n]_{p,q} = \frac{p^{n}−q^{n}}{p−q}. Proof. The proof follows from an easy computation involving an induction on n, and using Eqs. (7) and (8). Corollary 2. One can rewrite (10) as follows:

(XD_{p,q})X^{n} = X^{n}([n]_{p,q}N_{p} + q^{n}(XD_{p,q})).  (11)

Proposition 3. Let n be a nonnegative integer, then the following relation holds:

(XD_{p,q})^{n} = ∑_{k=0}^{n} S_{p,q}(n, k)X^{k}N_{p}^{n−k}D_{p,q}^{k},  (12)

where

S_{p,q}(n, k) = p^{n−k}q^{k−1}S_{p,q}(n − 1, k − 1) + [k]_{p,q}S_{p,q}(n − 1, k),  (13)

are the (p, q)-Stirling numbers of the second kind with the initial value S_{p,q}(0, 0) = 1, and consequently the numbers

B_{n}(p, q) = ∑_{k=0}^{n} S_{p,q}(n, k),  (14)

can be considered as the (p, q)-Bell numbers. Proof. The proof follows by induction with respect to n, using Eqs. (7), (9), and (11).

Remark 4. For p = 1, Eqs. (13) and (14) reduce to the q-Stirling numbers of the second kind (5) and q-Bell numbers (6), respectively.

Definition 5. The (p, q)-exponential function is defined by the formula:

e_{p,q}(x) := ∑_{n=0}^{∞} p^{\binom{n}{2}} \frac{x^{n}}{[n]_{p,q}!},

where [n]_{p,q}! = ∏_{i=1}^{n}[i]_{p,q}, and [0]_{p,q}! = 1.

Sadjang [10] showed that the (p, q)-exponential function e_{p,q}(x) satisfies the following differential equation:

D_{p,q}^{n}e_{p,q}(x) = p^{\binom{n}{2}}e_{p,q}(p^{n}x),  (15)

where D_{p,q}^{n} is the nth (p, q)-derivative.

Applying the (p, q)-exponential function e_{p,q}(x) to both sides of Eq. (12), and dividing by e_{p,q}(p^{n}x), we arrive at the identity:

\frac{1}{e_{p,q}(p^{n}x)}(XD_{p,q})^{n}e_{p,q}(x) = ∑_{k=0}^{n} S̃_{p,q}(n, k)x^{k} = B̃_{n}(p, q; x),  (16)

where S̃_{p,q}(n, k) = p^{\binom{k}{2}}S_{p,q}(n, k) can be considered as new versions of the (p, q)-Stirling numbers of the second kind, with B̃_{n}(p, q; x) being the associated (p, q)-Bell polynomials, which give for x = 1 the new version of the (p, q)-Bell numbers B̃_{n}(p, q; 1) = ∑_{k=0}^{n} S̃_{p,q}(n, k). Moreover, putting x = −1 in the Equation (16), we obtain (p, q)-Rényi numbers R_{n}(p, q) = ∑_{k=0}^{n}(−1)^{k}S̃_{p,q}(n, k) which reduce to the usual q-Rényi number [4] for p = 1.

3 Main result

We are now in a position to prove the main result of this paper.

Theorem 6. Let m and n be nonnegative integers. Then we have the following (p, q)- deformation of Spivey’s Bell number formula:

B̃_{n+m}(p, q; 1) = ∑_{k=0}^{m} ∑_{j=0}^{n} \binom{n}{j}S̃_{p,q}(m, k)[k]_{p,q}^{n−j}q^{jk}B̃_{j}(p, q; p^{n+m−j}).  (17)

Proof. Using Eqs. (12) and (11), we obtain

(XD_{p,q})^{n+m} = ∑_{k=0}^{m} S_{p,q}(m, k)(XD_{p,q})^{n}X^{k}N_{p}^{m−k}D_{p,q}^{k}

= ∑_{k=0}^{m} S_{p,q}(m, k)X^{k} ([k]_{p,q}N_{p} + q^{k}(XD_{p,q}))^{n} N_{p}^{m−k}D_{p,q}^{k}

= ∑_{k=0}^{m} ∑_{j=0}^{n} \binom{n}{j}S_{p,q}(m, k)[k]_{p,q}^{n−j}q^{jk}X^{k}N_{p}^{n−j}(XD_{p,q})^{j}N_{p}^{m−k}D_{p,q}^{k},

where in the last equality, we have used the binomial expansion. Applying the above identity to the (p, q)-exponential function e_{p,q}(x), and using Eqs. (15) and (16), we get the following:

(XD_{p,q})^{n+m}e_{p,q}(x) = ∑_{k=0}^{m} ∑_{j=0}^{n} \binom{n}{j}S_{p,q}(m, k)[k]_{p,q}^{n−j}q^{jk}X^{k}N_{p}^{n−j}(XD_{p,q})^{j}N_{p}^{m−k}D_{p,q}^{k}e_{p,q}(x)

= ∑_{k=0}^{m} ∑_{j=0}^{n} \binom{n}{j}S̃_{p,q}(m, k)[k]_{p,q}^{n−j}q^{jk}X^{k}N_{p}^{n−j}(XD_{p,q})^{j}N_{p}^{m−k}e_{p,q}(p^{k}x)

= ∑_{k=0}^{m} ∑_{j=0}^{n} \binom{n}{j}S̃_{p,q}(m, k)[k]_{p,q}^{n−j}q^{jk}X^{k}N_{p}^{n−j}(XD_{p,q})^{j}e_{p,q}(p^{m}x)

= ∑_{k=0}^{m} ∑_{j=0}^{n} \binom{n}{j}S̃_{p,q}(m, k)[k]_{p,q}^{n−j}q^{jk}X^{k}N_{p}^{n−j}B̃_{j}(p, q, p^{m}x)e_{p,q}(p^{m+j}x)

= ∑_{k=0}^{m} ∑_{j=0}^{n} \binom{n}{j}S̃_{p,q}(m, k)[k]_{p,q}^{n−j}q^{jk}x^{k}B̃_{j}(p, q; p^{n+m−j}x)e_{p,q}(p^{n+m}x).

Dividing both sides of the above identity by e_{p,q}(p^{n+m}x) and setting x = 1 gives

B̃_{n+m}(p, q; 1) = ∑_{k=0}^{m} ∑_{j=0}^{n} \binom{n}{j}S̃_{p,q}(m, k)[k]_{p,q}^{n−j}q^{jk}B̃_{j}(p, q; p^{n+m−j}).

Remark 7. For p = 1, Eq. (17) reduces to the q-deformation of Spivey’s Bell number formula (4), and for p = q = 1, it reduces to Spivey’s Bell number formula (3).

Proposition 8 (Dobinski formula). The (p, q)-Bell polynomials B̃_{n}(p, q; x), satisfy the following identity:

B̃_{n}(p, q; x) = \frac{1}{e_{p,q}(p^{n}x)} ∑_{k=0}^{∞} p^{\binom{k}{2}} \frac{[k]_{p,q}^{n}}{[k]_{p,q}!}x^{k}.  (18)

Consequently, the (p, q)-Bell numbers B̃_{n}(p, q; 1), are given by:

B̃_{n}(p, q; 1) = \frac{1}{e_{p,q}(p^{n})} ∑_{k=0}^{∞} p^{\binom{k}{2}} \frac{[k]_{p,q}^{n}}{[k]_{p,q}!}.

Proof. Since (XD_{p,q})x^{m} = [m]_{p,q}x^{m}, it follows that

(XD_{p,q})^{n}x^{m} = [m]_{p,q}^{n}x^{m},

and

(XD_{p,q})^{n}e_{p,q}(x) = ∑_{k=0}^{∞} p^{\binom{k}{2}} \frac{[k]_{p,q}^{n}}{[k]_{p,q}!}x^{k}.  (19)

Multiplying both sides of Eq. (19) with \frac{1}{e_{p,q}(p^{n}x)} and taking into account Eq. (16) proves the desired result.

Remark 9. When p = 1, Eq. (18) reduces to the Dobinski formula for q-Bell polynomials given by Katriel [6]. Furthermore, when p = q = 1, we obtain the Dobinski formula for the ordinary Bell polynomials:

B_{n}(x) = \frac{1}{e} ∑_{k≥0} \frac{k}{k!}x^{k}.

Therefore, we can think of Eq. (18) as the Dobinski formula for (p, q)-Bell polynomials B̃_{n}(p, q; x).

4 Final remarks

It seems likely that following the techniques due to Mangontarum [7], one can obtain an alternative proof of (17), using the creation, annihilation and number operators in the (p, q)- Fock space [2]. Our speculation stems from the following observation:

Let l^{2}(ℕ) be a Hilbert space with the standard orthonormal basis (δ_{n})_{n=0}^{∞}. We define the creation, annihilation and number operators as follows:

1. The creation operator a^{+} defined by

a^{+}δ_{n} = \sqrt{[n + 1]_{p,q}} · δ_{n+1}, n ≥ 0.

2. The annihilation operator (adjoint of a^{+}) a^{−} defined by

a^{−}δ_{0} = 0, a^{−}δ_{n} = \sqrt{[n]_{p,q}} · δ_{n−1}, n ≥ 1.

3. The number operator a^{∘} defined by

a^{∘}δ_{n} = p^{n}δ_{n}, n ≥ 0.

It follows by an easy computation that these operators satisfy the (p, q)-commutation relation

a^{−}a^{+} − qa^{+}a^{−} = a^{∘}.

5 Acknowledgments

Research partially supported by the National Agency for Academic Exchange (NAWA) POLONIUM project PPN/BIL/2018/1/00197/U/00021 and by the Polish National Science Center (NCN) grant 2016/21/B/ST1/00628.

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2000 Mathematics Subject Classification: Primary 11B73; Secondary 11B83, 05A19, 05A30. Keywords: (p, q)-Bell number, (p, q)-Stirling number, Spivey’s Bell number formula, (p, q)- Fock space, (p, q)-calculus.

(Concerned with sequences \underline{A000110} and \underline{A008277}.)

Received December 4 2019; revised versions received December 13 2019; February 7 2020; March 25 2020; May 4 2020; May 6 2020. Published in Journal of Integer Sequences, May 7 2020.

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