Differential equations for p, q-Touchard polynomials

Taekyun Kim; Orli Herscovici; Toufik Mansour; Seog-Hoon Rim

doi:10.1515/math-2016-0082

Article Open Access

Differential equations for p, q-Touchard polynomials

Taekyun Kim , Orli Herscovici , Toufik Mansour and Seog-Hoon Rim

Published/Copyright: November 11, 2016

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$Open Mathematics$

From the journal Open Mathematics Volume 14 Issue 1

Abstract

In this paper, we present differential equation for the generating function of the p, q-Touchard polynomials. An application to ordered partitions of a set is investigated.

Keywords: Exponential polynomials; Ordered partitions; Touchard polynomials

MSC 2010: 05A18; 05A48; llB48; llB47

1 Introduction

It is well known that the Touchard polynomials (exponential polynomials or Bell polynomials) T_n(x), play an important role in statistics and probability(see [1-14]). They are given by

Tn(x)=∑k=0nS(n;k)xk.

where S(n, k) is the Stirling number of second kind (see [6], [14]). Moreover, the Touchard polynomials are given by the generating function (see [6], [13])

ex(et−1)=∑n=0∞Tn(x)tnn!.

By using the combinatorial definitions of the Stirling and the Bell numbers and the fact that the generating (see [13]) function of the Bell numbers B_n is given by

eet−1=∑n=0∞Bntnn!,

we have that B_n = T_n(1).

The q-exponential function is given by (see [6], [14])

expq(x)=(1+(1−q)x)11−q,(1)

which corresponds to the degenerated exponential function of Carlitz [2] with λ = 1−q. Borges [1] showed that this q-exponential function has the following expansion in Taylor series

expq(x)=1+x+∑k≥1q(2q−1)(3q−2)…(kq−(k−1))xk+1(k+1)!.(2)

Following [6] and [3], we define the (p, q)-deformed Touchard polynomials, T_n(x|p, q), for integer n ≥ 0 by means of the generating function

Tp,q(x;t)=expp(x(expq(t)−1))=∑n=0∞Tn(x|p,q)tnn!,(3)

where deformed exponential functions exp_p(t) and exp_q(t) are given by (1). The first five (p, q)-deformed Touchard polynomials are

We denote the coefficient of x^j in T_n(x|p, q) by S_{p, q}(n, i) (see [3]). For instance, S_{p, q}(1,1)=1, S_p,q(2,1) = q and S_{p, q}(2,2) = p. Clearly, S_{p, q}(n, 1) = q(2q−1)(3q−2)…((n−1)q−(n−2)) and S_{p, q}(n, n) = p(2p− 1) (3p−2)…((n−1)p−(n−2)).

Note that Kim at al. [7] studied the linear differential equations arising from the Poisson-Charlier, actuarial, and Meixner polynomials. Also, in [11] the authors considered differential equations arising from the generating function of general modified degenerate Euler numbers. New identities were obtained for Laguerre polynomials [10] and for Bernoulli polynomials of the second kind [5] using linear and non-linear differential equations. Using differential equations, new identities for Changee and Euler polynomials were obtained in [8], for Frobenius-Euler polynomials in [4], and for degenerate Euler numbers in [9].

Here, we study the differential equations arising from the generating function of (p, q)-deformed Touchard polynomials. In particular, we show the following result.

Theorem 1.1

LetN ≥ 1. Then the differential equation

F(N)=∑i=1NSp,q(N,i)xi(1+(1−q)t)i1−q−NF1−i(1−p),

has a solutionF = T_{p, q}(x; t), whereF(N)=dNdtNTp,q(x;t).

2 Proofs and Applications

Let F = T_{p, q}(x;t) and define F(N)=dNdtNTp,q(x;t), for all N > 1. Observe that

F=Tp,q(x;t)=(1−(1−p)x+(1−p)x(1+(1−q)t)11−q)11−p.

Note that ddt(F1−p)=(1−p)x(1+(1−q)t)11−q−1. Hence,

F(1)=x(1+(1−q)t)11−q−1Fp,F(2)=qx(1+(1−q)t)11−q−2Fp+px2(1+(1−q)t)21−q−2F1−2(1−p).

Let us assume that

F(N)=∑i=1NaN,i(p,q)xi(1+(1−q)t)i1−q−NF1−i(1−p).

Clearly, a_1,1(p, q) = 1, a_2,1(p, q) = q and a_2,2(p, q) = p.

By differentiating F^(N) with respect to t, we obtain

F(N+1)=ddtF(N)=∑i=1N(i/(1−q)−N)(1−q)aN,i(p,q)xi(1+(1−q)t)i1−q−N−1F1−i(1−p)+∑i=1N(1/(1−p)−i)(1−p)xaN,i(p,q)xi(1+(1−q)t)i+11−q−N−1F1−(i+1)(1−p),

which implies

F(N+1)=∑i=1(i−N(1−q))aN,i(p,q)xj(1N+(1−q)t)i1−q−N−1F1−i(1−p)+∑i=2N+1(1−i(1−p))aN,i−1(p,q)xj(1+(1−q)t)i1−q−N−1F1−i(1−p).

Note that we assume that F(N+1)=∑i=1N+1aN+1,i(p,q)xj(1+(1−q)t)i1−q−N−1F1−j(1−p), so by comparing the coefficients of xi(1+(1−q)t)i1−q−N−1F1−i(1−p), we obtain

aN+1,i(p,q)=(i−N(1−q))aN,i(p,q)+(1−(i−1)(1−p))aN,i−1(p,q)(4)

with the initial conditions a_1,1(p, q) = 1 and a_{N, i}(p, q) = 0 whenever N < i or i < 0.

Lemma 2.1

For all 1 ≤ i ≤ N, a_{N, i}(p, q) is the coefficient ofx^jinT_N(x|p, q).

Proof

Define Aj(t)=∑N≥iaN,i(p,q)tNN!. Then, by multiplying (4) by tNN!, and summing over N ≥ i, we obtain

(1+(1−q)t)ddtAi(t)−iAi(t)=Ai−1(t)−(i−1)(1−p)Ai−1(t)

with A₀(t) = 1.

Define A(x;t)=∑i≥0Aj(t)xj. By multiplying the last recurrence by xⁱ and summing over i ≥ 0, we have

(1+(1−q)t)ddtA(x;t)−x(1−(1−p)x)ddxA(x;t)=xA(x;t)

with A(x; 0) = 1. Solving the partial differentiate equation for A(x; t), gives that there exists a function f(y) such that

A(x;t)=f1−(1−p)xx(1+(1−q)t)−11−q(1+(1−q)t)1(1−q)(1−p)x11−p.

By using the initial condition A(x; 0) = 1, we obtain that f(y)=(y+1−p)11−p. Hence,

A(x;t)=1−(1−p)xx(1+(1−q)t)−11−q+1−p11−p(1+(1−q)t)1(1−q)(1−p)x11−p=((1−(1−p)x)(1+(1−q)t)−11−q+(1−p)x)11−p(1+(1−q)t)1(1−q)(1−p)=(1−(1−p)x+(1−p)x(1+(1−q)t)11−q)11−p=(1+(1−p)x(expq(t)−1))11−p=expp(x(expq(t)−1))=Tp,q(x;t),

which completes the proof. □

Thus, by Lemma 2.1, we obtain that

F(N)=∑i=1NSp,q(N,i)xiN(1+(1−q)t)i1−q−NF1−i(1−p),

where S_{p, q}(N, i) is the coefficient of xⁱ in T_N(x|p, q). This completes the proof of Theorem 1.1. □

Note that for p = 1, Theorem 1.1 gives

F(N)|p=1=∑i=1NS1,qN(N,i)xi(1+(1−q)t)i1−q−NF,

where, by the main result of [12] and (4), we have

S1,q(N,i)=∑j=1i∏ℓ=1N−1(j+ℓ(q−1))∏ℓ=1,ℓ≠ji(j−ℓ).

These numbers have the following combinatorial interpretation. An ordered partition of a set [n] = {1, 2,…, n} is a set of nonempty disjoint subsequences B₁/B₂/…/B_k, called blocks, whose union is [n] (For partitions of a set [n], see [13]). Note that we can assume that min B₁ < min B₂ < … < min B_k. For example, the ordered partition of the set [3] are given by 123, 132, 213, 231, 312, 321, 12/3, 21/3, 13/2, 31/2, 1/23, 1/32, 1/2/3. We denote the set of all ordered partitions of the set [n] with k blocks by OPn,k. Define OPn=⋃k=0nOPn,k.

Let π be any ordered partition in OPn,k. Let π⁽ⁱ⁾ be the ordered partition that is obtained from π by removing all the elements i + 1, i + 2, …, n from π. An element i is called a closer if there exists a block B in π⁽ⁱ) such that i is the rightmost element of B and B contains at least two elements. Otherwise, it is called an opener. For example, if π = 152/437/68 then 8, 7, 2 is closers and 6, 5, 4, 3, 1 openers.

Let b_j(N, m) to be the number of ordered partitions of [N] with exactly m closers and i blocks. We define bN,i(q)=∑m=0Nbj(N,m)qm.

Lemma 2.2

LetN ≥ 1. Thenb_1,1(q) = 1, b_{N, i}(q) = 0 wheneveri > Nori < 0, and

bN+1,i(q)=bN,i−1(q)+(i+Nq)bN,i(q),1≤i≤N+1.

Proof

Clearly the initial conditions are held. Let π be any ordered partition of [N + 1] with exactly 2 ≤ i ≤ N blocks and let us write an equation for b_{N + 1, i}(q). If the element N + 1 is form a block, then it should be the rightmost block of π and in this case we have a contribution of b_{N, i−1}(q) . Otherwise, the element N + 1 belongs to a block, say B_j with 1 ≤ j ≤ i. In the case N + 1 is rightmost element of B_j we have a contribution b_{N, i}(q), and in the case N + 1 is not rightmost element of B_j we obtain a contribution of |B_j|qb_{N, i}(q). Hence,

bN+1,i(q)=bN,i−1(q)+∑j=1i(1+q|Bj|)bN,i(q)=bN,i−1(q)+(i+Nq)bN,i(q),

as required. □

By (4) and Lemma 2.2, we obtain the following result.

Proposition 2.3

For all 1 ≤ i ≤ N, S_1,q(N, i) = b_{N, i}(q−1).

Recall that T1,q(x;t)=∑n≥0Tn(x|1,q)tnn!. Clearly,

F(N)=dNdtNT1,q(x;t)=∑n≥0Tn+N(x|1,q)tnn!.

Hence, by Theorem 1.1, we have

F(N)=∑i=1N∑ℓ≥0S1,q(N,i)xj(i1−q−N)ℓ(1−q)ℓtℓℓ!∑m≥0Tm(x|1,q)tmm!=∑n≥0∑i=0N∑ℓ=0nnℓi1−q−Nℓ(1−q)ℓS1,q(N,i)xjTn−ℓ(x|1,q)tnn!,

where (c)ℓ={c|−1}ℓ. By comparing the coefficients of tnn!, we obtain the following result.

Corollary 2.4

For allN ≥ 1,

Tn+N(x|1,q)=∑i=0N∑ℓ=0nnℓi1−q−Nℓ(1−q)ℓS1,q(N,i)xjTn−ℓ(x|1,q).

Acknowledgement

This research was supported by Kyungpook National University Bokhyeon Research Fund, 2015. Taekyun Kim was appointed as a chair professor at Tianjin polytechnic University by Tianjin city in China from Aug 2015 to Aug 2019. The second author was supported by the Ministry of Science and Technology, Israel.

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Received: 2016-7-20

Accepted: 2016-10-3

Published Online: 2016-11-11

Published in Print: 2016-1-1

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