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Abstract
In 2002, Eu, Liu and Yeh introduced new Taylor expansions of the generating function of Catalan and Motzkin numbers. And they presented that this Taylor style expansion can be applied to more generating functions satisfying some relations (Advances in Applied Mathematics, 29 (2002) 345–357). In this paper, we focus on this Taylor expansion of the generating function of Catalan-like numbers, which are common generalizations of many classic counting coefficients, such as the Catalan numbers, the Motzkin numbers and the Schröder numbers. We present the recurrence relations of the coefficients and bivariate generating functions of the remainders of the new Taylor expansion of the generating function of Catalan-like numbers.
Public Interest Statement
Eu, Liu and Yeh introduced new Taylor expansions, whose remainders involve the function itself, of the generating function of Catalan and Motzkin numbers. In this paper, we focus on this Taylor expansion of the generating function of Catalan-like numbers, which are the 0th column of special cases of Riordan arrays. Riordan arrays play an important unifying role in enumerative combinatorics. There have been quite a few papers concerned with combinatorics of Riordan arrays. Our concern is one class of special interesting Riordan arrays—the recursive matrix introduced by Aigner. The 0th column of such recursive matrix includes many classical combinatorial sequences, such as the Catalan numbers, the Motzkin numbers, the large and little Schröder numbers. We present the recurrence relations of the coefficients and bivariate generating functions of the remainders of the new Taylor expansion of the generating function of Catalan-like numbers.
1. Introduction
The Taylor expansion of a real function f(x), which is infinitely differential, at the real number 0 is(1)
(1)
where is the nth remainder, which is the error incurred in approximating the function f(x) by its
th Taylor polynomial
.
Traditionally, the remainders of Taylor expansions play a central role in the theory of functions, numberical approximations, asymptotic expansions, etc. They are used mainly for quantitative or numerical purposes (Eu, Liu, & Yeh, Citation2002). Unlike the usual Taylor expansions, the remainders in Taylor expansions considered in this paper involve the generating function itself. Such expansions are quite different from the usual binomial expansions or continued fraction expansions but are not exceptions for combinatorial structures. Eu et al. (Citation2002) introduced these new Taylor expansions for the Catalan numbers and the Motzkin numbers
. They showed that
and
where the and
are recursively defined polynomials. This new Taylor expansions can be generalized to the Catalan-like numbers (Eu et al., Citation2002).
The Catalan-like numbers considered in this paper are the 0th columns of special cases of Riordan arrays. Riordan arrays play an important unifying role in enumerative combinatorics (Shapiro, Getu, Woan, & Woodson, Citation1991). A (proper) Riordan array, denoted by (g(x), f(x)), is an infinite lower triangular matrix whose generating function of the kth column is for
where
and
. A Riordan array
can also be characterized by two sequences
and
such that
(2)
(2)
for (see Cheon, Kim, & Shapiro, Citation2012; He & Sprugnoli, Citation2009 for instance). Call
and
the A- and Z-sequences of R, respectively. The 0th column of such a Riordan array includes many classical combinatorial sequences, such as the Catalan numbers, the Motzkin numbers, the large and the little Schröder numbers. There have been quite a few papers concerned with combinatorics of Riordan arrays (see Cheon et al., Citation2012; Ehrenfeucht, Harju, ten Pas, & Rozenberg, Citation1998; He & Sprugnoli, Citation2009; Shapiro et al., Citation1991 for instance). Our concern in the present paper is one class of special interesting Riordan arrays—the recursive matrix introduced by Aigner (Citation1999,Citation2001). Let p, s, t be three nonnegative numbers. Denote by
, the Riordan array with
and
. More precisely,
(3)
(3)
Following Aigner (Citation1999,Citation2001), the matrix R(p; s, t) is called the recursive matrix, and the numbers are called the Catalan-like numbers corresponding to
, where
The Catalan-like numbers unify many famous counting coefficients. For example, the numbers are
(1) | the Catalan numbers | ||||
(2) | the Motzkin numbers | ||||
(3) | the large Schröder numbers | ||||
(4) | the little Schröder numbers | ||||
(5) | the restricted hexagonal numbers |
2. The Catalan-like numbers’ Taylor expansion
Let be the nth Catalan-like numbers. Denote by
the generating function of Catalan-like numbers. Then by the theory of Riordan array, we have
(4)
(4)
Let So C(x) satisfies the following recurrence relation
Also let .
Then we have
After arrangement successively, we have(5)
(5)
Then following Eu et al. (Citation2002), Taylor expansions for the generating function of Catalan-like numbers are(6)
(6)
where and
are polynomials, and
is the nth remainder.
So in order to get the uniqueness of and
, it suffices to prove that the equation
only has the trivial solution for polynomials a(y) and b(y). Note that
is a continuous function. For any y, there exist different , so that
. By inserting
into
, we can get
. As to the polynomial, a and b must be the constant zero.
Since(7)
(7)
we set
where
Now, we replace C with . So we have
Then we set(8)
(8)
where
We replace C with . And we have
(9)
(9)
So inserting (Equation 9) into (Equation 8), we get
Then we set
where
After arrangement successively, we can get the nth Taylor expansion for the generating function of Catalan-like numbers as follows (Eu et al., Citation2002).
where
and for
and
. Now we present the main result of this paper based on the method proposed by Eu et al. (Citation2002).
Theorem 1
Let
be the nth Taylor expansion for the generating fuction of Catalan-like numbers defined by Equation (3). Then we have the following.
(i) |
| ||||
(ii) |
| ||||
(iii) |
| ||||
(iv) |
|
Proof
It is trivial for . Let
Replacing each
in
th Taylor expansion of Equation (6) with
we get
Note that
Hence comparing coefficients of each term of the remainder on the above two equations, we can get(10)
(10)
where
(i) | If | ||||
(ii) | If | ||||
(iii) | If | ||||
(iv) | If |
In the proof of Theorem (1), we can get recurrence relations (Equation 10) of and
with
and
. So we can obtain a relation between
and
according to (iii).
Corollary 2
The two functions and
satisfy
for with
Proof
Since
and
we have
Corollary 3
The generating functions
have closed forms:
Proof
Since
and
we have(11)
(11)
where(12)
(12)
and(13)
(13)
Now inserting (Equation 12)–(Equation 14) into (Equation 11), we can get
Also since
and
we have(14)
(14)
where(15)
(15)
(16)
(16)
(17)
(17)
and(18)
(18)
Now inserting (Equation 16)–(Equation 19) into (Equation 15), we have(19)
(19)
Multiplying by y on both sides of (Equation 20) and then after arrangement, we have
So we get
3. Remarks
In the paper, we study the remainders of new Taylor expansions for the generating function of Catalan-like numbers. Since the Catalan-like numbers, such as the Motzkin numbers (Aigner, Citation1998; Donaghey & Shapiro, Citation1977; Sulanke, Citation2001), the Catalan numbers (Aigner, Citation2001; He, Citation2013; Mahmoud & Qi, Citation2016), the large and little Schröder numbers (Ehrenfeucht et al., Citation1998; Qi, Shi, & Guo, Citation2016a,Citation2016b; Stanley, Citation1997) naturally appear in the combinatorial objects, their Taylor expansions can be interpreted in distinct combinatorial ways.
Acknowledgements
The authors thank the anonymous referees for their constructive comments and helpful suggestions which have greatly improved the original manuscript.
Additional information
Funding
Notes on contributors
Lily Li Liu
Lily Li Liu is working as an associate professor in School of Mathematical Science at Qufu Normal University in China. She is pursuing her PhD degree in Mathematics from Dalian University of Technology in 2009. She has published several research papers in national and international journals. Her area of interest is inequalities, the unimodality property and the properties of combinatorial sequences.
Xiaoli Li
Xiaoli Li is a graduate student of the first author in School of Mathematical Science at Qufu Normal University in China. Her research interests are the properties of combinatorial sequences.
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