Implicit Divided Differences, Little Schröder Numbers, and Catalan Numbers

Georg Muntingh Centre of Mathematics for Applications Department of Mathematics University of Oslo P.O. Box 1053, Blindern N-0316, Oslo Norway

                      georgmu@math.uio.no

Abstract

Under general conditions, the equation g(x, y) = 0 implicitly defines y locally as a function of x. In this short note we study the combinatorial structure underlying a recently discovered formula for the divided differences of y expressed in terms of bivariate divided differences of g, by analyzing the number of terms a_{n} in this formula. The main result describes six equivalent characterizations of the sequence {a_{n}}.

1 Introduction

The act of taking differences and dividing is a recursive process with a wide range of applications in mathematics. In approximation theory, such divided differences can be viewed as a discrete analogue of derivatives; see the survey by de Boor [2]. Generalizations of univariate divided differences play a role in the theory of orthogonal polynomials, Macdonald polynomials in particular, and are used to define recurrences between Schubert polynomials in the algebraic geometry of flag varieties [11].

Although a classical topic dating back to Newton, interest in divided differences seems to have been revived recently. One point of view defines multivariate divided differences as the leading term of an interpolating polynomial and attempts to carry over properties from the univariate case [15, 17]. In addition, several formulas from (differential) calculus have been generalized to a ‘divided difference calculus’, including several new chain rules [6, 7, 21] akin to Faà di Bruno’s formula, and expressions for divided differences of inverse functions [8] and implicit functions [12, 14].

It is the goal of this paper to study the combinatorial structure underlying a formula by Muntingh and Floater [12], restated below as (5), for the divided differences of a function y implicitly defined by a relation g. More precisely, for some open intervals U, V ⊂ ℝ, let y : U ⟶ V be a function that is implicitly defined by a function g : U × V ⟶ ℝ via

g(x, y(x)) = 0,  \frac{∂g}{∂y} (x, y(x)) ≠ 0 ∀ x ∈ U.  (1)

Using a chain rule for divided differences, this relation induces relations between the divided differences of y and those of g and in particular yields (5). In this paper we study the number of terms a_{n} in this formula.

In the following section, we recall the definition of a divided difference, a recurrence relation for the divided differences of y, and the explicit formula (5). Next, in Section 3, we first derive a generating function from the recurrence relation and then prove six equivalent characterizations of the sequence {a_{n}}. We end with an analysis of the asymptotic behaviour of {a_{n}}.

2 Divided Differences

Let [x_{0}, …, x_{n}]f denote the divided difference [of order n] of a function f : (a, b) ⟶ ℝ at the distinct points x_{0}, …, x_{n} ∈ (a, b), which is recursively defined by [x_{0}]f := f(x_{0}) and

[x_{0}, …, x_{n}]f = \frac{[x_{1}, …, x_{n}]f − [x_{0}, …, x_{n−1}]f}{x_{n} − x_{0}} if n > 0.

For given indices i_{0}, i_{1}, …, i_{k} satisfying i_{0} < i_{1} < ⋯ < i_{k}, we shall shorten notation to [i_{0}i_{1} ⋯ i_{k}]f := [x_{i_{0}}, x_{i_{1}}, … x_{i_{k}}]f.

The above definitions generalize to bivariate divided differences as follows. This time, let f : U ⟶ ℝ be defined on some rectangle

U = (a_{1}, b_{1}) × (a_{2}, b_{2}) ⊂ ℝ^{2}.

Suppose we are given integers m, n ≥ 0, distinct points x_{0}, …, x_{m} ∈ (a_{1}, b_{1}), and distinct points y_{0}, …, y_{m} ∈ (a_{2}, b_{2}). The Cartesian product

{x_{0}, …, x_{m}} × {y_{0}, …, y_{n}}

defines a rectangular grid of points in U. The [bivariate] divided difference of f at this grid, denoted by

[x_{0}, …, x_{m}; y_{0}, …, y_{n}]f,  (2)

can be defined recursively as follows. If m = n = 0, the grid consists of only one point (x_{0}, y_{0}), and we define [x_{0}; y_{0}]f := f(x_{0}, y_{0}) as the value of f at this point. In case m > 0, we can define (2) as

\frac{[x_{1}, …, x_{m}; y_{0}, …, y_{n}]f − [x_{0}, …, x_{m−1}; y_{0}, …, y_{n}]f}{x_{m} − x_{0}} ,

Figure 1: The dissections of a convex polygon with vertices labeled 0, 1, 2, 3, 4. The bottomright dissection has faces (0, 1, 4), (1, 2, 4), and (2, 3, 4).

or if n > 0, as

\frac{[x_{0}, …, x_{m}; y_{1}, …, y_{n}]f − [x_{0}, …, x_{m}; y_{0}, …, y_{n−1}]f}{y_{n} − y_{0}} .

If both m > 0 and n > 0 the divided difference (2) is uniquely defined by either recurrence relation. Similarly to the univariate case, we shorten the notation for bivariate divided differences to

[i_{0}i_{1} ⋯ i_{s}; j_{0}j_{1} ⋯ j_{t}]f := [x_{i_{0}}, x_{i_{1}}, …, x_{i_{s}}; y_{j_{0}}, y_{j_{1}}, …, y_{j_{t}}]f.

In what follows, assume that y and g are related by (1). Applying a chain rule for divided differences, one can derive [12]

[01]y = −\frac{[01; 1]g}{[0; 01]g} ,  (3)

and for any n ≥ 2 the recurrence relation

[01 ⋯ n]y = − ∑_{k=2}^{n} ∑_{0=i_{0}<···<i_{k}=n} ∑_{\begin{matrix}s=0 \\ s=i_{s}−i_{0} \end{matrix}}^{k} \frac{[01 ⋯ s; i_{s}i_{s+1} ⋯ i_{k}]g}{[0; 0n]g} ∏_{l=s+1}^{k}[i_{l−1}(i_{l−1} + 1) ⋯ i_{l}]y. (4)

The third summation might look a bit mysterious. It is a concise way of describing all s for which the increasing sequence 0 = i_{0} < i_{1} < ⋯ < i_{k} = n starts with at least s steps of 1, i.e., i_{1} − i_{0} = ⋯ = i_{s} − i_{s−1} = 1.

For n ≥ 2, this recurrence relation was “solved” to yield the explicit formula (5). In order to describe this formula, we introduce some notation for dissections of a convex polygon. With a sequence of labels 0, 1, …, n we associate the ordered vertices of a convex polygon. A dissection of a convex polygon is the result of connecting any pairs of nonadjacent vertices with straight line segments, none of which intersect. We denote the set of all such dissections of a polygon with vertices 0, 1, …, n by P(0, 1, …, n). The vertices 0, 1, …, n and line segments (between either adjacent or nonadjacent vertices) form the vertices and edges of a plane graph. As such, every dissection π ∈ P(0, 1, …, n) is described by its set F (π)

\begin{aligned}n & : & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ a_{n} & : & 1 & 1 & 3 & 13 & 71 & 441 & 2955 & 20805 & 151695 & 1135345 & 8671763 & 67320573 \end{aligned}

Table 1: The number of terms a_{n} in the right hand side of Equation (5) for small values of n, with the additional convention a_{0} = a_{1} = 1.

of [oriented] faces, which does not include the unbounded face. Each face f ∈ F (π) is represented by a subsequence f = (i_{0}, i_{1}, …, i_{k}) of the sequence (0, 1, …, n) of length at least three.

See Figure 1 for the 11 dissections of a convex polygon with vertices 0, 1, 2, 3, 4. For general n, this number is given by the little Schröder number [19] (or super-Catalan number)

\frac{1}{n − 1} ∑_{k=1}^{n−1} \binom{n − 1}{k}\binom{n − 1 + k}{k − 1},

which form sequence \underline{A001003} in The On-Line Encyclopedia for Integer Sequences (OEIS) [18]. The main result of [12] is the formula

[01 ⋯ n]y = ∑_{π∈P(0,...,n)} ∏_{(i_{0},...,i_{k})∈F (π)} ∑_{\begin{matrix}s=0 \\ s=i_{s}−i_{0} \end{matrix}}^{k} −\frac{[i_{0} ⋯ i_{s}; i_{s} ⋯ i_{k}]g}{[i_{0}; i_{0}i_{k}]g} ∏_{\begin{matrix}l=s+1 \\ i_{l}−i_{l−1}=1 \end{matrix}}^{k}(−\frac{[i_{l−1}i_{l}; i_{l}]g}{[i_{l−1}; i_{l−1}i_{l}]g} ) , (5)

which expresses divided differences of y solely in terms of bivariate divided differences of g. In a follow-up paper, this formula was generalized to a formula for divided differences of multivariate implicit functions [13, 14].

3 The Number of Terms

For any implicit divided differences [01 ⋯ n]y of any order n ≥ 2, we wish to analyze the number of terms a_{n} in the right hand side of Equation (5). Equation (3) yields the natural extension a_{1} = 1. With the additional convention a_{0} = 1, we can collect these integers into a sequence {a_{n}}. The sequences in this paper will always start at n = 0, unless stated otherwise. Alternatively, we can encode the integers a_{n} by an [ordinary] generating function, which is the formal power series ∑_{n=0}^{∞} a_{n}x^{n}. Table 1 lists a_{n} for small values of n, and more terms can be found as sequence \underline{A162326} in The OEIS.

To analyze {a_{n}}, we rewrite (4) in terms of the more standard notion of a composition. Let ℕ = {1, 2, …} denote the set of positive integers. A composition [of n of length k] is any sequence (j_{1}, …, j_{k}) ∈ ℕ^{k} satisfying j_{1} + ⋯ + j_{k} = n. The positive integers j_{1}, …, j_{k} are called the parts of the composition. Clearly these correspond to increasing sequences 0 = i_{0} < ⋯ < i_{k} = n via j_{l} = i_{l} − i_{l−1} for l = 1, …, k. Thus (4) is equivalent to

[01 ⋯ n]y = − ∑_{k=2}^{n} ∑_{compositions j_{1}+···+j_{k}=n} ∑_{\begin{matrix}s=0 \\ s=j_{1}+···+j_{s} \end{matrix}}^{k}  (6)

\frac{[01 ⋯ s; (s + j_{s+1}) ⋯ n]g}{[0; 0n]g} ∏_{l=s+1}^{k} [(j_{1} + ⋯ + j_{l−1}) ⋯ (j_{1} + ⋯ + j_{l})]y.

Proposition 1. The sequence {a_{n}} has associated generating function

G(x) := ∑_{n=0}^{∞} a_{n}x^{n} = \frac{5}{4} − \frac{1}{4}\sqrt{\frac{1 − 9x}{1 − x}}.  (7)

Proof. Replacing, in Equation (6), the divided differences of y of order n by the number of terms a_{n} yields

a_{n} = ∑_{k=2}^{n} ∑_{compositions j_{1}+···+j_{k}=n} ∑_{\begin{matrix}s=0 \\ s=j_{1}+···+j_{s} \end{matrix}}^{k} ∏_{l=1}^{k} a_{j_{l}}, n ≥ 2.

The third sum makes sure that any composition starting with precisely s ones will be counted s + 1 times. We can, therefore, replace this sum by a sum over the number t of initial parts in the composition that are one. Using that a_{1} = 1, we find

a_{n} = 1 + ∑_{k=2}^{n} ∑_{t=0}^{k−1} ∑_{compositions j_{1}+···+j_{k−t}=n−t} ∏_{l=1}^{k−t} a_{j_{l}},  n ≥ 2,

where the first term, 1, is included to make sure that the composition n = 1 + 1 + ⋯ + 1 is counted n + 1 times. The generating function G will therefore satisfy

G = \frac{1}{1 − x} + ∑_{k=2}^{∞} ∑_{t=0}^{k−1} x^{t}(G − 1)^{k−t}

= \frac{1}{1 − x} + (G − 1) ∑_{k=1}^{∞} ∑_{t=0}^{k} x^{t}(G − 1)^{k−t}

= \frac{1}{1 − x} + (G − 1) [\frac{1}{1 − x} \frac{1}{2 − G} − 1] .

Multiplying by (1 − x)(2 − G) and rewriting gives

2G^{2} − 5G + 2 + \frac{1}{1 − x} = 0.  (8)

Since G(0) = 1, we find that G is as in the Proposition.

Up to a constant and a rescaling, the generating function G(x) is exactly the magnetization M_{∞}(s) for the XYZ spin chain along a special line of couplings [4].

Before we state the next proposition, we need to recall some standard notions. The Gaussian hypergeometric series [with parameters a, b, c] is defined by

2F_{1}(a, b; c; z) = ∑_{n=0}^{∞} \frac{(a)_{n}(b)_{n}}{(c)_{n}} \frac{z^{n}}{n!} ,

where the Pochhammer symbol (·)_{n} is defined by

(a)_{n} = \{ \begin{matrix}1, & if n = 0; \\ a(a + 1) ⋯ (a + n − 1), & if n > 0. \end{matrix}

Hypergeometric series with adjacent parameter values are related [1, Section 15.2.11] by Gauss’ contiguous relation

(c − b)_{2}F_{1}(a, b − 1; c; z) + (2b − c + (a − b)z)_{2}F_{1}(a, b; c; z) + b(z − 1)_{2}F_{1}(a, b + 1; c; z) = 0. (9)

Moreover, as explained by Roy [16], sums of products of binomial coefficients can often be reduced to hypergeometric series, with the help of elementary identities like

(a)_{2k} = 2^{2k} (\frac{a}{2})_{k} (\frac{a + 1}{2})_{k} ,  \frac{n!}{(n − k)!} = (−1)^{k}(−n)_{k}.  (10)

Given a sequence {a_{n}}, define its binomial transform {b_{n}} by

b_{n} = ∑_{k=0}^{n} \binom{n}{k}a_{k}, n ≥ 0.

It is well known that if A(x) is the generating function of an integer sequence {a_{n}}, then

\frac{1}{1 − x}A (\frac{x}{1 − x})

is the generating function of its binomial transform. Although sometimes proved in the case that A(x) is analytic [3, Appendix], the statement holds for any formal power series A(x) [9].

For any n ≥ 0, let

C_{n} := \frac{1}{n + 1}\binom{2n}{n}

denote the n-th Catalan number, which form sequence \underline{A000108} in The OEIS. It is well known [10] that the sequence {C_{n}} has generating function

F (x) := \frac{1 − \sqrt{1 − 4x}}{2x}.  (11)

By means of the binomial transform, we shall relate {a_{n}} to {2^{n}C_{n}}, which form sequence \underline{A151374} in The OEIS.

Proposition 2. With the convention a_{0} = a_{1} = 1, the following statements are equivalent and hold.

(a) The number of terms in the r.h.s. of (5) is a_{n} for any n ≥ 2;

(b) The sequence {a_{n}} has generating function given by (7); (c) The integers a_{n} satisfy the quadratic recurrence relation

a_{n} = 1 + 2 ∑_{m=1}^{n−1} a_{m}a_{n−m}, n ≥ 2;

(d) The integers a_{n} satisfy the linear recurrence relation

na_{n} = (−14 + 10n)a_{n−1} + (18 − 9n)a_{n−2}, n ≥ 2;

(e) One has a_{n} = _{2}F_{1} (\frac{1}{2}, 1 − n; 2; −8) for any n ≥ 2;

(f) The sequence {a_{n+1}}_{n≥0} is the binomial transform of {2^{n}C_{n}}_{n≥0}.

Proof. Clearly (a) holds by definition of the {a_{n}}, and (a) ⟺ (b) by Proposition 1.

(b) ⟺ (c): Given that a_{0} = 1 and a_{1} = 1, the sequence {a_{n}} has generating function G given by (7) if and only if (8) holds, i.e., if and only if

0 = 2 (∑_{n=0}^{∞} a_{n}x^{n})^{2} − 5 ∑_{n=0}^{∞} a_{n}x^{n} + 2 + ∑_{n=0}^{∞} x^{n}

= ∑_{n=0}^{∞} (1 − 5a_{n} + 2 ∑_{m=0}^{n} a_{m}a_{n−m}) x^{n} + 2

= ∑_{n=2}^{∞} (1 − a_{n} + 2 ∑_{m=1}^{n−1} a_{m}a_{n−m}) x^{n},

which happens precisely when {a_{n}} satisfies (c).

(b) ⟹ (d): In the differential ring of formal power series ℤ[[x]] with derivation \frac{d}{dx}, the generating function G satisfies the differential equation

0 = (1 − 10x + 9x^{2})G′ + 4G − 5,

which can be written in terms of the {a_{n}} as

0 = (−5 + 4a_{0} + a_{1}) · 1 + (2a_{2} − 6a_{1}) · x

+ ∑_{n=2}^{∞} ((n + 1)a_{n+1} + (4 − 10n)a_{n} + 9(n − 1)a_{n−1}) · x^{n}.

Comparing coefficients yields (d).

(d) ⟹ (e): Substituting a = \frac{1}{2}, b = 2 − n, c = 2, z = −8 in Equation (9), one finds that 2F_{1} (\frac{1}{2}, 1 − n; 2; −8) satisfies the same recurrence relation as a_{n}, and must therefore be equal to a_{n}.

(e) ⟹ (f): The binomial transform {b_{n}} of {2^{n}C_{n}} is given by

b_{n} = ∑_{k=0}^{n} \binom{n}{k}2^{k} \frac{1}{k + 1}\binom{2k}{k}, n ≥ 0.

Using the identities (10) with a = 1, one finds

b_{n} = ∑_{k=0}^{n} \frac{(2k)!}{k!} · \frac{n!}{(n − k)!} · \frac{1}{(k + 1)!} · \frac{2^{k}}{k!}

= ∑_{k=0}^{n} \frac{2^{2k}(\frac{1}{2})_{k}(1)_{k}}{(1)_{k}} · (−1)^{k}(−n)_{k} · \frac{1}{(2)_{k}} · \frac{2^{k}}{k!}

= ∑_{k=0}^{n} \frac{(\frac{1}{2})_{k}(−n)_{k}}{(2)_{k}} \frac{(−8)^{k}}{k!}

= _{2}F_{1} (\frac{1}{2}, −n; 2; −8)

= a_{n+1}

for any n ≥ 0.

(f) ⟹ (b): Since {C_{n}} has generating function F (x) given by (11), the sequence {2^{n}C_{n}} has generating function

H(x) := F (2x) = \frac{1 − \sqrt{1 − 8x}}{4x},

and its binomial transform {a_{n+1}}_{n≥0} has generating function

∑_{n=0}^{∞} a_{n+1}x^{n} = \frac{1}{1 − x}H (\frac{x}{1 − x}) = \frac{1 − \sqrt{1 − \frac{8x}{1−x}}}{4x} = \frac{1 − \sqrt{\frac{1−9x}{1−x}}}{4x} .

We conclude that {a_{n}} has generating function

∑_{n=0}^{∞} a_{n}x^{n} = 1 + x (\frac{1 − \sqrt{\frac{1−9x}{1−x}}}{4x} ) = \frac{5}{4} − \frac{1}{4}\sqrt{\frac{1 − 9x}{1 − x}} = G(x).

We end this paper with a remark on the asymptotic behaviour of the sequence {a_{n}}. Flajolet and Sedgewick [5] describe how the asymptotic properties of a sequence can be deduced from the singularities of its generating function, i.e., points where the function ceases to be analytic. The location ρ of the singularity closest to the origin dictates an exponential growth ρ^{−n}, while the singularity type determines a subexponential growth factor C(n). Using the dictionary in Figure VI.5 in [5], the expansion

G(x) = \frac{5}{4} − \frac{3}{8\sqrt{2}}\sqrt{1 − 9x} + O ((x − \frac{1}{9})^{3/2})

around the singularity at ρ = \frac{1}{9} yields the asymptotic formula

C(n) · 9^{n},  C(n) = \frac{3}{16\sqrt{2π}} n^{−3/2} + O(n^{−5/2}),  (12)

Figure 2: The relative error 1 − C(n) · 9^{n}/a_{n} plotted against n.

for the coefficients a_{n}. Using Proposition 2(d) to quickly calculate the first 1000 entries in {a_{n}}, we compare (12) to a_{n} by plotting the relative error

1 − \frac{C(n) · 9^{n}}{a_{n}}

in Figure 2.

4 Acknowledgments

I am grateful to Paul Kettler, for kindly providing several useful comments on a draft of this paper, and to Paul Fendley and Emeric Deutsch, for conjecturing the generating function G(x) to me in a personal communication.

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2010 Mathematics Subject Classification: Primary 05A15; Secondary 26A24.

Keywords: divided difference, implicit function, polygon dissection, generating function, hypergeometric series, binomial transform.

(Concerned with sequences \underline{A000108}, \underline{A001003}, \underline{A151374}, and \underline{A162326}.)

Received April 12 2012; revised version received June 12 2012. Published in Journal of Integer Sequences, June 26 2012.

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