A Note on the Hankel Transform of the Central Binomial Coefficients

Mario Garcia Armas Facultad de Matemática y Computación Universidad de La Habana Havana, Cuba

                 marioga@matcom.uh.cu

B. A. Sethuraman Dept. of Mathematics California State University, Northridge Northridge, CA 91330 USA

                al.sethuraman@csun.edu

Abstract

We show that the n × n Hankel matrix formed from the successive even central binomial coefficients \binom{2l}{l}, l = 0, 1, … arises naturally when considering the trace form in the number ring of the maximal real subfield of suitable cyclotomic fields. By considering the trace form in two different integral bases of the number ring we get a factorization of this matrix which immediately yields the well-known zeroth and first Hankel transforms of the sequence.

1 Introduction

Given a sequence a_{l}, l = 0, 1, …, the n × n Hankel matrix H_{n}^{(k)}, k = 0, 1, …, formed from this sequence is the matrix

( \begin{matrix}a_{k} & a_{k+1} & a_{k+2} & … & a_{k+n−1} \\ a_{k+1} & a_{k+2} & a_{k+3} & … & a_{k+n} \\ a_{k+2} & a_{k+3} & a_{k+4} & … & a_{k+n+1} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ a_{k+n−1} & a_{k+n} & a_{k+n+1} & … & a_{k+2n−2} \end{matrix} )

The k-th Hankel Transform of the sequence a_{l}, l = 0, 1, … is the sequence of determinants d_{n}^{(k)} of the matrices H_{n}^{(k)} above, n = 1, 2, . . .. It is worth mentioning that some authors refer to the Hankel transform only as the sequence d_{n} = det H_{n}^{(0)} (see, for example, [7]).

Hankel matrices have been studied extensively, and connections between Hankel matrices and other areas of mathematics are well known (see [13] for a very nice survey of Hankel matrices especially in relation to combinatorics and coding theory). The term Hankel transform was introduced in Sloane’s sequence \underline{A055878} and first studied in [7]. Since then, there have been numerous further studies of Hankel transforms of sequences, for instance, [1, 3, 4, 5, 12].

Consider the particular sequence a_{l} = \binom{2l}{l}, l = 0, 1, …, and denote H_{n}^{(0)} by simply H_{n}. The main purpose of this paper is to show that the matrices H_{n} := H_{n}^{(0)} and H_{n}^{(1)} that define the zeroth and first Hankel transforms of this particular sequence arise very naturally when considering the trace form Tr(x, y) = Tr_{K/ℚ}(xy) on the number ring O_{K}, where K is the maximal real subfield of the 2^{N}-th cyclotomic field, for any N such that 2^{N} ≥ 8n. By considering the same trace form with respect to two different integral bases of O_{K}, we obtain in a very natural way a factorization of H_{n} as

H_{n} = B_{n}D_{n}B_{n}^{T}  (1)

where

B_{n} = ( \begin{matrix}1 & 0 & 0 & … & 0 \\ \binom{2}{1} & 1 & 0 & … & 0 \\ \binom{4}{2} & \binom{4}{1} & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ \binom{2(n−1)}{n−1} & \binom{2(n−1)}{n−2} & \binom{2(n−1)}{n−3} & … & 1 \end{matrix} )  (2)

and

D_{n} = ( \begin{matrix}1 & 0 & 0 & … & 0 \\ 0 & 2 & 0 & … & 0 \\ 0 & 0 & 2 & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & … & 2 \end{matrix} )  (3)

Similarly, we get the factorization

H_{n}^{(1)} = 2C_{n}C_{n}^{T}  (4)

where

C_{n} = ( \begin{matrix}1 & 0 & 0 & … & 0 \\ \binom{3}{1} & 1 & 0 & … & 0 \\ \binom{5}{2} & \binom{5}{1} & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ \binom{2n−1}{n−1} & \binom{2n−1}{n−2} & \binom{2n−1}{n−3} & … & 1 \end{matrix} )  (5)

These factorizations yield at once the following well-known result (see, for example, [1, 10, 11]):

Corollary 1. The zeroth Hankel transform d_{n}^{(0)} of the sequence \binom{2l}{l}, l = 0, 1, … is the sequence 2^{n−1}, n = 1, 2, …, and the first Hankel transform d_{n}^{(1)} is the sequence 2^{n}, n = 1, 2 . . ..

Of course, once the factorization has been guessed at, Eq. (1) and Eq. (4) can be proved by elementary means: our point here is only to show that the Hankel matrices H_{n} and H_{n}^{(1)} and the factorizations above arise completely naturally in number theory.

2 Trace Calculations in O_{K}

Given a positive integer N we write ω for the primitive 2^{N}-th root of unity e^{2πı/2^{N}}. We write θ for the element ω + ω^{−1}. We write θ_{j} (j = 0, 1, …,) for the element ω^{j} + ω^{−j}, so that θ_{1} = θ and θ_{0} = 2. We write L for the field ℚ(ω), and K for the real subfield ℚ(θ). Note that [L : ℚ] = 2^{N−1} and [K : ℚ] = 2^{N−2}. We write m for 2^{N−2}.

We begin by computing traces of the elements θ^{i} as well as of products θ_{i}θ_{j}. (Lemmas 2 and 3 also appear in [9], and are implicit in [2, Prop. 4.3].)

Lemma 2. For 1 ≤ s < 2 · m,

Tr_{K/ℚ}(θ^{s}) = \begin{cases}0, & if s is odd; \\ m\binom{s}{s/2}, & if s is even. \end{cases}

Proof. Observe that Tr_{K/ℚ}(θ^{s}) = Tr_{L/ℚ(ı)}(θ^{s}). Now expanding θ^{s} = (ω + ω^{−1})^{s} binomially, we find

θ^{s} = ∑_{j=0}^{s} \binom{s}{j}ω^{j}ω^{−(s−j)} = ∑_{j=0}^{s} \binom{s}{j}ω^{2j−s}

Notice that when s is odd, only odd powers of ω appear in this expansion. Since ω raised to any odd power is also a primitive 2^{N}-th root of unity, it has minimal polynomial x^{m} ± ı over ℚ(ı), and consequently, any odd power of ω has trace zero from L to ℚ(ı). It follows that Tr_{L/ℚ(ı)}(θ^{s}) = 0 when s is odd. (Notice that this is true for all odd s and not just those in the range of the statement of the lemma.)

When s is even, we first assume that s < m. Then, the terms in the expansion of θ^{s} above have even powers of ω that run through s, s − 2, …, 2, 0, −2, …, −(s − 2), −s. Given any nonzero even integer 2l in this set, we write it as 2^{e}a for some e and odd integer a. Then ω^{2l} is a primitive 2^{N−e}-root of unity, and [L : ℚ(ω^{2l})] = 2^{e}. Since, by assumption, e < N −2, ℚ(ω^{2l}) strictly contains ℚ(ı). Now, Tr_{L/ℚ(ı)}(ω^{2l}) = Tr_{ℚ(ω^{2l})/ℚ(ı)}Tr_{L/ℚ(ω^{2l})}(ω^{2l}) = 2^{e}Tr_{ℚ(ω^{2l})/ℚ(ı)}(ω^{2l}). Just as in the previous paragraph, Tr_{ℚ(ω^{2l})/ℚ(ı)}(ω^{2l}) is zero since the minimal polynomial of ω^{2l} is x^{2^{N−e−2}} ± ı. Hence, all nonzero powers of ω contribute nothing to the trace, so Tr_{L/ℚ(ı)}(θ^{s}) is m times the coefficient of the term ω^{0} which is \binom{s}{s/2}.

When 2 · m > s ≥ m, we need a small modification. The expansion of θ^{s} will have only even powers of ω as before, but continuing to write these powers as 2l, we will now also have powers where 2 · m > 2l ≥ m. We first consider the powers 2l > m: we factor ω^{m} out to find ω^{2l} = ıω^{2l−m}. Thus, Tr_{L/ℚ(ı)}(ω^{2l}) = ıTr_{L/ℚ(ı)}(ω^{2l−m}). From our assumptions we find that 2l − m is a positive even integer and that m > 2l − m, so the arguments of the previous paragraph show that this trace is zero. Thus, we are left with the terms ω^{m}, ω^{0}, and ω^{−m}. But Tr_{L/ℚ(ı)}(ω^{m}) = Tr_{L/ℚ(ı)}(ı) = mı, while Tr_{L/ℚ(ı)}(ω^{−m}) = Tr_{L/ℚ(ı)}(−ı) = −mı, so these two terms cancel each other out. Once again, we are left with the term ω^{0} whose trace is m\binom{s}{s/2}.

Lemma 3. For 1 ≤ j < 2m,

Tr_{K/ℚ}(θ_{j}) = 0  (6)

and for 1 ≤ i, j < m

Tr_{K/ℚ}(θ_{i}θ_{j}) = \begin{cases}0, & if i ≠ j; \\ 2m, & if i = j. \end{cases}  (7)

Proof. The proof of the first part is essentially contained in the proof of Lemma 2 above. We have Tr_{K/ℚ}(θ_{j}) = Tr_{L/ℚ(ı)}(θ_{j}) = Tr_{L/ℚ(ı)}(ω^{j} +ω^{−j}). We saw in that proof that Tr_{L/ℚ(ı)}(ω^{j}) = 0 for all 1 ≤ j < 2m except when j = m, so Tr_{L/ℚ(ı)}(θ_{j}) = 0 for all such j. When j = m, we have Tr_{L/ℚ(ı)}(ω^{m}) = mı and Tr_{L/ℚ(ı)}(ω^{−m}) = −mı. Hence Tr_{L/ℚ(ı)}(θ_{m}) = 0 as well.

For the second assertion, note that θ_{i}θ_{j} = θ_{i+j} +θ_{j−i} where we can assume without loss of generality that j − i ≥ 0. The result immediately follows from the calculations of Tr_{K/ℚ}(θ_{j}) above, noting that i + j < 2m, and θ_{0} = 2.

Note that O_{K} = ℤ[θ] (see [8, Exer. 35, Chap. 2] for instance). Expanding each power θ^{s} binomially and collecting terms we find

θ^{s} = \begin{cases}∑_{j=0}^{⌊s/2⌋} \binom{s}{j}θ_{s−2j}, & if s is odd; \\ ∑_{j=0}^{(s/2)−1} \binom{s}{j}θ_{s−2j} + \binom{s}{s/2}, & if s is even. \end{cases}  (8)

For any positive integer n, let B_{n} be as in (2), and C_{n} as in (5). Let

V_{e} = (1, θ^{2}, θ^{4}, …, θ^{m−2})^{T}
V_{o} = (θ, θ^{3}, θ^{5}, …, θ^{m−1})^{T}
W_{e} = (1, θ_{2}, θ_{4}, …, θ_{m−2})^{T}
W_{o} = (θ_{1} = θ, θ_{3}, θ_{5}, …, θ_{m−1})^{T}

Then Eq. (8) splits as two matrix relations:

V_{e} = B_{m/2}W_{e};  (9)
V_{o} = C_{m/2}W_{o}.  (10)

Since 1, θ, θ^{2}, …, θ^{m−1} is a ℤ-basis for O_{K}, and since B_{m} and C_{m} are integer matrices with determinant 1, these relations show that 1, θ_{1} = θ, θ_{2}, …, θ_{m−1} is also a ℤ-basis for O_{K}. But these relations show us even more: if we define

M_{e} = ℤ ⊕ ℤθ^{2} ⊕ ℤθ^{4} ⊕ ⋯ ⊕ ℤθ^{m−2} and  (11)
M_{o} = ℤ ⊕ ℤθ ⊕ ℤθ^{3} ⊕ ⋯ ⊕ ℤθ^{m−1}  (12)

then 1, θ_{2}, θ_{4}, …, θ_{m−2} is also a ℤ basis for M_{e}, and θ, θ_{3}, …, θ_{m−1} is also a ℤ basis for M_{o}.

Hence, since Tr_{K/ℚ} is ℤ linear, for any x ∈ M_{e}, x = b_{0} + b_{2}θ^{2} + ⋯ + b_{m−2}θ^{m−2} and any y ∈ M_{e}, y = ∑ c_{0} + c_{2}θ^{2} + ⋯ + c_{m−2}θ^{m−2}, the value of Tr(x, y) = Tr_{K/ℚ}(xy) is determined by the values of Tr_{K/ℚ}(θ^{i}θ^{j}), i, j = 0, 2 …, m − 2. By a similar reasoning, writing x and y in terms of the basis 1, θ_{2}, θ_{4}, …, θ_{m−2}, the values of Tr(x, y) on M_{e} is also determined by the values of Tr_{K/ℚ}(θ_{i}θ_{j}), i, j = 0, 2 …, m − 2. Lemmas 2 and 3 immediately give us the following result which connects our Hankel matrix to the trace form:

Corollary 4. The matrix (Tr_{K/ℚ}(θ^{i}θ^{j})) (i, j = 0, 2 …, m − 2) equals m times the Hankel matrix H_{m/2} and (Tr_{K/ℚ}(θ_{i}θ_{j}) (i, j = 0, 2 …, m − 2) equals m times the matrix D_{m/2} defined in Equation (3).

Similarly, by considering the values of Tr(x, y) on the ℤ module M_{o} in the two bases θ, θ^{3}, …, θ^{m−1} and θ, θ_{3}, …, θ_{m−1}, we have the following:

Corollary 5. The matrix (Tr_{K/ℚ}(θ^{i}θ^{j})) (i, j = 1, 3 …, m − 1) equals m times the Hankel matrix H_{m/2}^{(1)} and (Tr_{K/ℚ}(θ_{i}θ_{j}) (i, j = 1, 3 …, m − 1) equals 2m times the identity matrix.

Now observe that the matrix (θ^{i}θ^{j}) (i, j = 0, 2 …, m − 2) is just V_{e} · V_{e}^{T} (a product of n × 1 and 1 × n matrices), and that (θ_{i}θ_{j}) (i, j = 0, 2 …, m − 2) equals W_{e} · W_{e}^{T}. Similarly, (θ^{i}θ^{j}) (i, j = 1, 3 …, m − 1) equals V_{o} · V_{o}^{T} and (θ_{i}θ_{j}) (i, j = 1, 3 …, m − 1) equals W_{e} · W_{e}^{T}.

Equations (9) and (10), the ℤ bilinearity of Tr_{K/ℚ}, and Corollaries 4 and 5 now give us the following (here, given a matrix M, Tr_{K/ℚ}(M) stands for the matrix whose entries are the traces of the entries of M):

H_{m/2} = \frac{1}{m}(Tr_{K/ℚ}(θ^{i}θ^{j}))_{i,j=0,2...,m−2} = \frac{1}{m}Tr_{K/ℚ}(V_{e} · V_{e}^{T})
= \frac{1}{m}B_{m/2}Tr_{K/ℚ}(W_{e} · W_{e}^{T})B_{m/2}^{T}
= \frac{1}{m}B_{m/2}(Tr_{K/ℚ}(θ_{i}θ_{j})_{i,j=0,2...,m−2}B_{m/2}^{T}
= B_{m/2}D_{m/2}B_{m/2}^{T}.
H_{m/2}^{(1)} = \frac{1}{m}(Tr_{K/ℚ}(θ^{i}θ^{j}))_{i,j=1,3...,m−1} = \frac{1}{m}Tr_{K/ℚ}(V_{o} · V_{o}^{T})
= \frac{1}{m}C_{m/2}Tr_{K/ℚ}(W_{o} · W_{o}^{T})C_{m/2}^{T}
= \frac{1}{m}C_{m/2}(Tr_{K/ℚ}(θ_{i}θ_{j})_{i,j=1,3...,m−1}C_{m/2}^{T}
= 2C_{m/2}C_{m/2}^{T}.

Let G_{n}^{(k)} denote the Hankel matrices formed from the sequence of odd central binomial coefficients \binom{2l+1}{l}, l = 0, 1, . . .. Note that since \binom{2n}{n} = 2\binom{2n−1}{n−1}, we have the relation H_{n}^{(k+1)} = 2G_{n}^{(k)}.

The discussions above now immediately yield the following theorem:

Theorem 6. For any n ≥ 1 and any k ≥ 1, we have the factorizations:

i. H_{n} = B_{n}D_{n}B_{n}^{T}, and

ii. H_{n}^{(1)} = 2C_{n}C_{n}^{T}.

iii. G_{n} := G_{n}^{(0)} = C_{n}C_{n}^{T}.

iv. H_{n}^{(k)} = B_{n+k,n}D_{n}B_{n}^{T}, where B_{n+k,n} denotes the lower left n × n block of B_{n+k}.

v. G_{n}^{(k)} = \frac{1}{2}B_{n+k+1,n}D_{n}B_{n}^{T},

vi. H_{n}^{(k)} = 2C_{n+k−1,n}C_{n}^{T}. (Of course, when k = 1, this is the same as Part (ii) above.)

vii. G_{n}^{(k)} = C_{n+k,n}C_{n}^{T} .

Proof. We pick an N such that 2^{N} ≥ 8n, and work in the maximal real subfield K of the 2^{N}-th cyclotomic extension of ℚ. The equations preceding the statement of the theorem yield the factorizations H_{m/2} = B_{m/2}D_{m/2}B_{m/2}^{T} and H_{m/2}^{(1)} = 2C_{m/2}C_{m/2}^{T}. By the choice of N,

we have n ≤ m/2. Note that H_{n}, H_{n}^{(1)}, B_{n}, C_{n} and D_{n} are all just the upper left n×n blocks of the corresponding matrices H_{m/2}, H_{m/2}^{(1)}, B_{m/2}, C_{m/2} and D_{m/2}. Studying the upper left n × n blocks of the products B_{m/2}D_{m/2}B_{m/2}^{T} and C_{m/2}C_{m/2}^{T}, and noting the lower triangular nature of B_{m/2} and C_{m/2} and the diagonal nature of D_{m/2}, we get the first two factorizations of the theorem.

We now substitute n + k for n throughout in the first two factorizations, and observe that H_{n}^{(k)} is the lower left n × n block of H_{n+k}, as also the lower left n × n block of H_{n+k−1}^{(1)}. Studying the products B_{n+k}D_{n+k}B_{n+k}^{T} and C_{n+k−1}C_{n+k−1}^{T} yields the two factorizations in (iv) and (vi) as well.

The factorizations in (iii), (v), and (vii) are a direct consequence of the relation H_{n}^{(k+1)} = 2G_{n}^{(k)}.

Note that Factorization (iii) of G_{n} was described in [1, Prop. 6] by showing that odd binomial coefficients could be regarded as Catalan-like numbers. (In the notation of [1, Prop. 6], the odd binomial coefficients are C_{n}^{(3,2)}, the matrix Ã_{n} is our G_{n+1} and the matrix A_{n} is our C_{n+1}.)

Taking the determinants on both sides of Parts (i) and (ii) of Theorem 6 above yeilds Corollary 1.

Let c_{n}^{(k)} denote the determinant of G_{n}^{(k)}, and note that the relation H_{n}^{(k+1)} = 2G_{n}^{(k)} shows that d_{n}^{(k+1)} = 2^{n}c_{n}^{(k)}. We therefore also have the following:

Corollary 7. (See [13, Eq. 1.5], also [1, Prop. 6].) c_{n}^{(0)} = 1.

Example 8. Parts (iv) or (vi) of Theorem 6 show that the computation of d_{n}^{(k)} for k ≥ 2 can be accomplished by computing the determinants of B_{n+k,n} or C_{n+k,n}. Since the primary goal of this note is to establish the connection between the H_{n}^{(k)} and number theory we will not do this here, but we note that these can be computed using, for instance, the very general techniques described in [6, Thm. 26] (as can the determinants of the original H_{n}^{(k)} themselves!). Computing these determinants shows that for k ≥ 2

d_{n}^{(k)} = 2^{n} ∏_{1≤i≤j≤k−1} \frac{i + j − 1 + 2n}{i + j − 1}  (13)

(recovering, for example, [13, Eq. 1.5]).

3 Acknowledgements

The second author would like to thank B.M. Ábrego for introducing him to the literature on Hankel transforms, and to Frédérique Oggier for some preliminary discussions and computations. The second author was supported in part by an NSF grant.

References

[1] Martin Aigner, Catalan–like numbers and determinants, J. Combinat. Theory Ser. A, 87 (1999), 33–51.

[2] E. Bayer-Fluckiger and Gabriele Nebe, On the Euclidean minimum of some real number fields, J. Théo. Nombres Bordeaux, 17 (2005), 437–454.

[3] Marc Chamberland and Christopher French, Generalized Catalan numbers and generalized Hankel transformations, J. Integer Sequences, 10 (2007), Article 07.1.1.

[4] Aleksander Cvetković, Predrag Rajković, Milos Ivković, Catalan numbers, the Hankel transform, and Fibonacci numbers, J. of Integer Sequences, 5 (2002), Article 02.1.3.

[5] Ömer Egecioglu, Timothy Redmond and Charles Ryavec, Almost Product Evaluation of Hankel Determinants, Electronic J. Combinatorics, 15(1) R6, (2008). Text available

at http://www.combinatorics.org/Volume 15/Abstracts/v15i1r6.html.

[6] C.  Krattenthaler,  Advanced  determinant  calculus,  Seminaire

Lotharingien Combin. 42 (1999), Article B42q. Text available at

http://www.emis.de/journals/SLC/wpapers/s42kratt.html.

[7] John W. Layman, The Hankel transform and some of its properties, J. Integer Sequences, 4 (2001), Article 01.1.5.

[8] Daniel A. Marcus, Number Fields, Springer-Verlag, NY, 1977.

[9] Frédérique Oggier and B. A. Sethuraman, Constructions of orthonormal lattices and quaternion division algebras for totally real number fields, in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Proceedings, 17th international symposium, AAECC-17, Bangalore, India, December 2007, Springer Lecture Notes in Computer Science, LNCS 4851, pp. 138–147.

[10] Paul Peart, Hankel determinants via Stieltjes matrices, Congressus Numerantium, 144 (2000), 153–159.

[11] Christian Radoux, Calcul effectif de certains determinants de Hankel, Bull. Soc. Math. Belg., 31 Fasc. 1, serie B, 49–55.

[12] Michael Z. Spivey and Laura L. Steil, The k-binomial transform and the Hankel transform, J. Integer Sequences, 9 (2006), Article 06.1.1.

[13] Ulrich Tamm, Some aspects of Hankel matrices in coding theory and combinatorics, Electronic J. Combinatorics, 8(1) A1, (2001). Text available at

http://www.combinatorics.org/Volume 8/Abstracts/v8i1a1.html.

2000 Mathematics Subject Classification: Primary 11B83; Secondary 11R04. Keywords: Hankel transform, Hankel matrix, central binomial coefficients, trace forms, ring of algebraic integers.

(Concerned with sequence \underline{A055878}.)

Received November 10 2008; revised version received November 25 2008. Published in Journal of Integer Sequences, December 14 2008.

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