Mateus De

Jesus Gonc ¸alves

ON THE NUMEROLOGY OF TRIGONOMETRIC POLYNOMIALS

Keywords: 1991 Mathematics Subject Classification. Primary 42A05 ; Secondary 05A10 Chebyshev polynomials, power reduction formulas, trigonometric polynomials, Catalan triangle, pyramidal numbers, spread polynomials, Riordan arrays, super Catalan numbers, golden ratio

This note is about the observation that the various transition formulas between bases of trigonometric polynomials can be expressed in terms binomial coefficients. More specifically, we write the entries of the Chebyshev matrices T and U in terms of binomial coefficients. We remark that the inverses of certain submatrices of the Chebyshev matrix T correspond to classical power reduction formulas for trigonometric functions. Therefore the entries of T -1 are expressible in terms of binomial coefficients and powers of two. The inverses of certain submatrices of the Chebyshev matrix U appear in power reduction formulas involving the Catalan triangles. We could not spot them in the literature. As a corollary the Catalan triangles are interpreted as inverses of rather natural matrices of binomial coefficients. We explain the matrix inversions in terms of Riordan arrays. We solve the integral 2π 0 cos 2m (x) sin 2n (x)dx and deduce a hypergeometric identity from the Fourier expansion of cos 2m (x) and sin 2n (x). As a corollary we prove that super Catalan numbers are integers. We emphasize that from the point of view of these base changes between the trigonometric polynomials it is more natural to work with 2 cos(x) and 2 sin(x) instead of cos(x) and sin(x), since the transition matrices become invertible over the integers. We do a similar analysis for the spread polynomials. This enables us to make a conjecture of Goh and Wildberger more precise. In this exposition we give a mostly self-contained account of the matter, the prerequisites just being four semesters of calculus, linear algebra, elementary complex variables and the residue theorem.

Introduction

When teaching basic calculus and trigonometry the educator is faced with the challenge that he or she cannot employ in a systematic way binomial coefficients, as Pascal's triangle is said to be too complicated for the modern student. On the other hand, since the advent of the Wilf-Zeilberger [START_REF] Petkovšek | [END_REF] method binomial coefficients are considered to be too shallow to be a subject of serious research. In the end, nobody appears to be responsible for this old subject, except the Online Encyclopaedia of Integer Sequences (OEIS) [START_REF]The On-Line Encyclopedia of Integer Sequences[END_REF]. This invaluable resource contains no proofs however, but merely references. To find proofs of classical formulas in the literature can be a tedious task.

In this article we take the opportunity to discuss a number of trigonometric formulas that arise when doing base changes between the various bases of trigonometric polynomials. The formulas are all classical, even though the ones involving Catalan triangles we were not able to spot in the literature. We do not know of a reference where they are discussed in context and proofs are provided. Moreover, we try to exhibit the appearance of binomial coefficients as succinctly as possible and deduce the relevant simple hypergeometric identities behind the inversions. Among other things, we point out the close relation between the coefficients of the Chebyshev polynomials and higher dimensional pyramidal numbers, a link we stumbled into while contemplating on the OEIS. This is closely related to the observation that the base change formulas simplify when working with 2 cos(x) and 2 sin(x) instead of cos(x) and sin(x) (but remember to keep the 1 !). We do a similar analysis for Norman Wildberger's spread polynomials that play a central role in his rational trigonometry [START_REF] Wildberger | Divine proportions. Rational trigonometry to universal geometry[END_REF]. We express them in terms of the even-dimensional pyramidal numbers. The most elegant point of view on our matrix inversions is provided by the language of Riordan arrays [SSB + 22], relating generating functions and matrix multiplication. Since we got rid of the annoying factors of four, we cleared the view to make a conjecture of Goh and Wildberger on the factorization of spread polynomials more precise.

The paper is organized as follows. In Section 2 we review the definitions of the Chebyshev polynomials. In Section 3 we recall the the notion of a trigonometric polynomial. In Section 4 we discover the higher pyramidal numbers and binomial coefficients in the coeficients of the Chebyshev polynomials. In Section 5 we propose an easy way to deduce quickly the first few Chebyshev polynomials T n on a sheet of paper. In Section 6 (the only section in this paper where we refer to other sources) we recall the definition of the even and odd Catalan triangles. In Section 7 we provide proofs of the well-known power reduction formulas and not-so-well-known variants thereof. In Section 8 we relate the Chebyshev polynomials to the power reduction formulas via matrix inversion. In Section 9 we provide another proof of the matrix inversions using Riordan arrays. In Section 10 we deduce further hypergeometric identities evaluating Fourier integrals and deduce the integrality of the super Catalan numbers. In Section 11 we once again go through the laundry list of the paper for the case of the spread polynomials. In Section 12 elaborate on a conjecture of Goh and Wildberger and provide some empirical evidence.

Acknowledgements. The paper arose from HCH teaching Calculus and Linear Algebra. The more HCH dug into the material he came to the realize that it should be known to Louis Shapiro in some form or another. The paper is part of an undergraduate research project of MdJG. He is acknowledging financial support of PROFAEX. We greatly benefited from discussions with Daniel Herden. HCH wants to thank A.-K. Gallagher for disposing [START_REF] Pólya | Problems and theorems in analysis II. Theory of functions, zeros, polynomials, determinants, number theory, geometry[END_REF] on his bookshelf. HCH thanks his administration at UFRJ for forcing him to teach beginner's courses for a decade, turning him into an Elephant im Porzellanladen (He really tried to teach them Eqn. (10.2), but they do not want to listen!).

Statements and Declarations. The authors declare that they have no conflicts of interest.

Chebyshev polynomials

For convenience of the reader we review here some basic facts about the Chebyshev polynomials (see, e.g., [PS98, Exercise VI.2]). The Chebyshev polynomials of the first kind (T n (x)) n≥0 and of the second kind (U n (x)) n≥0 are defined by the recursions

T 0 (x) = 1, T 1 (x) = x, T n (x) = 2xT n-1 (x) -T n-2 (x) for n ≥ 2, (2.1) U 0 (x) = 1, U 1 (x) = 2x, U n (x) = 2xU n-1 (x) -U n-2 (x) for n ≥ 2. (2.2) Proposition 2.1. ∞ n=0 T n (x)t n = 1 -tx 1 -2tx + t 2 (2.3) ∞ n=0 U n (x)t n = 1 1 -2tx + t 2 (2.4) Proof. From 1 -2tx + t 2 n≥0 T n (x)t n = n≥0 T n (x)t n - n≥0 T n (x)2t n+1 x + n≥0 T n (x)t n+2 = n≥0 T n (x)t n - m≥1 T m-1 (x)2t m x + m≥2 T m-2 (x)t m = 1 + tx -2tx + m≥2 (T m (x) -2xT m-1 (x) + T m-2 (x)) t m = 1 -tx
we deduce Eqn. (2.3); the proof of (2.4) is similar.

Theorem 2.2. For each integer n ≥ 0 we have

T n (cos(θ)) = cos(nθ), T n (sin(θ)) = (-1) n/2 cos(nθ) if n is even sin(nθ) if n is odd , U n (cos(θ)) = sin((n + 1)θ) sin(θ) , U n (sin(θ)) = (-1) n/2 cos((n+1)θ) cos(θ) n even sin((n+1)θ) cos(θ)
n odd .

Proof. We give an inductive proof of T m (cos(θ)) = cos(mθ) and sin(θ)U m (cos(θ)) = sin(mθ). The start of the induction is obvious. Let us assume that these identities hold for m = n -1 and m = n -2. Let us put for simplicity

U m (x) = 0 = T m (x) for m < 0. From n≥0 xT n-1 (x) + n≥0 x 2 -1 U n-2 (x)t n = n≥0 xT n (x)t n+1 + n≥0 x 2 -1 U n (x)t n+2 = xt 1 -tx 1 -2tx + t 2 + t 2 x 2 -1 1 1 -2tx + t 2 = xt -t 2 1 -2tx + t 2 = -1 + 2tx -t 2 + 1 -xt 1 -2tx + t 2 = 1 -tx 1 -2tx + t 2 -1 = n≥1 T n (x)t n , we see that xT n-1 (x) + x 2 -1 U n-2 (x) = T n (x) for n ≥ 1. Similarly, we observe that n≥0 (T n (x)t n + xU n-1 (x))t n = n≥0 T n (x)t n + n≥0 xU n (x)t n+1 = 1 -tx 1 -2tx + t 2 + tx 1 -2tx + t 2 = n≥0 U n (x)t n , so that T n (x) + xU n-1 (x) = U n (x)
for n ≥ 0. But, using the Addition Theorem for cosine, cos(nθ) = cos(θ) cos((n -1)θ) -sin(θ) sin((n -1)θ) = cos(θ)T n-1 (cos(θ)) -sin 2 (θ)U n-2 (cos(θ)), the first relation proves T n (cos(θ)) = cos(nθ). Moreover, the second relation in combination with the Addition Theorem for sine

sin((n + 1)θ) = cos(θ) sin(nθ) + sin(θ) cos(nθ) = sin(θ)(cos(θ)U n-1 (cos(θ)) + T n (cos(θ)))
shows sin(θ)U n (cos(θ)) = sin((n + 1)θ).

Trigonometric polynomials

A trigonometric polynomial of degree ≤ d in the variable θ is a univariate real function of the form

a 0 + d n=1 a n cos(θ) + d n=1 b n sin(nθ) with a 0 , a i , b i ∈ R for i = 1, 2, .
. . , d. The (2d + 1)-dimensional real vector space of trigonometric polynomials of of degree ≤ d will be denoted by Trig ≤d . By 2π-periodicity an f ∈ Trig ≤d is uniquely determined by its restriction to the interval [-π, π]. There is a inner product

1 f, g = 1 2π π -π f (θ)g(θ)dθ
1 In Fourier analysis it is more convenient to work with the inner product 2 f, g .

making Trig ≤d into an euclidean vector space. There is an orthogonal decomposition Trig ≤d = Trig 0 ≤d ⊕ Trig 1 ≤d into even and odd trigonometric polynomials Trig

j ≤d := f ∈ Trig ≤d |f (-θ) = (-1) j f (θ) , j = 0, 1. It is well-known that 1 √ 2 cos(θ) √ 2 cos(2θ) √ 2 cos(3θ) . . . √ 2 cos(dθ)
forms an orthonormal basis for Trig 0 ≤d and √ 2 sin(θ) √ 2 sin(2θ) √ 2 sin(3θ) . . . √ 2 sin(dθ) forms an orthonormal basis for Trig 1 ≤d . In this paper we find it convenient to view a basis as a frame, i.e., a row of vectors with trivial kernel. We can understand the space of all trigonometric polynomials as the direct limit Trig := ∪ d≥0 Trig ≤d with a decomposition Trig = Trig 0 ⊕ Trig 1 , Trig j := ∪ d≥0 Trig j ≤d . It follows that 1 √ 2 cos(θ) √ 2 cos(2θ) √ 2 cos(3θ) . . . forms an orthonormal basis for Trig 0 and that √ 2 sin(θ) √ 2 sin(2θ) √ 2 sin(3θ) . . . forms an orthonormal basis for Trig 1 . There are a lot of other (non-orthogonal) bases for Trig j , j = 0, 1. In this paper we are concerned, e.g., with the bases cos 0 (θ) cos 1 (θ) cos 2 (θ) cos 3 (θ) . . . , sin 1 (θ) sin(θ) sin 2 (θ) sin(θ) sin 3 (θ) sin(θ) sin 4 (θ) sin(θ) . . . for Trig 0 and sin(x) sin 2 (x) sin 3 (x) . . . for Trig 1 . To verify that these form bases we refer the reader to the classical literature (see, e.g., [START_REF] Pólya | Problems and theorems in analysis II. Theory of functions, zeros, polynomials, determinants, number theory, geometry[END_REF]). Alternatively, the claim also follows from the calculations of Section 9.

The Chebyshev polynomials T n provide, for example, a base change between the bases cos 0 (θ) cos 1 (θ) cos 2 (θ) cos 3 (θ) . . . and 1 cos(θ) cos(2θ) cos(3θ) . . . , while the Chebyshev polynomials U n provide a base change between the bases sin 1 (θ) sin(θ) sin 2 (θ) sin(θ) sin 3 (θ) sin(θ) sin 4 (θ) sin(θ)

. . . and 1 cos(θ) cos(2θ) cos(3θ) . . . .

In the interest of keeping the entries of the base change matrices as simple as possible we advocate (see Section 8) for working with the basis 1 2 cos(θ) 2 cos(2θ) 2 cos(3θ) . . . instead of 1 cos(θ) cos(2θ) cos(3θ) . . . and with 2 sin(θ) 2 sin 2 (θ) 2 sin 3 (θ) . . . instead of sin(θ) sin 2 (θ) sin 3 (θ) . . . . For the kabbalist the Chebyshev polynomials themselves appear to be a bit convoluted, and are probably defined this way for historical reasons.

There is also a more modern variation of the subject. The so-called spread polynomials, introduced by Norman Wildberger (see [START_REF] Wildberger | Divine proportions. Rational trigonometry to universal geometry[END_REF]), mediate a base change between sin 2 (θ) sin 4 (θ) sin 6 (θ) . . . and sin 2 (θ) sin 2 (2θ) sin 2 (3θ) . . . . Again, from the point of view of numerology it seems more natural to work with 4 sin 2 (θ) 16 sin 4 (θ) 64 sin 6 (θ) . . . and 4 sin 2 (θ) 4 sin 2 (2θ) 4 sin 2 (3θ) . . . instead. The details of the base change are to be addressed in Section 11.

Chebyshev matrix and higher dimensional pyramidal numbers

We define the infinite matrix T = (T mn ) m,n≥0

T :=                  1 -1 1 -1 1 -1 • • • 1 -3 5 -7 9
• • • • • •                  , (4.1)
so that T 0 (x) T 1 (x) T 2 (x) . . . = 1 x x 2 . . . T , i.e., T n (x) = m≥0 T mn x m . We take the liberty to not annotate zero entries.

Let us start our analysis by making an empirical observation. For i ≥ 1 we record in a table the sequence p

[i] j j≥0
of nonzero entries of T ith row multiplied by (-1) is usually referred to as the i-dimensional pyramidal numbers (we are sloppy here about the index shifts). Its generating function p [i] (t) = j p

[i] j t j is simply

p [i] (t) = 1 + t (1 -t) i+1 = 2 -(1 -t) (1 -t) i+1 = 2 (1 -t) i+1 - 1 (1 -t) i = j≥0 2 i + j j - i + j -1 j t j .
The matrix

U =               1 -1 1 -1 1 2 -4 6 -8 10
              , (4.2)
on the other hand, is defined such that U n (x) = m≥0 U mn x m . For i ≥ 0 we record in a table the sequence of nonzero entries of the ith row of U multiplied by (-1) j 2 -i , , which apparently turns out to be the Pascal matrix. We phrase our observation as follows (this type of formulas is known for quite some time [START_REF] Clenshaw | Chebyshev series for mathematical functions[END_REF]).

i\j 0 1 2 3 4 5 • • • 0th row 1 1 1 1 1 1 • • • 1st row 1 2 3 4 5 6 2nd
Theorem 4.1. The Chebyshev polynomials are given by the formulas

T 2n (x) = (-1) n + n j=1 (-1) n+j 2 2j-1 p [2j] n-j x 2j , T 2n+1 (x) = n j=0 (-1) n+j 2 2j p [2j+1] n-j x 2j+1 , (4.3) U 2n (x) = n j=0 (-1) n+j 2 2j n + j 2j x 2j , U 2n+1 (x) = n j=0 (-1) n+j 2 2j+1 n + j + 1 2j + 1 x 2j+1 , (4.4)
where n ≥ 0 is an integer.

Proof. The first step is to cleanse Eqn. (4.1) from its powers of 2. The trick is to define P 0 (z) := 1, (4.5) P n (z) := 2T n (z/2) for n > 0 to ensure that P 0 (1) = 1 and P n (2 cos(θ)) = 2T n (cos(θ)) = 2 cos(nθ) for n > 0. We easily deduce the bivariate generating function

n≥0 P n (z)t n = 2 -zt 1 -zt + t 2 -1 = 1 -t 2 1 -zt + t 2 . (4.6)
To cleanse the Taylor coefficients from the signs we make the substitution

t → t/ √ -1, z → √ -1z to obtain 2 1 -t 2 1 -zt + t 2 → 1 + t 2 1 -zt -t 2 . (4.7)
We extend our definition of the pyramidal numbers to include also j p

[0] j t j = (1 + t)/(1 -t) = 1 + 2t + 2t 2 + 2t 3 + . . . . Let K ⊂ C × be compact and r = min ||t| -|t| -1 | | t ∈ K . If |z| < r it follows that | tz 1-t 2 | < 1 for all t ∈ K. Then we have for t ∈ K, |z| < r m,l≥0 p [m] l z m t m+2l = m≥0 1 + t 2 (1 -t 2 ) m+1 (tz) m = m≥0 1 + t 2 1 -t 2 tz 1 -t 2 m = 1 + t 2 1 -t 2 1 1 -tz 1-t 2 = 1 + t 2 1 -t 2 -tz .
This holds everywhere outside of the poles, and obviously also at

t = 0. But now n P n (z)t n = 2-t 2 1-zt+t 2 = j p [i] j t 2j (-z) i . Similarly, we define V n (z) := U n (z/2) for n ≥ 0, (4.8) so that V n (2 cos(θ)) = U n (cos(θ)) = sin((n + 1)θ)/ sin(θ) for n ≥ 0. Hence the bivariate generating function is n≥0 V n (z)t n = 1 1 -zt + t 2 .
To get rid of the signs we make the substitution

t → t/ √ -1, z → √ -1z to obtain 1 1-zt+t 2 → 1 1-zt-t 2 . The relation m,l≥0 m + l l z m t m+2l = m≥0 (tz) m (1 -t 2 ) m+1 = m≥0 1 1 -t 2 tz 1 -t 2 m = 1 1 -t 2 1 1 -tz 1-t 2 = 1 1 -t 2 -tz ,
holds for t ∈ K and |z| < r, hence everywhere outside the poles.

For practical matters, let us mention that for i ≥ 0 the coefficients of j p

[i] j t j are the difference pattern of the coefficients of j p

[i+1] j t j . This is because the generating functions differ by a factor 1 -t. We introduce the infinite matrix P = (P mn ) n,m≥0 so that P n (x) = m≥0 P mn x m (cf. Eqn. (4.6)).

Mnemonics

If you are stranded on a lonesome island and want to recall quickly the Chebyshev polynomials T n (x) without entering some tedious recursions we have a suggestion. Start with the row of odd numbers and write on top of it its difference pattern. Then put on the bottom the row of the sequence whose differences are the odd numbers, (i.e., the square numbers). Proceed the same way with the resulting row and continue recursively. Introduce alternating signs for the diagonals and fill the gaps with zeros. The result is P .

1 2 2 2 2 • • • 1 3 5 7 1 4 9 16 • • • → 1 2 2 2 2 • • • 1 3 5 7 1 4 9 16 1 5 14 1 6 20 1 7 1 8 1 1 • • • • • • → P =                 1 0 -2 0 2 0 -2 0 2 • • • 0 1 0 -3 0 5 0 -7 0 0 0 1 0 -4 0 9 0 -16 0 0 0 1 0 -5 0 14 0 0 0 0 0 1 0 -6 0 20 0 0 0 0 0 1 0 -7 0 0 0 0 0 0 0 1 0 -8 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 • • • • • •                
To obtain T turn all 2s in the 0th row into 1s and for each i ≥ 1 multiply the ith row with 2 i . There is another way to look at the coefficients of the Chebyshev polynomials that we would like to mention. We plan to discuss it in a more general context at another occasion. Namely, put

n k :=      T 00 /2 T 2k n if n > 0 is even, T 2k+1 n if n odd. Then n k satisfies the recurrence n k = a n -1 k + b n k -1 with a = 2, b = -1 and initial condition n 0 = 2 n-1 , 0 k = 1/2 if k = 0, (-1) k else.

Catalan triangles and the Fuß-Catalan numbers

The Catalan sequence (C n ) n≥0 can be defined by the Segner recursion

C 0 = 1, C n = n-1 k=0 C k C n-1-k .
In term of the generating function C(x) = n≥0 C n x n this can be written as xC = C 2 -1. On other words, if you have forgotten

C n but know C 0 , C 1 , . . . , C n-1 , let your computer expand ( n-1 k=0 C k x k ) 2 . The coefficient in front of x n-1 is C n .
By complementing the squares one can determine the two roots of C 2 -xC -1. This leads to the formula

C(x) = 1 - √ 1 -4x 2x , (6.1)
noting that only one root is a power series. From this one deduces

C n = 1 n+1 2n n = 2n n -2n-1 n
. The literature on the Catalan sequence is extensive (see, e.g., [Sta12, Sta15, GKP89, Kos09]). A closely related function is the generating function of the central binomial coefficients

B(x) = n≥0 2n n x n = C(x) 2 -C(x) = 1 √ 1 -4x . (6.2)
The Fuß-Catalan numbers are defined by

F m (p, r) := 1 mp + r mp + r m = r m(p -1) + r mp + r -1 m = r m mp + r -1 m -1 (6.3)
for integers m, p, r ≥ 0. They satisfy the important convolution property F m (p, s+r) = m k=0 F k (p, r)F m-k (p, s); see [GKP89, Eqn. (5.36)] or [START_REF] Riordan | Combinatorial identities[END_REF]. From this it follows that m≥0

F m (p, r)x m =   m≥0 F m (p, 1)x m   r , i.e., m≥0 F m (p, r)x m is the r-fold convolution of m≥0 F m (p, 1)x m . By inspection, m≥0 F m (1, 1)x m = (1-x) -1 and m≥0 F m (2, 1)x m = C(x).
The series m≥0 F m (p, 1)x m is also called the generalized binomial series. Let us record the first r-fold convolutions of C(x):

C(x) = 1 + x + 2x 2 + 5x 3 + 14x 4 + 42x 5 + 132x 6 + 429x 7 + . . . , C 2 (x) = 1 + 2x + 5x 2 + 14x 3 + 42x 4 + 132x 5 + 429x 6 + 1430x 7 + . . . , C 3 (x) = 1 + 3x + 9x 2 + 28x 3 + 90x 4 + 297x 5 + 1001x 6 + 3432x 7 + . . . , C 4 (x) = 1 + 4x + 14x 2 + 48x 3 + 165x 4 + 572x 5 + 2002x 6 + 7072x 7 + . . . , C 5 (x) = 1 + 5x + 20x 2 + 75x 3 + 275x 4 + 1001x 5 + 3640x 6 + 13260x 7 + . . . , C 6 (x) = 1 + 6x + 27x 2 + 110x 3 + 429x 4 + 1638x 5 + 6188x 6 + 23256x 7 + . . . , C 7 (x) = 1 + 7x + 35x 2 + 154x 3 + 637x 4 + 2548x 5 + 9996x 6 + 38760x 7 + . . . , C 8 (x) = 1 + 8x + 44x 2 + 208x 3 + 910x 4 + 3808x 5 + 15504x 6 + 62016x 7 + . . . . . .
The coefficients of the 2l-fold convolutions C 2l (x) for l ≥ 1 have been arranged by Louis Shapiro [START_REF] Shapiro | A Catalan triangle[END_REF] into an infinite lower triangular matrix B even that is usually referred to as the Catalan triangle. We will call it here the even Catalan triangle since we also need the odd Catalan triangle B odd . The latter is formed by arranging the (2l -1)-fold convolutions C 2l-1 (x) for l ≥ 1 into an infinite lower triangular matrix.

B even =           1 • • • 2 1 5 4 1 14 14 6 1 42 48 27 8 1 132 165 110 44 10 1 • • • • • •           , B odd =           1 • • • 1 1 2 3 1 5
9 5 1 14 28 20 7 1 42 90 75 35 9 1

• • • • • •           . (6.4)
The matrix entries are given by the formulas

B even ij = j i 2i i -j , B odd ij = 2j + 1 2i + 1 2i + 1 i -j , (6.5)
where i, j ≥ 1. One of the authors met B even ij already on another occasion [START_REF] Conca | Koszul properties of the moment map of some classical representations[END_REF]. We encounter the Catalan triangles in the next section.

Power reduction formulas

Theorem 7.1.

(1)

2 2n-1 cos 2n (θ) = 1 2 2n n + n k=1 2n n-k cos(2kθ), (2) 2 2n cos 2n+1 (θ) = n k=0 2n+1 n-k cos((2k + 1)θ), (3) 2 2n-1 sin 2n (θ) = 1 2 2n n + n k=1 (-1) k 2n n-k cos(2kθ), (4) 2 2n sin 2n+1 (θ) = n k=0 (-1) k 2n+1
n-k sin((2k + 1)θ).

Proof. The following proof of (1) we learnt from Daniel Herden.

n k=1 2n n -k cos(2kθ) = n k=1 2n n -k e 2k √ -1θ + e -2k √ -1θ 2 = e -2n √ -1θ n k=1 2n n -k e 2(n+k) √ -1θ + e 2(n-k) √ -1θ 2 = e -2n √ -1θ 2 n k=1 2n n -k e 2(n-k) √ -1θ + n k=1 2n n -k e 2(n+k) √ -1θ = e -2n √ -1θ 2 n k=1 2n n -k e 2(n-k) √ -1θ + n k=1 2n n + k e 2(n+k) √ -1θ = e -2n √ -1θ 2   - 2n n e 2n √ -1θ + j 2n j e 2j √ -1θ   = e -2n √ -1θ 2 1 + e 2 √ -1θ 2n - 1 2 2n n = 1 2 e √ -1θ + e - √ -1θ 2n - 1 2 2n n = 2 2n-1 cos 2n (θ) - 1 2 2n n .
By shifting the argument we deduce (3):

2 2n-1 sin 2n (θ) = 2 2n-1 cos 2n θ - π 2 = 1 2 2n n + n k=1 2n n -k cos 2k θ - π 2 = 1 2 2n n + n k=1 (-1) k 2n n -k cos(2kθ).
To show (2) along the lines of Herden's proof we calculate

n k=0 2n + 1 n -k cos((2k + 1)θ) = n k=0 2n + 1 n -k e (2k+1) √ -1θ + e -(2k+1) √ -1θ 2 = e -(2n+1) √ -1θ 2 n k=0 2n + 1 n -k e (2(n+k)+2) √ -1θ + n k=0 2n + 1 n -k e 2(n-k) √ -1θ = e -(2n+1) √ -1θ 2   2n+1 j=n+1 2n + 1 j e 2j √ -1θ + n j=0 2n + 1 j e 2j √ -1θ   = e -(2n+1) √ -1θ 2 2n+1 j=0 2n + 1 j e 2j √ -1θ = e (-2n-1) √ -1θ 2 1 + e 2 √ -1θ 2n+1 = 1 2 e √ -1θ + e - √ -1θ 2n+1 = 2 2n cos 2n+1 (θ).
By shifting the argument we deduce (4):

2 2n sin 2n+1 (θ) = 2 2n cos 2n+1 θ - π 2 = n k=0 2n + 1 n -k cos (2k + 1) θ - π 2 = n k=0 2n + 1 n -k cos (2k + 1)θ -kπ - π 2 = n k=0 (-1) k 2n + 1 n -k sin((2k + 1)θ).
The following banal calculation has a pretty matrix inversion as a consequence (see Eqn. (8.6)--(8.9)).

Corollary 7.2.

(1)

2 2n cos 2n (θ) = n k=0 B odd nk sin((2k+1)θ) sin(θ) , ( 2
) 2 2n-1 cos 2n-1 (θ) = n k=1 B even nk sin(2kθ) sin(θ) .
Proof. We derive (1) of Theorem 7.1:

-2 2n-1 2n sin(θ) cos 2n-1 (θ) = d dθ 2 2n-1 cos 2n (θ) = - n k=1 2n n -k 2k sin(2kθ) =⇒ 2 2n-1 cos 2n-1 (θ) = n k=1 k n 2n n -k sin(2kθ) sin(θ) ,
which shows (2). Similarly, taking the derivative of (2) of Theorem 7.1 we obtain:

-2 2n (2n + 1) sin(θ) cos 2n (θ) = d dθ 2 2n cos 2n+1 (θ) = - n k=0 2n + 1 n -k (2k + 1) sin((2k + 1)θ) =⇒ 2 2n cos 2n (θ) = n k=0 2k + 1 2n + 1 2n + 1 n -k sin((2k + 1)θ) sin(θ) , proving (1)
.

Hypergeometric identities from matrix inversions

In order to distill the relevant hypergeometric identities we choose to get rid of the superficial powers of 2. We put

κ(θ) := 2 cos(θ), σ(θ) := 2 sin(θ), ν m (θ) := sin((m + 1)θ) sin(θ) (8.1)
and work with the following bases 1 κ(θ) κ(2θ) κ(3θ) . . . ,

1 ν 1 (θ) ν 2 (θ) ν 3 (θ) . . .
for the even trigonometric polynomials and σ(θ) σ(2θ) σ(3θ) . . . for the odd trigonometric polynomials.

We are prepared to understand that T 2n (cos(θ)) = cos(2nθ) and Theorem 7.1 (1) can be interpreted as the following mutually inverse linear systems

(8.2) 1 κ(2θ) κ(4θ) κ(6θ) κ(8θ) • • • = κ 0 (θ) κ 2 (θ) κ 4 (θ) κ 6 (θ) κ 8 (θ) • • •          p [0] 0 -p [0] 1 p [0] 2 -p [0] 3 p [0] 4 • • • p [2] 0 -p [2] 1 p [2] 2 -p [2] 3 p [4] 0 -p [4] 1 p [4] 2 p [6] 0 -p [6] 1 p [8] 0 • • • • • •          , (8.3) κ 0 (θ) κ 2 (θ) κ 4 (θ) κ 6 (θ) κ 8 (θ) • • • = 1 κ(2θ) κ(4θ) κ(6θ) κ(8θ) • • •         0 0 2 1 4 2 6 3 8 4 • • • 2 0 4 1 6 2 8 3 4 0 6 1 8 2 6 0 8 1 8 0 • • • • • •         .
The equivalence of the relation T 2n (sin(θ)) = (-1) n cos(2nθ) and Theorem 7.1 (3) boils down to essentially the same matrix inversion.

Similary, T 2n+1 (cos(θ)) = cos((2n + 1)θ) can be compared with Theorem 7.1 (2):

(8.4) κ(θ) κ(3θ) κ(5θ) κ(7θ) κ(9θ) • • • = κ 1 (θ) κ 3 (θ) κ 5 (θ) κ 7 (θ) κ 9 (θ) • • •          p [1] 0 -p [1] 1 p [1] 2 -p [1] 3 p [1] • • • p [3] 0 -p [3] 1 p [3] 2 -p [3] p [5] 0 -p [5] 1 p [5] p [7] 0 -p [7] p [9] • • • • • •          , (8.5) κ 1 (θ) κ 3 (θ) κ 5 (θ) κ 7 (θ) κ 9 (θ) • • • = κ(θ) κ(3θ) κ(5θ) κ(7θ) κ(9θ) • • •         1 0 3 1 5 2 7 3 • • • 3 0 5 1 7 2 5 0 7 1 7 0 • • • • • •        
.

Essentially the same matrix inversion shows up when comparing T 2n+1 (sin(θ)) = (-1) n sin((2n + 1)θ) and Theorem 7.1 (4). Finally, U 2n (cos(θ)) = ν 2n (θ) is the inverse linear system to Corollary 7.2 (1)

(8.6) 1 κ 2 (θ) κ 4 (θ) κ 6 (θ) κ 8 (θ) • • • = 1 ν 2 (θ) ν 4 (θ) ν 6 (θ) ν 8 (θ) • • •         1 1 2 5 14 • • • 1 3 9 28 1 5 20 1 7 1 • • • • • •         , (8.7) 1 ν 2 (θ) ν 4 (θ) ν 6 (θ) ν 8 (θ) • • • = 1 κ 2 (θ) κ 4 (θ) κ 6 (θ) κ 8 (θ) • • •         0 0 -1 1 2 2 -3 3 • • • 2 0 -3 1 4 2 - 4 0 -5 1 6 0 - • • • • • •        
, and U 2n+1 (cos(θ)) = ν 2n+1 (θ) is the inverse linear system to Corollary 7.2 (2)

(8.8) κ κ 3 (θ) κ 5 (θ) κ 7 (θ) κ 9 (θ) • • • = ν 1 ν 3 (θ) ν 5 (θ) ν 7 (θ) ν 9 (θ) • • •         1 2 5 14 42 • • • 1 4 14 48 1 6 27 1 8 1 • • • • • •         , (8.9) ν 1 (θ) ν 3 (θ) ν 5 (θ) ν 7 (θ) ν 9 (θ) • • • = κ(θ) κ 3 (θ) κ 5 (θ) κ 7 (θ) κ 9 (θ) • • •         1 0 -2 1 3 2 -4 3 5 4 • • • 3 0 -4 1 5 2 -6 3 5 0 -6 1 7 2 7 0 -8 1 9 0 • • • • • •        
.

The transposed matrices of Eqn. (8.7) and (8.9) are the odd and the even Catalan triangles B odd and B even . The latter two matrix inversions were already observed by Louis Shapiro in [START_REF]Some open questions about random walks, involutions, limiting distributions, and generating functions[END_REF], and we gave here a trigonometric proof. In fact, all the matrices of this section are transposes of Riordan arrays. This can be used to explain all the inversions of this section; we do this in the next Section.

Interpretation in terms of the Riordan group

The Chebyshev matrix T itself cannot be interpreted as the transpose of a Riordan array, but the matrizes P, B even and B odd can. This means that the matrix inversions of the previous section can be understood as inversions in the so-called Riordan group. The calculations are based on Lagrange inversion and turn out to be surprisingly simple. Some of the statements of this section can be found in some form in the PhD thesis of Aoife Hennessy [START_REF] Hennessy | A study of Riordan arrays with applications to continued fractions, orthogonal polynomials and lattice paths[END_REF].

Let us review some basic ideas of the theory of Riordan arrays. The proofs of the facts reviewed here are actually quite accessible; we refer the reader to the monograph [SSB + 22]. We write R x for the formal power series with real coefficients. By R x × we mean the multiplicatively invertible formal power series.

The composition map

R x × x R x → R x that sends g(x) = n≥0 g n x n ∈ R x and f (x) = m≥1 f m x m ∈ x R x to n≥0 g n m≥1 f m x m n is denoted by (g, f ) → g • f . The space x R x × is
comprised of compositionally invertible formal power series. I.e., if f ∈ x R x × then the map R x → R x , g → g • f is invertible. The inversion is provided by the Langrange inversion formula: if

f (x) = x φ(x)
then the coefficients of the compositional inverse f are given by the formula (see, e.g., [Ges16])

[y n ] f (y n ) = 1 n x n-1 (φ(x)) n . (9.1) Definition 9.1. An infinite lower triangular matrix A = [a nk ] n,k≥0 ∈ R ∞×∞ is called a Riordan array if there exist (g, f ) ∈ R x × x R x × such that a nk = [x n ] g(x)f (x) k .
It is tradition to write, abusing notation, A = (g, f ), so that (g, f ) can be interpreted as a matrix or as a tuple. If g ∈ R x × then (g, f ) is called a proper Riordan array.

It turns out that R x × × x R x × with the composition (g 1 , f 1 ) * (g 2 , f 2 ) = (g 1 (g 2 • f 1 ), f 2 • f 1 ) forms a
group, the so-called Riordan group, with identity (1, x) and inverse (g, f ) -1 = 1 g•f , f . The Fundamental Theorem of Riordan Arrays links the Riordan group to matrix multiplication. Let us denote by h = h 0 h 1 h 2 . . .

T the infinite column vector associated to the formal power series

h(x) = n≥0 h n x n ∈ R x . Then the theorem says that (g, f ) h = -----→ g(h • f ). As a consequence one deduces the relation (g 1 , f 1 )(g 2 , f 2 ) = (g 1 , f 1 ) - → g 2 (g 1 , f 1 ) --→ g 2 f 2 (g 1 , f 1 ) --→ g 2 f 2 2 . . . = (g 1 , f 1 ) * (g 2 , f 2 ).
Lemma 9.2. Assume for i = 1, 2 the following inversions in the Riordan group

(g i , f ) -1 = (G i , F ). Then (g 1 g 2 , f ) -1 = (G 1 G 2 , F ) follows.
Proof. The assumption means (

g i , f ) * (G i , F ) = (1, x). It follows that (g 1 g 2 , f ) * (G 1 G 2 , F ) = (g 1 g 2 (G 1 G 2 • F ), f • F ) = (g 1 (G 1 • F )g 2 (G 2 • F ), f • F ) = (1, x).
The key observation we want to make now (it seems to be combinatorialist's folklore) is that

B odd = C, xC 2 , B even = C 2 , xC 2 , p [0] (-x), x p [2] (-x) p [0] (-x) = 1 -x 1 + x , x (1 + x) 2 , p [1] (-x), x p [3] (-x) p [1] (-x) = 1 -x (1 + x) 2 , x (1 + x) 2 .
are Riordan arrays. This permits to interprete Equations (8.2)-(8.9) in Section 8 as inversions in the Riordan group.

Theorem 9.3. Let C and B the generating functions of the Catalan numbers and of the central binomial coefficients, respectively. Then

(1) C, xC 2 -1 = 1 1+x , x (1+x) 2 , (2) C 2 , xC 2 -1 = 1 (1+x) 2 , x (1+x) 2 , (3) B, xC 2 -1 = 1-x 1+x , x (1+x) 2 , (4) BC, xC 2 -1 = 1-x (1+x) 2 , x (1+x) 2 .
Proof. First we show xC 2 (x) = x (1+x) 2 using formula (9.1). Namely, for n ≥ 1 we have

[y n ] x (1 + x) 2 (y) = 1 n x n-1 (1 + x) 2n = 1 n x n-1 j≥0 2n j x j = 1 n 2n n -1 = [x n ] xC 2 .
To prove (1) it remains to check that C • xC 2 = 1 + x. We do this by comparing

C = C • xC 2 • xC 2 with (1 + x) • xC 2 = 1 + xC 2 = 1 + C -1 = C, and remember that g → g • xC 2 is invertible.
To prove (3) we observe that 1-xC 2

j≥0 2j + n j -m x j = j≥0 1 2π √ -1 γ (1 + z) n+2j z j-m x j dz z = j≥0 1 2π √ -1 γ z m (1 + z) n (1 + z) 2 z x j dz z = 1 2π √ -1 γ z m (1 + z) n 1 1 -(1+z) 2 z x dz z = 1 2π √ -1 1 x γ z m (1 + z) n z x -(1 + z) 2 dz = - 1 2π √ -1 1 x γ z m (1 + z) n (1 + z) 2 -z x dz.
We substitute 1 + z = w, xw r = (w -1) d and write

j≥0 n + 2j k + j x j = - 1 2π √ -1 1 x γ+1 w n (w -1) m w 2 -w-1 x dw.
Factoring the quadratic polynomial in the denominator

w 2 - w -1 x = w - 1 2x 2 - 1 -4x 4x 2 = w - 1 - √ 1 -4x 2x w - 1 + √ 1 -4x 2x = (w -C(x)) w - 1 x + C(x)
and observing that C(x) -1 x + C(x) = -1 xB(x) we can apply now the Residue Theorem and deduce the claim. In fact the second pole is outside γ + 1 for |x| small enough.

2n n -m x n = (C -1) m B = x m BC 2m , n≥0 2n + 1 n -m x n = C(C -1) m B = x m BC 2m+1 .
Corollary 9.6.

(1)

1-x 2 (1+x) 2 -4 cos 2 (θ)x = 1 + 2 n≥1 cos(2nθ)x n , (2) 2(1-x) cos(θ) (1+x) 2 -4x cos 2 (θ) = n≥0 cos((2n + 1)θ)x n , (3) 1+x (1+x) 2 -4 cos 2 (θ) = n≥0 sin((2n+1)θ) sin(θ)
x n , (4)

2 cos(θ) (1+x) 2 -4x cos 2 (θ) = n≥0 sin(2(n+1)θ) sin(θ)
x n .

Proof. We put κ even

θ (x) := n≥0 κ 2 (θ)x n = 1 1-κ 2 (θ)
x and note that the right hand side of Equation (8.2)

is 1 -x 1 + x , x (1 + x) 2 ---→ κ even θ = ----------------→ 1 -x 1 + x 1 1 -κ 2 (θ) x (1+x) 2 = -------------→ 1 -x 2 (1 + x) 2 -xκ 2 (θ)
.

This shows

1-x 2 (1+x) 2 -κ 2 (θ)x = 1 + n≥1 κ(2nθ)
x n , which is equivalent to (1). Similarly, we put

κ odd θ (x) := n≥0 κ(θ) κ 2 (θ)x n = κ(θ) 1-κ 2 (θ)
x and express the right hand side of Equation (8.4) as -----------------→ 1 ------------→ (1 -x)κ(θ)

1 -x (1 + x) 2 , x (1 + x) 2 --→ κ odd θ = -
-x (1 + x) 2 κ(θ) 1 -κ 2 (θ) x (1+x) 2 = -
(1 + x) 2 -xκ 2 (θ)
.

This shows (1-x)κ(θ) (1+x) 2 -κ 2 (θ)x = n≥0 κ((2n + 1)θ)
x n , which is equivalent to (2). Writing the right hand side of Equation 8.7

1 1 + x , x (1 + x) 2 ---→ κ even θ = ----------------→ 1 1 + x 1 1 -κ 2 (θ) x (1+x) 2 = -------------→ 1 + x (1 + x) 2 -xκ 2 (θ) we prove 1+x (1+x) 2 -xκ 2 (θ) = n≥0 ν(2nθ)
x n , and hence (3). Finally, writing the right hand side of Equation (8.9) as 1 -----------------→ 1

(1 + x) 2 , x (1 + x) 2 --→ κ odd θ = -
(1 + x) 2 κ(θ) 1 -κ 2 (θ) x (1+x) 2 = -------------→ κ(θ) (1 + x) 2 -xκ 2 (θ) we deduce κ(θ) (1+x) 2 -xκ 2 (θ) = n≥0 ν((2n + 1)θ)
x n , which is equivalent to (4).

Hypergeometric identities from trigonometric intergrals

This section is not essential for the logic of the paper, but gives an example how the matrix inversions can be used. Our first aim is to deduce further hypergeometric identities by inspecting the trigonometric integral 2π 0 cos n (θ) sin m (θ)dθ. This could be evaluated using the beta function, but we prefer here an more elementary treatment. When n ≥ 1, m ≥ 0 we perform the partial integration

I n,m = 2π 0 cos n (θ) sin m (θ) dθ = cos n-1 (θ) sin m+1 (θ) m + 1 2π 0 + 2π 0 (n -1) sin(θ) cos n-2 (θ) sin m+1 (θ) m + 1 dθ.
We deduce (m + 1)I n,m = (n -1) I n-2,m+2 . From the substitution s = θ -π/2 we find

I n,m = 2π 0 cos n (θ) sin m (θ)dθ = 3π/2 -π/2 (-sin(s) n ) cos m (s)ds = (-1) n I m,n .
Evidently, I n,1 =

I n,0 = 0 if n odd 2π 2 -n n n/2 if n even . (10.1)
Hence I n,m = 0 if and only if n are m even. By induction we obtain for k ≥ l ≥ 0:

I 2k,2l = l m=1 (2m + 1) (2k + 2m -1) I 2(k+l),0 . But l m=1 (2k + 2m -1) = (2k+2l-1)!! (2k-1)!! and hence I 2k,2l = (2k-1)!! (2k+2l-1)!! (2l -1)!! I 2(k+l),0 . From (2k -1)!! (2k + 2l -1)!! (2l -1)!! = 2 k+l-1 (k + l -1)! (2k + 2l -1)! (2k -1)! 2 k-1 (k -1)! (2l -1)! 2 l-1 (l -1)! = 2 k+l-1 k-1 l! 2k+2l-1 2k-1 (2l)! (2l -1)! (l -1)! = k+l-1 k-1 2k+2l-1 2k-1 = k+l-1 l 2k+2l-1 2l
Remembering I 2k,2l = I 2l,2k , we finally deduce that for k, l ≥ 0

I 2k,2l = k+l-1 l 2k+2l-1 2l I 2(k+l),0 .
Combining this this with (10.1) we have proven the following.

Theorem 10.1. For integers k, l ≥ 0 we have

2π 0 cos 2k (θ) sin 2l (θ) dθ = 2π 2 -2(k+l) ( k+l-1 l )( 2(k+l) k+l ) ( 2(k+l)-1 2l
)

.

We evaluate

2π = 2π 0 cos 2 (θ) + sin 2 (θ) m dθ = k,l≥0 k+l=m k + l l 2π 0 cos 2k (θ) sin 2l (θ) dθ = 2π k,l≥0 k+l=m k + l l 2 -2(k+l) k+l-1 l 2(k+l) k+l 2k+2l-1 2l
and find as a corollary

2 2m 2m m = m l=0 m-1 l m l 2m-1 2l . (10.2)
The rest of this section we use our base change of the previous section to show that the infinite symmetric matrix M = [M kl ] k,l≥0 ,

M kl := 1 2π π -π κ 2k (θ)σ 2l (θ) dθ = k+l-1 l 2(k+l) k+l 2(k+l)-1 2l
.

has integer entries. On the way we deduce another hypergeometric identity (namely Eqn. (10.3) below).

For the kabbalist3 the matrix M looks actually a bit curious

M =             1
(2k + 2l)! (k + l)! 2 (2l)! (2k -1)! ! (2k + 2l -1)! (k + l -1)! (k -1)!l! = 2(k + l) (2l)! (2k)! (k + l)!(k + l) (k -1)!l!2k = (2l)! (2k)! (k + l)! (k -1)!l!k .
Moreover, according to computer algebra the matrix M admits the neat LU factorization

M = L diag(1, -2, 2, -2, 2, . . . )L T , (10.3)
where L is the lower triangular matrix 2i i-j i,j≥0

. This entails that the upper left n × n-block M (n) of M has determinant (-1) n/2 2 n-1 and that M is integer. Once again, the number sequence 1, 2, 2, 2, . . . pops up.

It turns out that with our preparations we can easily find an explanation for Eqn. (10.3). By the results of Section 8 we have

κ 0 (θ) κ 2 (θ) κ 4 (θ) κ 6 (θ) • • • = 1 κ(θ) κ(2θ) κ(4θ) • • • L T , σ 0 (θ) σ 2 (θ) σ 4 (θ) σ 6 (θ) • • • = 1 κ(θ) κ(2θ) κ(4θ) • • • diag(1, -1, 1, -1, . . . )L T .
Transposing and integrating we deduce

M = 1 2π π -π       κ 0 (θ) κ 2 (θ) κ 4 (θ) κ 6 (θ) • • •       σ 0 (θ) σ 2 (θ) σ 4 (θ) σ 6 (θ) • • • dθ = L       1 2π π -π       1 κ(2θ) κ(4θ) κ(6θ) • • •       1 κ(2θ) κ(4θ) κ(6θ) • • • dθ       diag(1, -1, 1, -1, . . . )L T = L diag(1, 2, 2, 2, . . . )diag(1, -1, 1, -1, . . . )L T .

Spread polynomials

The spread polynomials (S n (x)) n≥0 (see [START_REF] Wildberger | Divine proportions. Rational trigonometry to universal geometry[END_REF]) play a central role in Wildberger's Rational Trigonometry, an approach to plane geometry free of transcendental expressions. They encode rotations in the plane and are defined by the following recursion:

S 0 (x) = 0, S 1 (x) = x, S n (x) = 2(1 -2x)S n-1 (x) -S n-2 (x) + 2x.
Proposition 11.1. For every integer n ≥ 0 we have S n (x) = 1-Tn(1-2x) 2 .

Proof. We define

S n (x) := 1 -T n (1 -2x) 2 ( * ) n .
Obviously, S 0 (x) = 0, S 1 (x) = x. Assuming ( * ) n we want to deduce the recurrence for S n (x) from the recurrence for T n (x). We observe that ( * ) n is equivalent to

T n (1 -2x) = 1 -2S n (x) ( * * ) n .
We have

T n (1 -2x) = 2(1 -2x)T n-1 (1 -2x) -T n-2 (1 -2x)
and hence

S n (x) = 1 -2(1 -2x)T n-1 (1 -2x) + T n-2 (1 -2x) 2 .
since n ≥ 1.

Let Wild be the subspace of the space Trig 0 consisting of functions that vanish at θ = 0. Put ¡(θ) := 4 sin 2 (θ) and consider the two alternative bases for Wild:

¡(θ) ¡(2θ) ¡(3θ) ¡(4θ) ¡(5θ) . . . , (11.2) ¡(θ) ¡ 2 (θ) ¡ 3 (θ) ¡ 4 (θ) ¡ 5 (θ) . . . . (11.
3)

The idea is now to get rid of the powers of 4 in the coefficients of S n by putting Z n (x) := 4S n x 4 . We refer to the polynomials Z n (x) as the zpread polynomials. It follows that Z n (¡(θ)) = ¡(nθ), (11.4) and we will see in a moment that Z n (x) ∈ Z[x]. We introduce the infinite zpread matrix (Z mn ) m,n≥1 such that

Z 1 (x) Z 2 (x) Z 3 (x) . . . = x x 2 x 3 . . . Z.
It turns out Z is a matrix of even pyramidal numbers of dimension ≥ 2.

Theorem 11.5. For m, n ≥ 1 we have Z mn = (-1) m+1 p

[2m]

n-m . Proof. Let k ≥ 0 be an integer. From the generating function of the pyramidal numbers we deduce

n≥k p [2m] n-k t n = n≥0 p [2m] n t n+k = t k 1 + t (1 -t) 2m+1 , and hence obtain n≥m (-1) m+1 p [2m] n-m t n = t m 1 + t (1 -t) 2m+1 .
For the bivariate generating function this leads us to n,m≥1

(-1) m+1 p [2m] n-m x m t n = - m≥1 (-x) m t m (1 -t) 2m 1 + t 1 -t = 1 + t 1 -t xt (1-t) 2 1 + xt (1-t) 2 = 1 + t 1 -t xt (1 -t) 2 + xt , (11.5)
where we assume -1 < x, t < 1 and by default p

¡(θ) ¡ 2 (θ) ¡ 3 (θ) ¡ 4 (θ) ¡ 5 (θ) . . . = ¡(θ) ¡(2θ) ¡(3θ) ¡(4θ) ¡(5θ) . . .         1 4 15 56 210 • • • -1 -6 -28 -120 1 8 45 -1 -10 1 • • • • • •         , (11.7) ¡(θ) ¡(2θ) ¡(3θ) ¡(4θ) ¡(5θ) . . . = ¡(θ) ¡ 2 (θ) ¡ 3 (θ) ¡ 4 (θ) ¡ 5 (θ) . . .         1 4 9 16 25 • • • -1 -6 -20 -50 1 8 35 -1 -10 1 • • • • • •        
, the latter matrix being Z, whose entries are relatively small. The former equation can be rephrased as

¡(θ) n = n k=1 (-1) k-1 2n n -k ¡(kθ).
(11.8)

The zpread matrix Z can also be understood as the transpose of a Riordan array.

Theorem 11.6. The transpose of the zspread matrix Z is the Riordan array

Z T = 1+x (1-x) 3 , -x (1-x) 2 with inverse (Z T ) -1 = BC 2 , -xC 2 = C , -xC 2 .
Proof. From Lemma 9.2 and Theorem 9.3 item (2) and (3) we deduce

1 -x (1 + x) 3 , x (1 + x) 2 * BC 2 , xC 2 = (1, x).
To adjust the sign in front of x we argue as follows. Let ρ ∈ xR x × and consider the formal pullback ρ * : R x → R x , f → f • ρ, which is an automorphism of the algebra (R x , •). Let us denote the product of two Riordan arrays (g 1 , f 1 ) and (g 2 , f 2 ) as (g

1 , f 1 ) * (g 2 , f 2 ) = (g 1 (g 2 • f 1 ), f 2 • f 1 ) =: (G, F ). It then follows that (ρ * g 1 , ρ * f 1 ) * (g 2 , f 2 ) = ((g 1 (g 2 • f 1 )) • ρ, (f 2 • f 1 ) • ρ) = (ρ * G, ρ * F ) . In the special case, when ρ(x) = -x we deduce 1 + x (1 -x) 3 , -x (1 -x) 2 * BC 2 , xC 2 = (1, -x).
But (1, -x) is an idempotent, i.e., (1, -x) * 2 = (1, x). Hence we obtain

(1, x) = 1 + x (1 -x) 3 , -x (1 -x) 2 * BC 2 , xC 2 * (1, -x) = 1 + x (1 -x) 3 , -x (1 -x) 2 * BC 2 , -xC 2 .
The verification of the well-known fact BC 2 = C we leave to the reader.

We refer to the following formula as the spreadometric series since it specializes to (1

-x 2 ) -1 when θ = π/2. Corollary 11.7. 1+x 1-x sin 2 (θ) (1-x) 2 +4x sin 2 (θ) = n≥0 sin 2 ((n + 1)θ)x n .
Proof. We use the language of Riordan arrays. Let us put ¡ θ (x) := n≥0 ¡(θ)(¡(θ)x) n = ¡(θ) 1-¡(θ)x . The right hand side of Equation (11.7) can be understood as

Z T -→ ¡ θ = 1 + x (1 -x) 3 , - x (1 -x) 2 -→ ¡ θ = ------------------→ 1 + x (1 -x) 3 ¡(θ) 1 + ¡(θ) x (1-x) 2 = ------------------→ 1 + x 1 -x ¡(θ) (1 -x) 2 + x ¡ (θ) .
Hence Equation (11.7) means

1 + x 1 -x ¡(θ) (1 -x) 2 + x ¡(θ) = n≥0 ¡((n + 1)θ)x n .
The claim follows by writing this in terms of sin 2 (θ).

To memorize S n (x) one can proceed as the follows. Start with the square numbers and put below it the sequence whose differences are made by the square numbers. Underneath that sequence put the sequence whose differences are made by the latter. Continue recursively. In the resulting pattern take every second row and arrange them into a triangle. The spread matrix S is then made by multiplying the mth row with (-4) m-1 , keeping in mind that the first row carries the row index m = 1: 1 4 9 16 25 36 49

• • • • • •            
.

To obtain Z one has to multiply the mth row with (-1) m-1 instead. The transpose of the spread matrix is the Riordan array S T = 1+x

(1-x) 3 , -4x

(1-x) 2 , whose inverse BC 2 , -xC 2 /4 does of course not have integer entries.

12. A conjecture of Goh and Wildberger Goh and Wildberger [START_REF] Goh | Spread polynomials, rotations and the butterfly effect[END_REF][START_REF] Wildberger | Advice for maths exploration -Chebyshev and spread polynumbers: the remarkable Goh factorization[END_REF] formulated an interesting conjecture concerning the factorizations of the spread polynomials over the ring Z[x]. We take here the liberty to formulate it in terms of the zpread polynomials and speculate a bit about numbers appearing in the coefficients of the factors. Using the zpread polynomials those become less obscure. As spread polynomials are encoding rotations in the plane5 we find these unexpected phenomena intriguing. In order to convince the reader we provide some empirical evidence: 5 -5x + x 2 2 (1 -x) 2 7 -14x + 7x 2 -x 3 2 2 -4x + x 2 2 9 10 11 12 3 -9x + 6x 2 -x 3 2 1 -3x + x 2 2 11 -55x + 77x 2 -44x 3 + 11x 4 -x 5 2 1 -4x + x 2 2 13 14 15 13 -91x + 182x 2 -156x 3 + 65x 4 -13x 5 + x 6 2 1 -6x + 5x 2 -x 3 2 1 -8x + 14x 2 -7x 3 + x 4 2 16 17 2 -16x + 20x 3 -8x 3 + x 4 2 17 -204x + 714x 2 -1122x 3 + 935x 4 -442x 5 + 119x -17x 7 + x 8 2 .

It comes as a surprise that the coefficients of ψ d (x) seem to almost coincide with certain columns of the array (11.9). Up to shifts, there are merely differences in the linear and/or constant coefficients. The column index in the array relevant for ψ d (x) seems to be the coefficient in front of -x φ(d)/2-1 . From the empirical data we guess it is φ(d) -µ(d) (see entry A053139 in [START_REF]The On-Line Encyclopedia of Integer Sequences[END_REF]). We would like to mention that it seems to be the case that for each prime p we have Φ p (x) = Φ 2p (2 -x). Furthermore, the evaluations of the Φ n (x) at the integers x = 0, 1, 2, 3, 4 look interesting. One may ask which of those values Φ n (x) appear for infinitely many n.

Clearly, for each n ≥ 1 we have that Z n maps the square [0, 4] 2 into itself and all the zeros of Z n (x) have to be in [0, 4]. The question for what x ∈ [0, 4] there are distinct n, m such that Z n (x) = Z m (x) deserves further attention. With regards to this let us mention some empirical observations. Conjecture 12.2. Let n ≥ 1 be an integer an ϕ = 1 + √ 5 /2 be the golden ratio. Then (1) 2 = Z n (2) if and only if n ≡ 1 mod 2,

  = B is the generating function for the central binomial coefficients. Now (2) and (4) follow from (1) and (3) by Lemma 9.2. Theorem 9.4. Let n, m be integers with m ≥ 0. Then j≥0 2j+n j-m x j = BC n (C -1) m . Proof. We generalize here a trick from [Pap99, Exercise 8.6]. Note that for a simple closed curve around 0 ∈ C we have for N, M ∈ Z that N M = x ∈ C with |x| < dist(0, γ) we can swap summation and integral in the following calculation:

  Corollary 9.5. B, xC 2 = 2n n-m n,m≥0 , BC, xC 2 = 2n+1 n-m n,m≥0 . Proof. According to the Theorem 9.4 we have n≥0

2π0

  cos n (θ) sin(θ) dθ = -cos n+1 (t) n+1 2π 0 = 0. Moreover, from the power reduction formulas (see Theorem 7.1) we obtain

  l < 0. The last expression in Equation (11.5) coincides with 4S x 4 , t . The mutual inverse linear systems corresponding to Corollary 11.4 turn out to be (11.6)

Conjecture 12. 1 .

 1 There are Φ d (x) ∈ Z[x], d ≥ 1, with deg Φ d (x) = φ(d) such that for each n ≥ 1 we have Z n (x) = d|n Φ d (x). For d ≥ 3 the polynomial Φ d (x) = ψ d (x) 2 is the square of an irreducible polynomial ψ d (x) = Z[x] with constant term ψ d (0) > 0.This constant term evaluates to ψ d (0) = d|n n d µ(d) (see entry A014963 in [OEI23].) Here φ(d) denotes the Euler totient function and µ(d) the Möbius µ-function. If p ≥ 5 is a prime then ψ p (1) = (-1) φ(p)/2 .

  x) x 4 -x (3 -x) 2 (2 -x) 2

  j 2 -i+1 . The 0th row does not yet fit into the

	pattern.
			i\j	0 1 2 3	4	5	• • • Entry in OEIS
		1th row 1 3 5 7	9	11 • • •	A005408
		2nd row 1 4 9 16 25 36	A000290
		3rd row 1 5 14 30 55 91	A000330
		4th row 1 6 20 50 105 196	A002415
		5th row 1 7 27 77 182 378	A005585
			• • •			• • •
	The number sequence p	[i] j	j≥0

  The first row is comprised of the central binomial coefficients and the second row of doubled Catalan numbers. The lower triangular part of M appears in the OEIS as entry https://oeis.org/A182411. Let us mention other empirical observations about the entries of M . It appears that M kl = (2k)!(2l)! k!l!(k+l)! and the textbook [UH39] asks in Exercise 4. on p.103 to show that this is an integer. Nowadays these numbers are called super Catalan numbers (see [Ges92, Bor08]) 4 . That our M kl are the super Catalan numbers can be easily verified

	2	6	20 70 252 924 • • •	
	2 6 20 10 12 20 40 90 220 2 4 10 28 84 264 4 6 12 28 72 198 70 28 28 40 70 140 308 252 84 72 90 140 252 504 924 264 198 220 308 504 924	          	.
	• • •		• • •

This is almost the generating function of the Lucas numbers L(z) = n≥0 Lnz n = 2-tz 1-tz-t 2 .

In the words of Fields medalist Edward Witten, M stands for Magical, Mystery or Membrane.

In[START_REF] Limanta | Super Catalan numbers and Fourier summation over finite fields[END_REF] the rational numbers in Theorem 10.1 have been named circular super Catalan numbers.

That is, they come straight from the almighty.

From ( * * ) n-1 and ( * * ) n-2 we now obtain

Corollary 11.2 (M. Hirschhorn). The bivariate generating function of the spread polynomials is given by

Corollary 11.3. For each integer n ≥ 0 we have S n (sin 2 (θ)) = sin 2 (nθ).

Proof. From the Addition Theorem cos(2z) = 1 -2 sin 2 (z) and Proposition 11.1 we deduce

The spread matrix S = (S mn ) m,n≥1 is then defined such that S 1 (x) S 2 (x) S 3 (x) . . . = x x 2 x 3 . . . S.

Corollary 11.4. For each integer n ≥ 1 we have 2 2n-2 sin 2n (θ) = n k=1 (-1) k-1 2n n-k sin 2 (kθ). Proof. We use Theorem 7.1(3) to deduce from Proposition 11.1

The first two terms cancel each other. In fact, we have (-1) j 2n j = 1 2 (-1) n (1 -1) 2n = 0,


A computational model for the investigation of phonon-limited charge transport in 2D systems J. Li * , C. Delerue † , Y.M. Niquet * , F. Triozon ‡ and L. Genovese * * L Sim, SP2M, UMR-E CEA/UJF-Grenoble 1, INAC, Grenoble, France † IEMN -Department ISEN, UMR CNRS 8520, Lille, France ‡ CEA-LETI, Campus MINATEC, 17 rue des Martyrs, 38054, Grenoble, France A computational model has been developed for phonon-limited charge transport in 2D systems. Currently, this model is used to study transport in graphene and silicon thin films. The results obtained so far show a good agreement with experiments.

The model is based on an atomistic approach considering full electronic band structure and all phonon branches. The electronic band structure is calculated with a tight-binding model (Fig. 1). The phonon dispersion is computed with the Vanderbilt force field model [1] for silicon thin films, and with a force constant method [2] for graphene using parameters fitted to the density function theory (Fig. 2). All possible electron-phonon scattering processes are searched in the energy window relevant for transport by checking if any phonon satisfies energy and momentum conservation rules. The rate of each scattering process is calculated by Fermi golden rule. These scattering rates are then used in Boltzmann transport equation to compute the response of the electron distribution to a low electric field, then the resistivity and the mobility are determined.

The tight-binding parameters of graphene and their dependence on strain have been extracted from ab-initio calculations. Generally, for graphene on a oxidized substrate, the resistivity consists of three major components:

ρ(T ) = ρ 0 + ρ ph (T ) + ρ SO (T ), (1)
where T is temperature, ρ 0 is due to impurities, ρ ph is due to the intrinsic phonons of graphene, and ρ SO is due to the surface optical phonons of the substrate [3]. Here, graphene on SiO 2 was considered. Fig. 3 compares the results of the simulation (represented by a green dotted line) with the experimental data [4] (represented by red markers) in a wide range of temperature and at three different electron densities.

It is clear that they agree very well with each other. Fig. 4 shows the temperature dependence of ρ ph and ρ SO . It can be concluded that the intrinsic phonon scattering dominates the temperature dependent resistivity of graphene at sufficiently high carrier concentration. Fig. 5 shows the carrier density dependence of ρ ph and ρ SO at 300K. ρ ph is asymmetric respect to the charge neutrality point, because the carrier-phonon coupling is not the same for hole states and electron states.

In silicon thin films, it is known that the conduction band acoustic deformation potential D ac has to be increased from D ac ∼ 10 eV in bulk to D ac ∼ 15 eV in thin films to match the experimental electron mobility [5]. We are therefore exploring this problem from a tight-binding perspective to get a better insight into the underlying physics. So far, we have investigated the mobility in [001] silicon thin film (thickness < 5 nm) at low carrier density (Fig. 6). More comprehensive data on thicker films and as a function of carrier density/effective electric field will be discussed at the conference.

This work was supported by the French National Research Agency (ANR QUASANOVA). Part of the calculations were run on the TGCC/Curie machine thanks to allocations from PRACE and GENCI.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Electronic band structure of a 4 nm thick silicon film calculated with tight-biding method.

Fig. 3 .

 3 Fig.3. Comparison of simulations (dotted lines by setting ρ0 at the solid lines) and experimental data (red markers)[4].

Fig. 4 .Fig. 5 .

 45 Fig. 4. ρ ph (square makers) and ρSO (circle markers) versus temperature at electron densities 1 and 13.6 × 10 12 cm 2 .

Fig. 6 .

 6 Fig. 6. Low field mobility of [001] silicon thin film in [1 10] (xx) and [110] (yy) directions. The gray dotted line is the mobility of bulk silicon.


Aymeric Magisson

Edwin B Clatworthy

Sajjad Ghojavand

Philippe Bazin

Valérie Ruaux

Eddy Dib

Svetlana Mintova

CO 2 adsorption behaviour of nanosized CHA zeolites synthesised in the presence of barium or calcium cations

Keywords: nanozeolite, chabazite, CO 2 adsorption, alkaline-earth metal cations, gas separation

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Zeolites are microporous solids possessing molecule-sized systems of channels and cages containing extra-framework cations and Brønsted acid sites. These characteristics confer remarkable properties which have led to their application in key adsorption, catalysis and cation exchange processes. 1 Over the last few decades there has been particular focus on the potential application of zeolites for the separation and capture of CO 2 in response to the increasing atmospheric concentration of greenhouse gases from anthropogenic sources. [2][3][4] Among the various solid adsorbents applicable for CO 2 separation, including metal organic frameworks 5 , carbon-based adsorbents 6,7 and ionic liquids 8 , zeolites possess an advantage of having tuneable capacity and selectivity towards CO 2 due to the relatively stronger interaction between CO 2 and the extra-framework cations compared to other small molecules such as N 2 and CH 4 . 9 The CO 2 adsorption behaviour of zeolites is influenced by the zeolite composition and its framework structure, directly related to the cationic environment and aluminium distribution of the zeolite. 10,11 The use of organic structure-directing agents (OSDAs) to synthesise specific zeolite topologies, compositions or particle morphologies is expensive due to the cost of the organic molecule which must be removed and destroyed by calcination after synthesis. Alternatively, zeolite syntheses using exclusively inorganic metal cations can direct the formation of different zeolite topologies and morphologies while lowering the number of synthetic steps. [12][13][14] Moreover, significantly higher surface area and reduced diffusion path length for guest molecules can be achieved by reducing the discrete particle size of the zeolite to the nanoscale, greatly improving their separation performance. 15 The use of alkali-metal cations in the zeolite precursor mixture has been investigated extensively [16][17][18][19][20] , but the impact of alkaline-earth metal cations and their influence as inorganic structure directing agents (ISDAs) has received less attention. [21][22][START_REF] Vega-Vila | Reaction Chemistry & Engineering[END_REF][START_REF] Rivera | [END_REF][25] This is likely a consequence of their poor solubility in highly basic media compared to alkali-metal cations. An important consideration is that alkaline-earth metal cations have a higher charge density than alkali-metal cations while possessing similar ionic radii. 26 This makes them interesting ISDA candidates because fewer extra-framework cations are required to equilibrate the charge imbalance created by the presence of framework aluminium. 27 The pioneering work of Barrer and co-workers explored the hydrothermal chemistry of mixtures of silicon and aluminium precursors (Si sol and glasses, Al hydroxide and alumina gels) in the presence of either Ca 2+ , Sr 2+ or Ba 2+ cations. [28][29][30] Their experiments yielded aluminosilicate products of varying purity and reproducibility, including chabazite zeolite in the presence of Sr 2+ . Similarly, the hydrothermal treatment of kaolin with solutions containing Ba(OH) 2 was found to result in the formation of zeolites; chabazite was formed when using mixtures of KOH and Ba(OH) 2 . 31,32 The influence of Sr 2+ cations on the aluminium distribution in the chabazite framework and crystallisation rate was investigated recently, revealing that the synthesis time could be reduced from weeks to hours. 33 Amongst the available literature, it has been reported that calcium-or barium-exchanged X-type faujasites and chabazites exhibit better adsorption capacity at very low CO 2 pressures, 34,35 yet a higher N 2 capacity was also observed for barium-exchanged X-type faujasites and chabazites at low pressures (<30 kPa). 26,36 In order to investigate the changes on both the synthesis and adsorption properties of nanozeolites prepared in the presence of alkaline-earth metal cations, we considered the chabazite framework because it exhibits a high adsorption capacity, and depending on the type of extra-framework cations present, a high selectivity towards CO 2 over CH 4 or N 2 . 37 The CHA framework is described by a threedimensional pore network containing super cages of (t-cha) connected by a double six ring (t-hpr) tiling. The diffusion of molecules through the cages is permitted by eight-membered rings (8MRs) which have a crystallographic free diameter of 3.8×3.8 Å. 38 Here we describe the synthesis of two nanosized chabazite zeolites employing higher chargedensity alkaline-earth metal cations (Ca 2+ , Ba 2+ ) in the synthetic precursor mixture (Na + , K + and Cs + ) previously reported. The N 2 and CO 2 adsorption behaviour of the Ca-, Ba-and Ref-CHA zeolites is investigated. An optimisation of the hydrothermal treatment (HT) was performed over different times to investigate the rate of crystallisation of the nanosized zeolites. The physical and chemical properties of the nanosized CHA zeolites were comprehensively characterised and the effects of the alkaline-earth metal cations were examined with reference to the nanosized CHA zeolite (Ref-CHA) prepared with Na + , K + and Cs + extra-framework cations only. 39

Material and Synthesis.

LUDOX AS-40 colloidal silica (SiO 2 , Sigma-Aldrich, 40 wt% suspension), sodium aluminate (NaAlO 2 , 50-56% Al 2 O 3 , 40-45% Na 2 O Sigma-Aldrich) with an approximation of the minimum aluminium content of 50% and Na 2 O of 43%, sodium hydroxide pellets (NaOH, Thermo Scientific, 98%), potassium hydroxide flakes (KOH, Sigma-Aldrich, 90%), caesium hydroxide solution (CsOH, Sigma-Aldrich, 50%), calcium hydroxide powder (Ca(OH) 2 , Sigma-Aldrich, 95%) and barium hydroxide powder (Ba(OH) 2 •8H 2 O, Thermo Scientific, 98%) were used as received. All water used was MilliQ water from the Direct-8 ZR0Q008WW by Merck Millipore. The synthesis of CHA nanozeolite in the presence of alkaline-earth metals was performed in a sealed 60 mL polypropylene bottle. Sodium aluminate (0.594 g) was dissolved in MilliQ water (4.153 g) with rapid stirring. After solubilisation of the sodium aluminate, sodium hydroxide (1.704 g), potassium hydroxide (0.778 g), and a solution of caesium hydroxide (0.503 g) were added sequentially. This was then followed by the addition of either calcium or barium hydroxide (0.065 g or 0.152 g, respectively). After 2 h of stirring at room temperature, colloidal silica (10.0 g) was then added dropwise with rapid stirring. The alkali aluminosilicate colloidal suspension was then aged with rapid stirring for 7 days. The molar ratio of the initial Ca 2+ -containing alkali aluminosilicate colloidal suspension is 0. Another synthesis of CHA nanozeolite was performed, identical in all aspects with the one described above except for a lowered amount of potassium hydroxide corresponding to an equivalent of 1.3 moles of K 2 O. Sodium aluminate (0.594 g) was dissolved in MilliQ water (4.175 g) with rapid stirring. After solubilisation of the sodium aluminate, sodium hydroxide (1.704 g), potassium hydroxide (0.674 g), and a solution of caesium hydroxide (0.499 g) were added sequentially. This was then followed by the addition of either calcium or barium hydroxide (0.065 g or 0.152 g, respectively). The rest of the protocol is identical to the one described first.

20Cs 2 O • 1.5K 2 O • 6.0Na 2 O • 0.20CaO • 16.0SiO 2 • 0.

Characterisation Techniques.

Powder X-Ray Diffraction (PXRD) patterns were collected on Anton Parr XRDynamic 500 diffractometer with a graphite monochromator in Bragg-Brentano configuration, Cu Kα 1 and Kα 2 radiation (λ = 1.5406, 1.5444 Å, 40 kV, 50 mA), and Pixos 2000 detector. The patterns were collected between 2θ of 5 and 60° with a step size of 0.03° and time per step of 492.606 s.

Inductively coupled plasma mass spectrometry (ICP-MS) measurements were performed on a 7900 ICP-MS from Agilent Technologies. 50 mg of sample was subjected to a digestive media (0.5 ml aqua regia at 1:3 HNO 3 :HCl V/V, and 3 ml HF 40%) then heated to 110 °C for 1 hour before being neutralised with 2 g of boric acid and made up to 100 ml with milli-Q water.

Scanning Electron Microscopy (SEM) images were taken using a Tescan Mira I LMH under 30 kV. Energy Dispersive X-Ray (EDX) measurements were recorded using two Bruker XFLASH 6/30 EDX cameras.

Magic-angle spinning nuclear magnetic resonance (MAS NMR) spectra of 27 Al and 29 Si nuclei were recorded with a single pulse on a Bruker Avance 500 MHz (11.7 T) spectrometer using 4 mm outer diameter zirconia rotors with a MAS frequency of 12 kHz. Radiofrequency pulses of 2.2 µs (30° flip angle) were used for the acquisition of the 29 Si MAS NMR spectra with a recycle delay of 20 s, tetramethylsilane (TMS) was used as a chemical shift reference. Radiofrequency pulses of 1.25 µs (15° flip angle) were used for the acquisition of the 27 Al MAS NMR spectra with a recycle delay of 1 s, aluminium nitrate (Al(NO 3 ) 3 ) was used as a chemical shift reference.

N 2 adsorption and desorption isotherms were recorded at -196 °C on a Micromeritics 3Flex Surface Characterisation unit, and CO 2 adsorption isotherms were recorded at 0, 25 and 50 °C on a Micromeritics ASAP 2020 Surface Characterisation unit. The samples were outgassed under vacuum at 350 °C for 8 hours prior to the analysis. The specific surface area was calculated using the Brunauer-Emmett-Teller equation applying the Rouquerol criteria, the micropore volumes were determined using the non-local density functional theory (NLDFT) method. [START_REF] Rouquerol | Studies in Surface Science and Catalysis[END_REF][START_REF] Brunauer | [END_REF] Calculations of the isosteric enthalpies of adsorption were performed using the Clausius-Clapeyron equation in VersaWin.

Thermogravimetric analysis (TGA) was performed using a STA 449 F3 Jupiter analyser, from 25 to 35 °C and held for 10 min with a flow of air (40 mL•min -1 ), from 35 to 800 °C at a heating rate of 10 °C•min -1 , held at 800 °C for 10 min under air after reaching the target temperature, then cooled down to 25 °C at 20 °C•min -1 .

In situ Fourier transform infrared (FTIR) spectroscopic measurements were performed using a self-supported pellet of a zeolite sample (approximately 20 mg with a diameter of 16 mm), recorded on a Thermo Scientific Nicolet iS50 FTIR spectrometer equipped with an MCT detector, at a spectral resolution of 4 cm -1 and 16 or 64 scans in absorption mode for the kinetic analysis and isotherm respectively. The experiments were performed in a "Carroucell" cell , attached to a vacuum system (greater than 10 -5 mbar) that allows up to 12 different samples to be analysed with the same conditions. 42 The cell is equipped with a heating system to activate the samples at 350 °C prior to the measurements, at a rate of 2.7 °C•min -1 , followed by heating at 350 °C for 4 h under high vacuum. All IR spectra related to CO 2 adsorption were recorded at room temperature. The molar absorption coefficients used to obtain both physisorption and chemisorption isotherms were 16 and 40 cm•µmol -1 , respectively. 42 The quantitative amount of CO 2 adsorbed on the chabazite samples was determined using the peak area of the adsorbate, the molar absorption coefficient, and the thickness of the sample wafer.

Results and Discussion

The pure phase OSDA-free reference CHA nanozeolite was prepared using a mixture of inorganic alkali-metal cations where the Na + facilitates the solubilisation of silica and the stabilisation of the colloidal suspension, the K + promotes the formation of the secondary building units (SBUs), and the Cs + facilitates the stabilisation of the long-range crystalline order of the framework. 39 The PXRD of the reference chabazite (Ref-CHA) was compared to a simulated phase of a fully crystalline chabazite (space group R-3m); no crystalline phases other than chabazite were observed (Figure 1). The Ca-CHA and Ba-CHA nanosized zeolite samples synthesised in the presence of alkaline-earth metal hydroxides (Ca 2+ and Ba 2+ ) are compared with the reference nanosized zeolite Ref-CHA. A mixture of alkali-metal cations (Na + , K + and Cs + ) only is used to direct the synthesis of the Ref-CHA nanosized zeolite. Because of the similar ionic radii of Ca 2+ and Na + , and those of Ba 2+ and K + , we may reasonably expect a partial substitution of the sodium and potassium in the unit cell by calcium and barium cations, respectively. 26 Moreover, the choice of a low calcium or barium cation ratio in the precursor mixture is in consideration of the solubility limits of those two alkaline-earth metal cation hydroxides in aqueous solutions. To study the kinetics of the crystallisation of the Ca-CHA and Ba-CHA zeolites, the hydrothermal treatment at 90 °C was performed over three different time periods (2, 4, and 10 h). The samples were compared to the reference chabazite which achieves full crystallinity after 7 h of HT (Ref-CHA). 39 The PXRD patterns of the Ca-CHA and Ba-CHA are presented in Figure 1. After 2 h of HT the sample Ca-CHA remains amorphous indicated by the absence of Bragg peaks. After 4 h of HT the relative crystallinity is equal to 49% and reaches 100% after 10 h (Figure 1a, Table S1). The crystallinity was determined by comparing the integrated area of the peaks at 9.36 °, 12.8 °, 30.45 ° and 31.44 ° 2 (Figure S1, Figure S2). In comparison, after 2 h of HT for the Ba-CHA sample, evidence of crystallisation is observed by the emergence of some weak Bragg peaks (Figure 1b). At 4 h of HT the PXRD pattern of the Ba-CHA indicates the completion of crystallisation. To verify that the Ba-CHA after 4 h was fully crystallised a sample was prepared with 10 h of HT. PXRD analysis revealed very little change in the intensity of the diffraction peaks indicating that 4 h of HT was enough to crystallise the nanosized chabazite in the presence of barium cations, with a relative crystallinity of 97% (Table S1, Figure 1). The reference sample (Ref-CHA) subjected to 4 h of HT was also synthesised to compare with the reference zeolite prepared with 7 h of HT. The PXRD pattern of the reference zeolite obtained after 4 h of HT shows the exact same Bragg peaks with near identical crystallinity (95%) to the reference nanosized CHA zeolite after 7 h of HT (Figure S3). In comparison, a reference chabazite subjected to 2 h of HT treatment was found to be completely amorphous, similar to the Ca-CHA zeolite obtained after 2 h of HT. From the PXRD results, calcium cations clearly have an inhibiting effect on the rate of crystallisation while barium cations appear to slightly improve the rate of crystallisation of the nanosized chabazite. This is coherent with literature work realised on a low silica zeolite. 25 The morphology of the fully crystalline samples Ba-CHA (4 h HT) and Ca-CHA (10 h HT) were also evaluated and compared to the Ref-CHA nanosized chabazite by SEM (Figure 2). For all of the samples, identical discrete spherical particles 150-300 nm in diameter composed of aggregated crystallites of 50-60 nm were observed. This is consistent with our previous observations investigating the effect of the ageing time of the alkali-metal aluminosilicate precursor suspension on the discrete particle size of CHA zeolites. 37,39,43 These results suggest that for the synthesis of nanozeolites conducted at high cation concentration and alkalinity, the effect of ageing time on the particle size remains largely unaffected by the addition of alkaline-earth metal cations employed at the concentrations used here. In comparison, the introduction of Ba 2+ to the synthesis mixture of LTL has been shown to result in a reduction of the zeolite crystal size from 2.5 µm to 140 nm. 44 Additional images at different magnifications are presented in Figure S4.

The unit cell composition and the alkaline-earth metal extra-framework cation content of the nanosized chabazite samples were determined by ICP-MS and EDX (Table 1 and Table S2). The Ca-CHA sample has similar Na + and K + content to the reference zeolite, but a decrease in the Cs + content is observed. A negligible amount of precipitated Ca(OH) 2 was identified by EDX mapping of the Ca-CHA sample; the impurity persisted after extensive washings steps (Figure S5). This can be explained by the low solubility of calcium hydroxide in the precursor suspension. In the sample Ba-CHA some of the Na + cations were substituted for Ba 2+ , while the amount of K + and Cs + cations remains unchanged compared to the Ref-CHA. In contrast to sample Ca-CHA, no precipitated Ba(OH) 2 in sample Ba-CHA was observed (Figure S5). The distribution and amount of cations of the Ca-CHA and Ba-CHA samples determined by ICP-MS and EDX analyses are similar. The presence of alkaline-earth metal cations reduces the amount of sodium in both the Ca-CHA and Ba-CHA samples. This suggests, that for the mixed cation (Na + , K + , Cs + ) system employed here, Ca 2+ and Ba 2+ may play a similar primary role to Na + , rather than the primary roles of K + or Cs + , by Al and low molecular weight Si species. This may be rationalised by the similar or higher charge density of Na + , Ca 2+ and Ba 2+ (24, 52 and 23 C•mm -3 ) cations compared to K + and Cs + (11 and 6 C•mm -3 ) cations, creating stronger interactions with negatively charged Al and Si species. 45 The observation of a similar K + cation content across all of the samples suggests that Ca 2+ and Ba 2+ cations do not act as inorganic directing agents in the synthesis of nanosized chabazite.

Due to the relatively low amount of alkaline-earth metal cations in the chabazite unit cell, two samples were prepared with a reduced amount of K 2 O (1.3 moles equivalents) to determine whether the Ca 2+ or Ba 2+ loading could be increased. The decision to decrease the K 2 O content, rather than the NaO 2 content, was made with consideration of the mixed cation (Na + , K + , Cs + ) system tolerating a lower K 2 O ratio. 43 This permits maintaining the same ageing and hydrothermal treatment conditions while avoiding the formation of other structural phases. The PXRD results, elemental composition, thermogravimetric analysis and N 2 sorption isotherms are presented in the Supporting Information (Figures S6-S9 and Table S3). The PXRD patterns of the chabazites prepared with 1.3 K 2 O in the presence of either Ca 2+ or Ba 2+ present identical peaks with slightly less intensity compared to the 1.5 K 2 O samples. A small amount of phillipsite is present in the Ca 2+ -containing chabazite, whereas the Ba 2+ -containing chabazite exhibits reflections consistent with the chabazite structure only. The elemental composition of these two samples is similar to the fully crystalline samples described before, but with slightly lower amounts of extra-framework cations; the amount of alkaline-earth metal extra-framework cations is not increased. The results indicate that the synthesis of chabazite in the mixed cation (Cs + , K + , Na + ) system is not receptive towards the incorporation of alkaline-earth metal cations into the crystalline zeolite product, including when the amount of K + present in the synthesis is decreased. The elemental composition of the samples is confirmed by ICP-MS analysis.

The Si/Al ratios of the chabazite samples determined by ICP-MS, EDX and 29 Si MAS NMR are summarized in Table S4. The Si/Al ratio of the Ref-CHA based on ICP-MS and EDX is equal to 2.2 and 2.0, respectively which is consistent with our previous work, and similar to the value calculated based on the 29 Si MAS NMR using the Engelhardt equation. 39,46 For the Ca-CHA sample a slight increase in the Si/Al ratio was reported based by EDX while the Ba-CHA exhibited a similar Si/Al ratio to the Ref-CHA. The Si/Al ratio determined by 29 Si MAS NMR was consistent across all samples. The deconvoluted 29 Si MAS NMR spectra of the Ca-CHA and Ba-CHA in comparison to the Ref-CHA are presented in Figure 3. The 29 Si MAS NMR spectra for all chabazite samples present five peaks at -90, -94, -99, -105 and -110 ppm corresponding to a silicon coordinated as

(Si[AlO] 4 ), (Si[SiO][AlO] 3 ), (Si[SiO] 2 [AlO] 2 ), (Si[SiO] 3 [AlO]
) and (Si[SiO] 4 ) aluminium atoms respectively. [START_REF] Ghojavand | Advanced Sustainable Systems[END_REF][START_REF] Ghojavand | [END_REF] The overall broadening of all the Q 4 species and mainly the Q 4 (nAl) where n is bigger than 2 in the Ca-CHA and Ba-CHA samples compared to the Ref-CHA, suggests an additional angular distortion in the framework. This is due to the partial substitution of monovalent cations by bivalent cations that are only stable when two or more aluminium are present in their surroundings. This effect is less pronounced for sample Ba-CHA, because of the higher electronegativity of Ca 2+ compared to Ba 2+ leading to a larger distortion of the negatively charged aluminium tetrahedra in the framework. 27 Al MAS NMR spectra of the Ref-CHA, Ca-CHA and Ba-CHA show a single peak around 57.9 ppm, indicating that all the aluminium atoms are tetrahedrally coordinated; no extra-framework aluminium is present (Figure S10).

The dehydration of Ca-, Ba-and Ref-CHA nanosized zeolites was followed by thermogravimetric analysis (Figure 4). The majority of the mass loss occurs between 35 °C and 250 °C attributed to the removal of water adsorbed on the external surface and within the pores of the zeolite. Above 300 °C the mass loss is minimal and appears to recommence decreasing above 350 °C, corresponding to 1.3% for Ca-CHA and 0.9% for Ba-CHA but only 0.6% for the Ref-CHA. This is likely due to the condensation of silanols and is consistent with the difference in the total population of silanols of the three zeolite samples as determined by FTIR spectroscopy, vide infra.

The total mass loss of all three samples at 300 °C is similar: 15.2%, 16.0% and 16.4% for the Ca-CHA, Ref-CHA and Ba-CHA, respectively. Nitrogen sorption analysis was performed to investigate the effect of the presence of Ca 2+ and Ba 2+ on the textural properties of the nanosized CHA samples. Isotherms measured at -196 °C are presented in Figure 5 and the corresponding BET specific surface areas and the microporous volumes are summarised in Table S5. The influence of the different alkali-metal cations on the N 2 and CO 2 adsorption behaviour of nanosized and micron-sized CHA has been reported in earlier works. 35,49 The BET specific surface areas and micropore volumes for all of the chabazite nanozeolite samples presented here are low due to cation gating (trapdoor) behaviour. The increase of the BET specific surface area of the Ca-CHA (290 m²•g -1 ) relative to the Ref-CHA (136 m 2 •g -1 )

can be attributed to the reduced amount of Cs + acting as a door-keeper cation, thus N 2 molecules are less restricted at accessing the pores at -196 °C. 50,51 In comparison, the slight increase for the Ba-CHA (169 m²•g -1 ) may be due to the stronger interaction of N 2 with Ba 2+ . 35 The BET specific surface areas of the Ca-CHA (172 m²•g -1 ) and Ba-CHA (155 m²•g -1 ) synthesised with 1. To understand the influence of Ca 2+ or Ba 2+ extra-framework cations on the CO 2 adsorption behaviour of the nanosized CHA, volumetric adsorption and in situ FTIR analyses were performed.

The CO 2 adsorption isotherms measured at 0, 25 and 50 °C are presented in Figure 6. The adsorption capacities are summarized in Table S6. The CO 2 adsorption behaviour of the nanosized CHA zeolite samples is broadly similar at 25 and 50 °C; some differences are observed for the isotherms recorded at 0 °C. The Ref-CHA exhibited the highest CO 2 capacity closely followed by the alkaline-earth metal containing CHA nanozeolites; the Ba-CHA synthesised with a 1.5 K 2 O molar oxide ratio exhibited a notably lower adsorption capacity at 0 °C compared to the other samples. Similar behaviour has been observed for micron-sized Ba-containing CHA, however, the degree of Ba 2+ exchange was much higher. 35 The marked differences between the Ba-CHA samples may be due to the differences in the nature of the adsorption sites at 0 °C ascribed to differences in the Ba 2+ site preferences. 52 For the Ca-containing nanosized CHA both samples exhibit similar adsorption enthalpy profiles offset by ~5.5-5.7 kJ/mol; the higher values for the Ca-CHA (1.5K 2 O) are likely due to the presence of a relatively greater amount of higher charge density cations (Ca 2+ , Na + and K + ) compared to Cs. Higher contents of Ca 2+ and Ba 2+ and determination of the cation locations by Rietveld analysis may provide further insight, but is beyond the scope of this study. For all of the CHA nanozeolite samples, it is expected that all of the Cs + extra-framework cations are located within the 8MRs, while Na + and Ca 2+ are located within the t-cha cages near the double six ring prisms. 37,53 Since Ba 2+ possesses a similar ionic radius to K + it is likely both cations fill positions near the double six ring prisms, and possibly within the 8MRs. 52 The CO 2 adsorption results suggest that the presence of alkaline-earth metal cations in the mixed (Cs + , K + , Na + ) cation CHA may reduce the adsorption enthalpy, but is dependent on their loading and presumably spatial arrangement.

The silanol region of FTIR spectra of activated Ca-CHA, Ba-CHA and Ref-CHA samples prior to CO 2 delivery is presented in Figure 7a; two major silanol peaks are observed. The O-H stretching frequency of the isolated or weakly bonded silanols in the range 3760-3700 cm -1 and the O-H stretching frequency of the moderately and strongly bonded silanols in the range 3700-3100 cm -1 are present. 54,55 The Ref-CHA sample has the lowest amount of silanols (464 cm -1 •g -1 ), determined by integrating a wide range of silanol sites across the entire range 3800-3100 cm -1 . Two bands at 3742 and 3726 cm -1 are clearly visible in the weakly H-bonded silanol region, while the Hbonded silanols of moderate strength are less intense for the Ref-CHA compared to those of the alkaline-earth metal cation containing chabazites. 56 The total concentration of silanol sites shows slight variations across the Ca-CHA, Ba-CHA and Ref-CHA samples (655, 632 and 644 cm -1 •g -1 , respectively) which may be a reflection of the slight variations in extra-framework cation content.

The CO 2 adsorption on the samples was investigated by in situ FTIR spectroscopy in a custom cell allowing simultaneous analysis under identical conditions (pressure and temperature); the full spectra are presented in the supporting information. 42 Two experiments were performed across a low pressure range of CO 2 , between near-vacuum and 13.3 kPa. The physisorbed CO 2 species are quantitatively evaluated based on the gas-phase asymmetric stretching (ν 3 ) bands at 2346 cm -1 and 2280 cm -1 that correspond to the physisorbed 12 CO 2 and 13 CO 2 species, respectively. 57 The chemisorbed CO 2 is identified by the complex series of weak bands in the range 1750-1300 cm - 1 . 56,58 After the first dose of CO 1609, 1381 and 1329 cm -1 are observed. Other bands at 1720 and 1351 cm -1 become more pronounced after delivery of 0.08 kPa CO 2 , while the bands at 1402 cm -1 and 1560 cm -1 appear at 1.06 kPa CO 2 ; the bands at 1641 and 1740 cm -1 are less intense. The FTIR spectra of Ba-CHA are similar to those of the Ref-CHA. A pair of bands at 1681 and 1352 cm -1 along with bands at 1655, 1381, and 1314 cm -1 emerge after the first CO 2 delivery. After introducing 0.01 kPa of CO 2 , a weak band at 1559 cm -1 appears. At 0.27 kPa CO 2 the band at 1742 cm -1 appears while the bands at 1720 and 1330 cm -1 are less intense. Above 2.27 kPa CO 2 , the bands at 1408 and 1640 cm -1 become more intense.

Changes in the silanol region (3760-3675 cm -1 ) of the IR spectra are observed as well. A decrease of the band at 3741 cm -1 and increase of bands at 3703 cm -1 (ν 3 + ν 1 ) and 3597 cm -1 (ν 3 + 2ν 2 ) are characteristic for weak interactions between the CO 2 and the hydroxyl groups of the activated zeolites (Figure S12 and S13). Above low pressure of CO 2 (>1 kPa), the 12 CO 2 physisorption band is saturated and cannot be used for quantification. Consequently, the band of the 13 CO 2 isotope with a natural abundance of 1.11% was used to determine the physisorbed CO 2 . 59 The three isotherms of each sample, representing the physisorbed, chemisorbed, and total amount of adsorbed CO 2 , measured at 25 °C are presented in Figure 7b. In the low-pressure range (0-1 kPa), samples Ca-CHA and Ba-CHA have very similar CO 2 uptake to the Ref-CHA zeolite. For the physisorbed CO 2 species, a clear trend is observed as follows Ca-CHA > Ref-CHA > Ba-CHA which may be ascribed to the higher charge density of Ca 2+ . Interestingly an opposite trend is seen for the CO 2 chemisorbed species. At equilibrium after 90 minutes, the proportion of chemisorbed species over the total amount of CO 2 adsorbed corresponds to 10.9% for the Ca-CHA, 12.7% for the Ref-CHA and 13.7% for the Ba-CHA (Table S7), which is significantly higher than the ratio of CO 2 chemisorbed species over total adsorbed species at higher pressures. The value of the Ref-CHA is comparable to that of the as-prepared chabazite in previous work at near-identical pressure. 49 The high relative amount of chemisorbed species at low pressure is due to the coordination of CO 2 to strongly adsorbing sites. 60 Alkaline-earth metal-containing zeolites have received less attention compared to their alkali-metal counterparts, despite their potential for various applications due to the tuneable adsorption behaviour. However, incorporating alkaline-earth metals into zeolites through direct synthesis methods remains challenging. Microscopic and PXRD analyses reveal that Ba-CHA nanocrystals synthesized for 4 hours exhibit higher crystallinity compared to reference nanosized chabazite (Ref-CHA), while maintaining the same size and morphology. This implies that only 4 hours of HT are needed to fully crystallise nanosized CHA in the presence of Ba 2+. On the other hand, the crystallisation rate of nanosized Ca-CHA is slower and require 10 hours of HT. Incorporating alkaline-earth metal cations into the aluminosilicate precursor mixture allows for the substitution of some Na + and Cs + cations, resulting in increased N 2 adsorption and higher specific BET surface area. While the crystallite size and particle morphology remain similar across all of the samples, the presence of alkaline-earth metal cations creates subtle differences to the silanol environments.

Conclusions

Description of the CO 2 adsorption behaviour and isosteric enthalpy of adsorption is challenging due to the complexity of four different cations present in the chabazite crystals, however, the majority of the Ca-and Ba-containing samples exhibit high CO 2 adsorption capacity. An interesting effect of the alkaline-earth metal cations is observed in the nanosized chabazite samples under a dilute CO 2 atmosphere (<1 kPa). The Ca-CHA zeolite exhibits a higher amount of physisorbed CO 2 and a lower amount of carbonates, while the opposite effect is observed for the Ba-CHA sample. This study demonstrates that extra-framework alkaline-earth metal cations can be incorporated into the CHA zeolite using a direct synthesis approach, while also affecting the rate of zeolite crystallisation. The use of alkaline-earth metal cations for preparation of nanosized zeolites could be expanded to other structures with potential applications in gas separation processes.

  70Al 2 O 3 • 140.0H 2 O, and the Ba 2+ -containing alkali aluminosilicate colloidal suspension is 0.20Cs 2 O • 1.5K 2 O • 6.0Na 2 O • 0.20BaO • 16.0SiO 2 • 0.70Al 2 O 3 • 140.0H 2 O. A static hydrothermal treatment was performed for 2, 4 and 10 h in a preheated oven at 90 °C. The recovered material was washed with hot (90 °C) MilliQ water by centrifugation until the pH of the supernatant was neutral. The washed nanozeolite was then freeze-dried overnight using a Beta 1-8 LSCbasic from Martin Christ.

Figure 1 .

 1 Figure 1. PXRD patterns of the nanosized chabazite zeolite samples (a) Ca-CHA, (b) Ba-CHA synthesized for 2, 4 and 10 h.

Figure 2 .

 2 Figure 2. SEM images of (a) Ref-CHA (7 h HT) (b) Ref-CHA (4 h HT) (c) Ca-CHA (10 h) (d) Ba-CHA (4 h). Scale bar is 1 µm.

Figure 3 :

 3 Figure 3: 29 Si MAS NMR spectra of samples (a) Ba-CHA (orange) and (b) Ca-CHA (red) in comparison to the Ref-CHA (black).

  3 K 2 O are slightly higher than that of the Ref-CHA made with 1.5 K 2 O. The difference between the two Ca 2+containing CHA can be explained by the changes in the Cs + extra-framework cation content, and the absence of Ca(OH) 2 in the 1.3 K 2 O sample. The micropore volume of each of the samples is low; a minor increase is observed for the Ca-CHA sample synthesised in the presence of 1.5 K 2 O ascribed to the lower extra-framework Cs + content.

Figure 4 .

 4 Figure 4. TG (solid) and dTG (dotted) curves of nanosized samples Ca-CHA (orange), Ba-CHA (red) and Ref-CHA (black).

Figure 5 .

 5 Figure 5. N 2 adsorption and desorption isotherms of the nanosized Ref-CHA (black), Ca-CHA (orange), and Ba-CHA (red) samples measured at -196 °C. Filled and empty squares represent the adsorption and desorption isotherms respectively.

Figure 6 :

 6 Figure 6: CO 2 adsorption isotherms recorded at (a) 0 °C, (b) 25 °C, and (c) 50 °C on the nanosized CHA zeolites: Ref-CHA (black circles), Ca-CHA 1.5K 2 O (oranges squares), Ba-CHA 1.5K 2 O (red diamonds), Ca-CHA 1.3K 2 O (blue triangles), and Ba-CHA 1.3K 2 O (green inverted triangles).

Figure 7 .

 7 Figure 7. (a) FTIR spectra in the silanol region (3250-3800 cm -1 ) of activated Ref-CHA (black), Ca-CHA (orange) and Ba-CHA (red), prior to CO 2 addition. (b) In situ FTIR CO 2 adsorption isotherms of nanosized samples Ref-CHA (black), Ca-CHA (orange) and Ba-CHA (red) recorded at 1 kPa CO 2: physisorbed (dashed line) and chemisorbed (dotted line) CO 2 . The spectra of samples were recorded at 25 °C after activation at 350 °C.

Table 1 .

 1 Chemical composition of Ref-CHA, Ca-CHA and Ba-CHA determined by ICP-MS. *Due to the co-existence of Ca(OH)2 precipitates the amount of Ca 2+ cations present in the CHA unit cell was estimated.

	Sample	Unit cell formula (ICP-MS)
	Ref-CHA	(Na 2.5 K 5.7 Cs 3.1 )Si 24.7 Al 11.3 O 72
	Ca-CHA*	(Ca 0.5x Na 3-x K 6.0 Cs 2.7 )Si 24.3 Al 11.7 O 72
	Ba-CHA	(Ba 0.5 Na 1.6 K 5.6 Cs 3.2 )Si 24.6 Al 11.4 O 72

  2 , the Ref-CHA sample presents a pair of bands at 1683 and 1350 cm -1 , as well as weaker bands at 1720, 1654, 1380 and 1330 cm -1 . At 0.26 kPa, additional bands appear at 1641, 1560 and 1411 cm -1 . Above 0.80 kPa these bands become more distinct, including the appearance of a weak band at 1740 cm -1 . The bands in the FTIR spectra of samples Ca-CHA and Ba-CHA are similar to the Ref-CHA, however, the evolution behaviour and final intensities are different, indicating changes to the distribution of the coordination environments of the chemisorbed CO 2 . For sample Ca-CHA, the intensity of the main chemisorption bands appears more evenly distributed. After the initial introduction of CO 2 (0.002 kPa), bands at 1683, 1653,

Acknowledgements

Co-funded by the European Union (ERC, ZEOLIghT, 101054004). Views and opinions expressed are, however, those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. The authors acknowledge the support of the Centre for Zeolites and Nanoporous Materials, Label of Excellence, Normandy Region (CLEAR) and Marie Lozier for performing adsorption measurements.