Sums of powers of Catalan triangle numbers

Miana, Pedro J.; Ohtsuka, Hideyuki; Romero, Natalia

doi:10.1016/j.disc.2017.05.006

Cited by 9 publications

(10 citation statements)

References 22 publications

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“…Remark The recurrence relation () in Proposition 4.4 can be rewritten as

4^{k} B_{a, k} = 4^{k - 1} B_{a - 1, k - 1} + 2 (4^{k - 1} B_{a, k - 1}) + 4^{k - 1} B_{a + 1, k - 1} for k \in N and a \in Z,

that is, the numbers

4^{k} B_{a, k}

satisfy the same recurrence relation as elements of the Catalan triangle which Shapiro introduced in [23] and as elements of other Catalan triangles, for example, see [20, 21]. …”

Section: Moment Analysis On the Diagonalmentioning

confidence: 98%

“…These sums satisfy a much simpler recurrence relation than the coefficients b l a,k themselves, where B a,k = B −a,k as a result of the symmetry property (4.3). [20,21].…”

Section: Moment Analysis On the Diagonalmentioning

confidence: 99%

“…Since this fixes the boundary values of our recursion, it suffices to verify that () satisfies the recurrence relation (). This can be done by observing that the combinatorial numbers

C_{m, l}

defined, for

m \in N

and

l \in {double-struckN}_{0}

, by

C_{m, l} = \frac{m - 2 l}{m} (\frac{0pt}{m})

satisfy the recurrence relation

C_{m + 2, l + 1} = C_{m, l - 1} + 2 C_{m, l} + C_{m, l + 1} for m, l \in N,

see [20, Proposition 2.1], and by noting that

C_{2 k + 1, k + a} = (\frac{0pt}{2 k}) - (\frac{0pt}{2 k}) and C_{2 k + 1, k + a + 1} = (\frac{0pt}{2 k}) - (\frac{0pt}{2 k}) .

Thus, the recurrence relation () is a consequence of (), and we obtain that

4^{k} B_{a, k} = \frac{1}{2} (C_{2 k + 1, k + a} - C_{2 k + 1, k + a + 1}) .

Finally, we conclude that, for

k \in

…”

Section: Moment Analysis On the Diagonalmentioning

confidence: 99%

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A semicircle law and decorrelation phenomena for iterated Kolmogorov loops

Habermann

2020

J. London Math. Soc.

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We consider a standard one-dimensional Brownian motion on the time interval [0,1] conditioned to have vanishing iterated time integrals up to order N. We show that the resulting processes can be expressed explicitly in terms of shifted Legendre polynomials and the original Brownian motion, and we use these representations to prove that the processes converge weakly as N → ∞ to the zero process. This gives rise to a polynomial decomposition for Brownian motion. We further study the fluctuation processes obtained through scaling by √ N and show that they converge in finite-dimensional distributions as N → ∞ to a collection of independent zero-mean Gaussian random variables whose variances follow a scaled semicircle. The fluctuation result is a consequence of a limit theorem for Legendre polynomials which quantifies their completeness and orthogonality property. In the proof of the latter, we encounter a Catalan triangle.

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“…Remark The recurrence relation () in Proposition 4.4 can be rewritten as

4^{k} B_{a, k} = 4^{k - 1} B_{a - 1, k - 1} + 2 (4^{k - 1} B_{a, k - 1}) + 4^{k - 1} B_{a + 1, k - 1} for k \in N and a \in Z,

that is, the numbers

4^{k} B_{a, k}

satisfy the same recurrence relation as elements of the Catalan triangle which Shapiro introduced in [23] and as elements of other Catalan triangles, for example, see [20, 21]. …”

Section: Moment Analysis On the Diagonalmentioning

confidence: 98%

“…These sums satisfy a much simpler recurrence relation than the coefficients b l a,k themselves, where B a,k = B −a,k as a result of the symmetry property (4.3). [20,21].…”

Section: Moment Analysis On the Diagonalmentioning

confidence: 99%

“…Since this fixes the boundary values of our recursion, it suffices to verify that () satisfies the recurrence relation (). This can be done by observing that the combinatorial numbers

C_{m, l}

defined, for

m \in N

and

l \in {double-struckN}_{0}

, by

C_{m, l} = \frac{m - 2 l}{m} (\frac{0pt}{m})

satisfy the recurrence relation

C_{m + 2, l + 1} = C_{m, l - 1} + 2 C_{m, l} + C_{m, l + 1} for m, l \in N,

see [20, Proposition 2.1], and by noting that

C_{2 k + 1, k + a} = (\frac{0pt}{2 k}) - (\frac{0pt}{2 k}) and C_{2 k + 1, k + a + 1} = (\frac{0pt}{2 k}) - (\frac{0pt}{2 k}) .

Thus, the recurrence relation () is a consequence of (), and we obtain that

4^{k} B_{a, k} = \frac{1}{2} (C_{2 k + 1, k + a} - C_{2 k + 1, k + a + 1}) .

Finally, we conclude that, for

k \in

…”

Section: Moment Analysis On the Diagonalmentioning

confidence: 99%

See 1 more Smart Citation

A semicircle law and decorrelation phenomena for iterated Kolmogorov loops

Habermann

2020

J. London Math. Soc.

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“…They play a very interesting role to describe the sums of cubes of Catalan triangle numbers, as the next result shows. See proofs and more details in ( [22], Section 3).…”

Section: Moments Of Squares and Cubes Of Catalan Triangle Numbersmentioning

confidence: 99%

“…In previous papers, the authors have considered some particular cases of these sums: for j ¼ 1 and m ¼ 0 in [14], for j ¼ 2 in [12,13], and for j ¼ 3 and m ¼ 0 in [22]. In [7], the authors solved a conjecture posed in [22] about divisibility properties in the case m ¼ 0. However, there are no results in the literatures for moments for j > 2.…”

Section: Introductionmentioning

confidence: 99%

Moments of Catalan Triangle Numbers

Miana¹,

Romero²

2020

Number Theory and Its Applications

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In this chapter, we consider the Catalan numbers, C n ¼ 1 nþ1 2n n , and two of their generalizations, Catalan triangle numbers, B n,k and A n,k , for n, k ∈ . They are combinatorial numbers and present interesting properties as recursive formulae, generating functions and combinatorial interpretations. We treat the moments of these Catalan triangle numbers, i.e., with the following sums: P n k¼1 k m B j n,k , P nþ1 k¼1 2k À 1 ðÞ m A j n,k , for j, n ∈  and m ∈  ∪ 0 fg. We present their closed expressions for some values of m and j. Alternating sums are also considered for particular powers. Other famous integer sequences are studied in Section 3, and its connection with Catalan triangle numbers are given in Section 4. Finally we conjecture some properties of divisibility of moments and alternating sums of powers in the last section.

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Further identities on Catalan numbers

Chu

2018

Discrete Mathematics

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Sums of powers of Catalan triangle numbers

Cited by 9 publications

References 22 publications

A semicircle law and decorrelation phenomena for iterated Kolmogorov loops

A semicircle law and decorrelation phenomena for iterated Kolmogorov loops

Moments of Catalan Triangle Numbers

Further identities on Catalan numbers

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