The Generating Function of Ternary Trees and Continued Fractions

Gessel, Ira M.; Xin, Guoce

doi:10.37236/1079

Cited by 42 publications

(54 citation statements)

References 21 publications

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“…in the ideal generated by (4) and (5). The coefficient p 2 does not vanish for j = 2n, and thus, together with the additional initial value c 2,4 = 1, the recurrence (6) allows to compute the values c n,2n .…”

Section: An Old Problem By George Andrewsmentioning

confidence: 99%

Advanced Computer Algebra for Determinants

Koutschan

Thanatipanonda

2013

Ann. Comb.

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Abstract. We prove three conjectures concerning the evaluation of determinants, which are related to the counting of plane partitions and rhombus tilings. One of them was posed by George Andrews in 1980, the other two were by Guoce Xin and Christian Krattenthaler. Our proofs employ computer algebra methods, namely, the holonomic ansatz proposed by Doron Zeilberger and variations thereof. These variations make Zeilberger's original approach even more powerful and allow for addressing a wider variety of determinants. Finally, we present, as a challenge problem, a conjecture about a closed-form evaluation of Andrews's determinant. Mathematics Subject Classification (2010). Primary 33F10; Secondary 15A15, 05B45.

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Section: An Old Problem By George Andrewsmentioning

confidence: 99%

Advanced Computer Algebra for Determinants

Koutschan

Thanatipanonda

2013

Ann. Comb.

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“…There are three simple rules to transform the determinant [D(x, y)] n to another determinant. See [12] for applications. The constant rules are clear.…”

Section: Basic Rulesmentioning

confidence: 99%

“…The proof of the remaining three equalities is almost the same, except that we need to use (13) instead of (12).…”

Section: 2mentioning

confidence: 99%

“…Our point of departure is that such lattice paths have quadratic generating functions, so that Sulanke-Xin's continued fraction method applies to evaluate their Hankel determinants. In [18] Sulanke and Xin used Gessel-Xin's continued fraction method [12] to evaluate the Hankel determinants for lattice paths with step set {(1, 1), (3,0), (1, −1)}. Proposition 1.…”

Section: Introductionmentioning

confidence: 99%

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Hankel determinants and shifted periodic continued fractions

Wang

Xin

Zhai³

2019

Advances in Applied Mathematics

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Sulanke and Xin developed a continued fraction method that applies to evaluate Hankel determinants corresponding to quadratic generating functions. We use their method to give short proofs of Cigler's Hankel determinant conjectures, which were proved recently by Chang-Hu-Zhang using direct determinant computation. We find that shifted periodic continued fractions arise in our computation. We also discover and prove some new nice Hankel determinants relating to lattice paths with step set {(1, 1), (q, 0), ( −1, −1)} for integer parameters m, q, . Again shifted periodic continued fractions appear.Mathematic subject classification: Primary 15A15; Secondary 05A15, 11B83.

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“…Example 7.5. As an illustration of the type of series Theorem 7.4 refers to consider the following expansion from[20][Example 9.2] …”

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confidence: 99%

The Structure of Bivariate Rational Hypergeometric Functions

Cattani¹,

Dickenstein²,

Villegas³

2010

International Mathematics Research Notices

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Abstract. We describe the structure of all codimension-two lattice configurations A which admit a stable rational A-hypergeometric function, that is a rational function F all whose partial derivatives are non zero, and which is a solution of the A-hypergeometric system of partial differential equations defined by Gel'fand, Kapranov and Zelevinsky. We show, moreover, that all stable rational A-hypergeometric functions may be described by toric residues and apply our results to study the rationality of bivariate series whose coefficients are quotients of factorials of linear forms.

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The Generating Function of Ternary Trees and Continued Fractions

Cited by 42 publications

References 21 publications

Advanced Computer Algebra for Determinants

Advanced Computer Algebra for Determinants

Hankel determinants and shifted periodic continued fractions

The Structure of Bivariate Rational Hypergeometric Functions

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