Some determinants of path generating functions, II

Krattenthaler, Christian; Yaqubi, Daniel

doi:10.48550/arxiv.1802.05990

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Repeated Application Of Proposition 6 Gives1

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2019

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“…Another possible research project is to study the nice Hankel determinants in [9] and [16], where the generating functions are quadratic. Can we generalize the step set and still obtain nice Hankel determinants?…”

Section: Repeated Application Of Proposition 6 Givesmentioning

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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Section: Repeated Application Of Proposition 6 Givesmentioning

confidence: 99%

“…For instance, Motzkin numbers count lattice paths from (0, 0) to (n, 0) with step set {(1, 1), (1, 0), (1, −1)} that never go below the horizontal axis. Partial Motzkin paths (similar to Motzkin paths but from (a, 0) to (n, b)) were considered in [9] and [16], where many nice determinant formulas were discovered.…”

Section: Introductionmentioning

confidence: 99%