scholarly journals Hankel determinants of sums of consecutive weighted Schröder numbers

2012 ◽  
Vol 437 (9) ◽  
pp. 2285-2299 ◽  
Author(s):  
Sen-Peng Eu ◽  
Tsai-Lien Wong ◽  
Pei-Lan Yen
Keyword(s):  
Hankel Determinants ◽  
Schröder Numbers

scholarly journals Hankel determinants of linear combinations of moments of orthogonal polynomials, II

2021 ◽  
Author(s):  
C. Krattenthaler
Keyword(s):  
Orthogonal Polynomials ◽  
Matrix Theory ◽  
Binomial Coefficients ◽  
Linear Recurrence ◽  
Hankel Determinants ◽  
Motzkin Numbers ◽  
Schröder Numbers ◽  
Delannoy Numbers ◽  
Dodgson Condensation ◽  
Full Proof

AbstractWe present a formula that expresses the Hankel determinants of a linear combination of length $$d+1$$ d + 1 of moments of orthogonal polynomials in terms of a $$d\times d$$ d × d determinant of the orthogonal polynomials. This formula exists somehow hidden in the folklore of the theory of orthogonal polynomials but deserves to be better known, and be presented correctly and with full proof. We present four fundamentally different proofs, one that uses classical formulae from the theory of orthogonal polynomials, one that uses a vanishing argument and is due to Elouafi (J Math Anal Appl 431:1253–1274, 2015) (but given in an incomplete form there), one that is inspired by random matrix theory and is due to Brézin and Hikami (Commun Math Phys 214:111–135, 2000), and one that uses (Dodgson) condensation. We give two applications of the formula. In the first application, we explain how to compute such Hankel determinants in a singular case. The second application concerns the linear recurrence of such Hankel determinants for a certain class of moments that covers numerous classical combinatorial sequences, including Catalan numbers, Motzkin numbers, central binomial coefficients, central trinomial coefficients, central Delannoy numbers, Schröder numbers, Riordan numbers, and Fine numbers.

scholarly journals A Simple Proof of the Aztec Diamond Theorem

10.37236/1915 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Sen-Peng Eu ◽  
Tung-Shan Fu
Keyword(s):  
Simple Proof ◽  
Lattice Paths ◽  
Hankel Determinants ◽  
Domino Tilings ◽  
Aztec Diamond ◽  
Schröder Numbers

Based on a bijection between domino tilings of an Aztec diamond and non-intersecting lattice paths, a simple proof of the Aztec diamond theorem is given by means of Hankel determinants of the large and small Schröder numbers.

The sharp bounds of the second and third Hankel determinants for the class 𝓢𝓛*

2020 ◽  
Vol 70 (4) ◽  
pp. 849-862
Author(s):  
Shagun Banga ◽  
S. Sivaprasad Kumar
Keyword(s):  
Triangle Inequality ◽  
Starlike Functions ◽  
Sharp Bound ◽  
The Novel ◽  
Sharp Bounds ◽  
Hankel Determinants ◽  
Lemniscate Of Bernoulli ◽  
Carathéodory Functions ◽  
The Right

AbstractIn this paper, we use the novel idea of incorporating the recently derived formula for the fourth coefficient of Carathéodory functions, in place of the routine triangle inequality to achieve the sharp bounds of the Hankel determinants H3(1) and H2(3) for the well known class 𝓢𝓛* of starlike functions associated with the right lemniscate of Bernoulli. Apart from that the sharp bound of the Zalcman functional: $\begin{array}{} |a_3^2-a_5| \end{array}$ for the class 𝓢𝓛* is also estimated. Further, a couple of interesting results of 𝓢𝓛* are also discussed.

Hankel determinants of second and third order for the class 𝓢 of univalent functions

2021 ◽  
Vol 71 (3) ◽  
pp. 649-654
Author(s):  
Milutin Obradović ◽  
Nikola Tuneski
Keyword(s):  
Unit Disc ◽  
Univalent Functions ◽  
Upper Bounds ◽  
Third Order ◽  
Hankel Determinants

Abstract In this paper we give the upper bounds of the Hankel determinants of the second and third order for the class 𝓢 of univalent functions in the unit disc.

Third Hankel determinants for two classes of analytic functions with real coefficients

2021 ◽  
Vol 33 (4) ◽  
pp. 973-986
Author(s):  
Young Jae Sim ◽  
Paweł Zaprawa
Keyword(s):  
Analytic Functions ◽  
Univalent Functions ◽  
Upper Bounds ◽  
Hankel Determinant ◽  
Lower And Upper Bounds ◽  
Hankel Determinants ◽  
The Third ◽  
Typically Real ◽  
Class Of Functions ◽  
Typically Real Functions

Abstract In recent years, the problem of estimating Hankel determinants has attracted the attention of many mathematicians. Their research have been focused mainly on deriving the bounds of H 2 , 2 {H_{2,2}} or H 3 , 1 {H_{3,1}} over different subclasses of 𝒮 {\mathcal{S}} . Only in a few papers third Hankel determinants for non-univalent functions were considered. In this paper, we consider two classes of analytic functions with real coefficients. The first one is the class 𝒯 {\mathcal{T}} of typically real functions. The second object of our interest is 𝒦 ℝ ⁢ ( i ) {\mathcal{K}_{\mathbb{R}}(i)} , the class of functions with real coefficients which are convex in the direction of the imaginary axis. In both classes, we find lower and upper bounds of the third Hankel determinant. The results are sharp.

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