Some Aspects of Hankel Matrices in Coding Theory and Combinatorics

Abstract

Hankel matrices consisting of Catalan numbers have been analyzed by various authors. Desainte-Catherine and Viennot found their determinant to be $\prod_{1 \leq i \leq j \leq k} {{i+j+2n}\over {i+j}}$ and related them to the Bender - Knuth conjecture. The similar determinant formula $\prod_{1 \leq i \leq j \leq k} {{i+j-1+2n}\over {i+j-1}}$ can be shown to hold for Hankel matrices whose entries are successive middle binomial coefficients ${{2m+1} \choose m}$. Generalizing the Catalan numbers in a different direction, it can be shown that determinants of Hankel matrices consisting of numbers ${{1}\over {3m+1}} {{3m+1} \choose m}$ yield an alternate expression of two Mills – Robbins – Rumsey determinants important in the enumeration of plane partitions and alternating sign matrices. Hankel matrices with determinant 1 were studied by Aigner in the definition of Catalan – like numbers. The well - known relation of Hankel matrices to orthogonal polynomials further yields a combinatorial application of the famous Berlekamp – Massey algorithm in Coding Theory, which can be applied in order to calculate the coefficients in the three – term recurrence of the family of orthogonal polynomials related to the sequence of Hankel matrices.

Heat flow, heat content and the isoparametric property

“…. } (they coincide with the polynomials defined in (1.8) in [26]). Taking into account (57) and (58), to prove Lemma 24 it is then sufficient to show the following fact.…”

“…Remark. We point out that the expression of α n and e (1) n taken from Corollary 2.1, equation (2.6) of [26].…”

“…in the continued fraction expansion of 1 − xF (x), where F (x) = ∞ m=0 c m+k x m (see(1.14),(1.19) and(1.20) in[26]):α 1 = q 1 and, for n ≥ 1 : α (k) n+1 = q (k) n+1 + e (k) n , β (k) n = q (k) n · e (k) n ,(60)For our sequence c m = 2m+1 m the corresponding coefficients have been computed in Corollary 2.1, equation (2.6):q (k) n = (2n + 2k)(2n + 2k + 1) (2n + k − 1)(2n + k) , e (k) n = (2n − 1)(2n) (2n + k)(2n + k + 1). 1)(2n) (2n + 1)(2n + 2)…”

The Generating Function of Ternary Trees and Continued Fractions

“…However, if A(x) can be expressed as a continued fraction, then there is a very nice formula. This is the case for g(x): Tamm [28] observed that g(x) has a nice continued fraction expression, which is a special case of Gauss's continued fraction. We introduce some notation to explain Tamm's approach.…”

Advanced determinant calculus: A complement

“…Using this theorem, Tamm[181, Theorem 3.1] observed that from Gauß' continued fraction for the ratio of two contiguous2 F 1 -series one can deduce several interesting binomial Hankel determinant evaluations, some of them had already been found earlier by Egecioglu, Redmond and Ryavec [53, Theorem 4] while working on polynomial Riemann hypotheses. Gessel and Xin [75] undertook a systematic analysis of this approach, and they arrived at a set of 18 Hankel determinant evaluations, which I list as (5.48)-(5.65) in the theorem below.…”