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Register a new userHankel Continued fractions and Hankel determinants of the Euler numbers
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Guo-Niu Han
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The Euler numbers occur in the Taylor expansion of $tan(x)+sec(x)$. Since Stieltjes, continued fractions and Hankel determinants of the even Euler numbers, on the one hand, of the odd Euler numbers, on the other hand, have been widely studied separately. However, no Hankel determinants of the (mixed) Euler numbers have been obtained and explicitly calculated. The reason for that is that some Hankel determinants of the Euler numbers are null. This implies that the Jacobi continued fraction of the Euler numbers does not exist. In the present paper, this obstacle is bypassed by using the Hankel continued fraction, instead of the $J$-fraction. Consequently, an explicit formula for the Hankel determinants of the Euler numbers is being derived, as well as a full list of Hankel continued fractions and Hankel determinants involving Euler numbers. Finally, a new $q$-analog of the Euler numbers $E_n(q)$ based on our continued fraction is proposed. We obtain an explicit formula for $E_n(-1)$ and prove a conjecture by R. J. Mathar on these numbers.
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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 Ciglers 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 ${(1,1),(q,0), (ell-1,-1)}$ for integer parameters $m,q,ell$. Again shifted periodic continued fractions appear.
For a real number $t$, let $r_ell(t)$ be the total weight of all $t$-large Schr{o}der paths of length $ell$, and $s_ell(t)$ be the total weight of all $t$-small Schr{o}der paths of length $ell$. For constants $alpha, beta$, in this article we derive recurrence formulae for the determinats of the Hankel matrices $det_{1le i,jle n} (alpha r_{i+j-2}(t) +beta r_{i+j-1}(t))$, $det_{1le i,jle n} (alpha r_{i+j-1}(t) +beta r_{i+j}(t))$, $det_{1le i,jle n} (alpha s_{i+j-2}(t) +beta s_{i+j-1}(t))$, and $det_{1le i,jle n} (alpha s_{i+j-1}(t) +beta s_{i+j}(t))$ combinatorially via suitable lattice path models.
In this paper we shall survey the various methods of evaluating Hankel determinants and as an illustration we evaluate some Hankel determinants of a q-analogue of Catalan numbers. Here we consider $frac{(aq;q)_{n}}{(abq^{2};q)_{n}}$ as a q-analogue of Catalan numbers $C_{n}=frac1{n+1}binom{2n}{n}$, which is known as the moments of the little q-Jacobi polynomials. We also give several proofs of this q-analogue, in which we use lattice paths, the orthogonal polynomials, or the basic hypergeometric series. We also consider a q-analogue of Schroder Hankel determinants, and give a new proof of Moztkin Hankel determinants using an addition formula for ${}_2F_{1}$.
We prove evaluations of Hankel determinants of linear combinations of moments of orthogonal polynomials (or, equivalently, of generating functions for Motzkin paths), thus generalising known results for Catalan numbers.
Given three nonnegative integers $p,q,r$ and a finite field $F$, how many Hankel matrices $left( x_{i+j}right) _{0leq ileq p, 0leq jleq q}$ over $F$ have rank $leq r$ ? This question is classical, and the answer ($q^{2r}$ when $rleqminleft{ p,qright} $) has been obtained independently by various authors using different tools (Daykin, Elkies, Garcia Armas, Ghorpade and Ram). In this note, we study a refinement of this result: We show that if we fix the first $k$ of the entries $x_{0},x_{1},ldots,x_{k-1}$ for some $kleq rleqminleft{ p,qright} $, then the number of ways to choose the remaining $p+q-k+1$ entries $x_{k},x_{k+1},ldots,x_{p+q}$ such that the resulting Hankel matrix $left( x_{i+j}right) _{0leq ileq p, 0leq jleq q}$ has rank $leq r$ is $q^{2r-k}$. This is exactly the answer that one would expect if the first $k$ entries had no effect on the rank, but of course the situation is not this simple. The refined result generalizes (and provides an alternative proof of) a result by Anzis, Chen, Gao, Kim, Li and Patrias on evaluations of Jacobi-Trudi determinants over finite fields.
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